Cyclotomic polynomial: Difference between revisions
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=== Another approach === |
=== Another approach === |
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As noted in the [http://jsoftware.com/pipermail/programming/2022-March/060209.html J programming forum], we can improve the big-O character of this algorithm by using the [[ |
As noted in the [http://jsoftware.com/pipermail/programming/2022-March/060209.html J programming forum], we can improve the big-O character of this algorithm by using the [[Fast Fourier transform]] for polynomial multiplication and division. |
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<lang J>NB. install'math/fftw' |
<lang J>NB. install'math/fftw' |
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roundreal =: [: <. 0.5 + 9&o. |
roundreal =: [: <. 0.5 + 9&o.</lang> |
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This variation for polynomial division is only valid when there's no remainder to be concerned with (which is the case, here). |
This variation for polynomial division is only valid when there's no remainder to be concerned with (which is the case, here). |
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Revision as of 20:52, 8 March 2022
You are encouraged to solve this task according to the task description, using any language you may know.
The nth Cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial of largest degree with integer coefficients that is a divisor of x^n − 1, and is not a divisor of x^k − 1 for any k < n.
- Task
- Find and print the first 30 cyclotomic polynomials.
- Find and print the order of the first 10 cyclotomic polynomials that have n or -n as a coefficient.
- See also
- Wikipedia article, Cyclotomic polynomial, showing ways to calculate them.
- The sequence A013594 with the smallest order of cyclotomic polynomial containing n or -n as a coefficient.
C++
<lang cpp>#include <algorithm>
- include <iostream>
- include <initializer_list>
- include <map>
- include <vector>
const int MAX_ALL_FACTORS = 100000; const int algorithm = 2; int divisions = 0;
//Note: Cyclotomic Polynomials have small coefficients. Not appropriate for general polynomial usage. class Term { private:
long m_coefficient; long m_exponent;
public:
Term(long c, long e) : m_coefficient(c), m_exponent(e) {
// empty
}
Term(const Term &t) : m_coefficient(t.m_coefficient), m_exponent(t.m_exponent) {
// empty
}
long coefficient() const {
return m_coefficient;
}
long degree() const {
return m_exponent;
}
Term operator -() const {
return { -m_coefficient, m_exponent };
}
Term operator *(const Term &rhs) const {
return { m_coefficient * rhs.m_coefficient, m_exponent + rhs.m_exponent };
}
Term operator +(const Term &rhs) const {
if (m_exponent != rhs.m_exponent) {
throw std::runtime_error("Exponents not equal");
}
return { m_coefficient + rhs.m_coefficient, m_exponent };
}
friend std::ostream &operator<<(std::ostream &, const Term &);
};
std::ostream &operator<<(std::ostream &os, const Term &t) {
if (t.m_coefficient == 0) {
return os << '0';
}
if (t.m_exponent == 0) {
return os << t.m_coefficient;
}
if (t.m_coefficient == 1) {
if (t.m_exponent == 1) {
return os << 'x';
}
return os << "x^" << t.m_exponent;
}
if (t.m_coefficient == -1) {
if (t.m_exponent == 1) {
return os << "-x";
}
return os << "-x^" << t.m_exponent;
}
if (t.m_exponent == 1) {
return os << t.m_coefficient << 'x';
}
return os << t.m_coefficient << "x^" << t.m_exponent;
}
class Polynomial { public:
std::vector<Term> polynomialTerms;
Polynomial() {
polynomialTerms.push_back({ 0, 0 });
}
Polynomial(std::initializer_list<int> values) {
if (values.size() % 2 != 0) {
throw std::runtime_error("Length must be even.");
}
bool ready = false;
long t;
for (auto v : values) {
if (ready) {
polynomialTerms.push_back({ t, v });
} else {
t = v;
}
ready = !ready;
}
std::sort(
polynomialTerms.begin(), polynomialTerms.end(),
[](const Term &t, const Term &u) {
return u.degree() < t.degree();
}
);
}
Polynomial(const std::vector<Term> &termList) {
if (termList.size() == 0) {
polynomialTerms.push_back({ 0, 0 });
} else {
for (auto t : termList) {
if (t.coefficient() != 0) {
polynomialTerms.push_back(t);
}
}
if (polynomialTerms.size() == 0) {
polynomialTerms.push_back({ 0, 0 });
}
std::sort(
polynomialTerms.begin(), polynomialTerms.end(),
[](const Term &t, const Term &u) {
return u.degree() < t.degree();
}
);
}
}
Polynomial(const Polynomial &p) : Polynomial(p.polynomialTerms) {
// empty
}
long leadingCoefficient() const {
return polynomialTerms[0].coefficient();
}
long degree() const {
return polynomialTerms[0].degree();
}
bool hasCoefficientAbs(int coeff) {
for (auto term : polynomialTerms) {
if (abs(term.coefficient()) == coeff) {
return true;
}
}
return false;
}
Polynomial operator+(const Term &term) const {
std::vector<Term> termList;
bool added = false;
for (size_t index = 0; index < polynomialTerms.size(); index++) {
auto currentTerm = polynomialTerms[index];
if (currentTerm.degree() == term.degree()) {
added = true;
if (currentTerm.coefficient() + term.coefficient() != 0) {
termList.push_back(currentTerm + term);
}
} else {
termList.push_back(currentTerm);
}
}
if (!added) {
termList.push_back(term);
}
return Polynomial(termList);
}
Polynomial operator*(const Term &term) const {
std::vector<Term> termList;
for (size_t index = 0; index < polynomialTerms.size(); index++) {
auto currentTerm = polynomialTerms[index];
termList.push_back(currentTerm * term);
}
return Polynomial(termList);
}
Polynomial operator+(const Polynomial &p) const {
std::vector<Term> termList;
int thisCount = polynomialTerms.size();
int polyCount = p.polynomialTerms.size();
while (thisCount > 0 || polyCount > 0) {
if (thisCount == 0) {
auto polyTerm = p.polynomialTerms[polyCount - 1];
termList.push_back(polyTerm);
polyCount--;
} else if (polyCount == 0) {
auto thisTerm = polynomialTerms[thisCount - 1];
termList.push_back(thisTerm);
thisCount--;
} else {
auto polyTerm = p.polynomialTerms[polyCount - 1];
auto thisTerm = polynomialTerms[thisCount - 1];
if (thisTerm.degree() == polyTerm.degree()) {
auto t = thisTerm + polyTerm;
if (t.coefficient() != 0) {
termList.push_back(t);
}
thisCount--;
polyCount--;
} else if (thisTerm.degree() < polyTerm.degree()) {
termList.push_back(thisTerm);
thisCount--;
} else {
termList.push_back(polyTerm);
polyCount--;
}
}
}
return Polynomial(termList);
}
Polynomial operator/(const Polynomial &v) {
divisions++;
Polynomial q;
Polynomial r(*this);
long lcv = v.leadingCoefficient();
long dv = v.degree();
while (r.degree() >= v.degree()) {
long lcr = r.leadingCoefficient();
long s = lcr / lcv;
Term term(s, r.degree() - dv);
q = q + term;
r = r + v * -term;
}
return q; }
friend std::ostream &operator<<(std::ostream &, const Polynomial &);
};
std::ostream &operator<<(std::ostream &os, const Polynomial &p) {
auto it = p.polynomialTerms.cbegin();
auto end = p.polynomialTerms.cend();
if (it != end) {
os << *it;
it = std::next(it);
}
while (it != end) {
if (it->coefficient() > 0) {
os << " + " << *it;
} else {
os << " - " << -*it;
}
it = std::next(it);
}
return os;
}
std::vector<int> getDivisors(int number) {
std::vector<int> divisiors;
long root = (long)sqrt(number);
for (int i = 1; i <= root; i++) {
if (number % i == 0) {
divisiors.push_back(i);
int div = number / i;
if (div != i && div != number) {
divisiors.push_back(div);
}
}
}
return divisiors;
}
std::map<int, std::map<int, int>> allFactors;
std::map<int, int> getFactors(int number) {
if (allFactors.find(number) != allFactors.end()) {
return allFactors[number];
}
std::map<int, int> factors;
if (number % 2 == 0) {
auto factorsDivTwo = getFactors(number / 2);
factors.insert(factorsDivTwo.begin(), factorsDivTwo.end());
if (factors.find(2) != factors.end()) {
factors[2]++;
} else {
factors.insert(std::make_pair(2, 1));
}
if (number < MAX_ALL_FACTORS) {
allFactors.insert(std::make_pair(number, factors));
}
return factors;
}
long root = (long)sqrt(number);
long i = 3;
while (i <= root) {
if (number % i == 0) {
auto factorsDivI = getFactors(number / i);
factors.insert(factorsDivI.begin(), factorsDivI.end());
if (factors.find(i) != factors.end()) {
factors[i]++;
} else {
factors.insert(std::make_pair(i, 1));
}
if (number < MAX_ALL_FACTORS) {
allFactors.insert(std::make_pair(number, factors));
}
return factors;
}
i += 2;
}
factors.insert(std::make_pair(number, 1));
if (number < MAX_ALL_FACTORS) {
allFactors.insert(std::make_pair(number, factors));
}
return factors;
}
std::map<int, Polynomial> COMPUTED; Polynomial cyclotomicPolynomial(int n) {
if (COMPUTED.find(n) != COMPUTED.end()) {
return COMPUTED[n];
}
if (n == 1) {
// Polynomial: x - 1
Polynomial p({ 1, 1, -1, 0 });
COMPUTED.insert(std::make_pair(1, p));
return p;
}
auto factors = getFactors(n);
if (factors.find(n) != factors.end()) {
// n prime
std::vector<Term> termList;
for (int index = 0; index < n; index++) {
termList.push_back({ 1, index });
}
Polynomial cyclo(termList);
COMPUTED.insert(std::make_pair(n, cyclo));
return cyclo;
} else if (factors.size() == 2 && factors.find(2) != factors.end() && factors[2] == 1 && factors.find(n / 2) != factors.end() && factors[n / 2] == 1) {
// n = 2p
int prime = n / 2;
std::vector<Term> termList;
int coeff = -1;
for (int index = 0; index < prime; index++) {
coeff *= -1;
termList.push_back({ coeff, index });
}
Polynomial cyclo(termList);
COMPUTED.insert(std::make_pair(n, cyclo));
return cyclo;
} else if (factors.size() == 1 && factors.find(2) != factors.end()) {
// n = 2^h
int h = factors[2];
std::vector<Term> termList;
termList.push_back({ 1, (int)pow(2, h - 1) });
termList.push_back({ 1, 0 });
Polynomial cyclo(termList);
COMPUTED.insert(std::make_pair(n, cyclo));
return cyclo;
} else if (factors.size() == 1 && factors.find(n) != factors.end()) {
// n = p^k
int p = 0;
int k = 0;
for (auto iter = factors.begin(); iter != factors.end(); ++iter) {
p = iter->first;
k = iter->second;
}
std::vector<Term> termList;
for (int index = 0; index < p; index++) {
termList.push_back({ 1, index * (int)pow(p, k - 1) });
}
Polynomial cyclo(termList);
COMPUTED.insert(std::make_pair(n, cyclo));
return cyclo;
} else if (factors.size() == 2 && factors.find(2) != factors.end()) {
// n = 2^h * p^k
int p = 0;
for (auto iter = factors.begin(); iter != factors.end(); ++iter) {
if (iter->first != 2) {
p = iter->first;
}
}
std::vector<Term> termList;
int coeff = -1;
int twoExp = (int)pow(2, factors[2] - 1);
int k = factors[p];
for (int index = 0; index < p; index++) {
coeff *= -1;
termList.push_back({ coeff, index * twoExp * (int)pow(p, k - 1) });
}
Polynomial cyclo(termList);
COMPUTED.insert(std::make_pair(n, cyclo));
return cyclo;
} else if (factors.find(2) != factors.end() && ((n / 2) % 2 == 1) && (n / 2) > 1) {
// CP(2m)[x] = CP(-m)[x], n odd integer > 1
auto cycloDiv2 = cyclotomicPolynomial(n / 2);
std::vector<Term> termList;
for (auto term : cycloDiv2.polynomialTerms) {
if (term.degree() % 2 == 0) {
termList.push_back(term);
} else {
termList.push_back(-term);
}
}
Polynomial cyclo(termList);
COMPUTED.insert(std::make_pair(n, cyclo));
return cyclo;
}
// General Case
if (algorithm == 0) {
// slow - uses basic definition
auto divisors = getDivisors(n);
// Polynomial: (x^n - 1)
Polynomial cyclo({ 1, n, -1, 0 });
for (auto i : divisors) {
auto p = cyclotomicPolynomial(i);
cyclo = cyclo / p;
}
COMPUTED.insert(std::make_pair(n, cyclo));
return cyclo;
} else if (algorithm == 1) {
// Faster. Remove Max divisor (and all divisors of max divisor) - only one divide for all divisors of Max Divisor
auto divisors = getDivisors(n);
int maxDivisor = INT_MIN;
for (auto div : divisors) {
maxDivisor = std::max(maxDivisor, div);
}
std::vector<int> divisorExceptMax;
for (auto div : divisors) {
if (maxDivisor % div != 0) {
divisorExceptMax.push_back(div);
}
}
// Polynomial: ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor
auto cyclo = Polynomial({ 1, n, -1, 0 }) / Polynomial({ 1, maxDivisor, -1, 0 });
for (int i : divisorExceptMax) {
auto p = cyclotomicPolynomial(i);
cyclo = cyclo / p;
}
COMPUTED.insert(std::make_pair(n, cyclo));
return cyclo;
} else if (algorithm == 2) {
// Fastest
// Let p ; q be primes such that p does not divide n, and q q divides n.
// Then CP(np)[x] = CP(n)[x^p] / CP(n)[x]
int m = 1;
auto cyclo = cyclotomicPolynomial(m);
std::vector<int> primes;
for (auto iter = factors.begin(); iter != factors.end(); ++iter) {
primes.push_back(iter->first);
}
std::sort(primes.begin(), primes.end());
for (auto prime : primes) {
// CP(m)[x]
auto cycloM = cyclo;
// Compute CP(m)[x^p].
std::vector<Term> termList;
for (auto t : cycloM.polynomialTerms) {
termList.push_back({ t.coefficient(), t.degree() * prime });
}
cyclo = Polynomial(termList) / cycloM;
m = m * prime;
}
// Now, m is the largest square free divisor of n
int s = n / m;
// Compute CP(n)[x] = CP(m)[x^s]
std::vector<Term> termList;
for (auto t : cyclo.polynomialTerms) {
termList.push_back({ t.coefficient(), t.degree() * s });
}
cyclo = Polynomial(termList);
COMPUTED.insert(std::make_pair(n, cyclo));
return cyclo;
} else {
throw std::runtime_error("Invalid algorithm");
}
}
int main() {
// initialization std::map<int, int> factors; factors.insert(std::make_pair(2, 1)); allFactors.insert(std::make_pair(2, factors));
// rest of main
std::cout << "Task 1: cyclotomic polynomials for n <= 30:\n";
for (int i = 1; i <= 30; i++) {
auto p = cyclotomicPolynomial(i);
std::cout << "CP[" << i << "] = " << p << '\n';
}
std::cout << "Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient:\n";
int n = 0;
for (int i = 1; i <= 10; i++) {
while (true) {
n++;
auto cyclo = cyclotomicPolynomial(n);
if (cyclo.hasCoefficientAbs(i)) {
std::cout << "CP[" << n << "] has coefficient with magnitude = " << i << '\n';
n--;
break;
}
}
}
return 0;
}</lang>
- Output:
Task 1: cyclotomic polynomials for n <= 30: CP[1] = x - 1 CP[2] = x + 1 CP[3] = x^2 + x + 1 CP[4] = x^2 + 1 CP[5] = x^4 + x^3 + x^2 + x + 1 CP[6] = x^2 - x + 1 CP[7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[8] = x^4 + 1 CP[9] = x^6 + x^3 + 1 CP[10] = x^4 - x^3 + x^2 - x + 1 CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[12] = x^4 - x^2 + 1 CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 CP[16] = x^8 + 1 CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[18] = x^6 - x^3 + 1 CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[20] = x^8 - x^6 + x^4 - x^2 + 1 CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[24] = x^8 - x^4 + 1 CP[25] = x^20 + x^15 + x^10 + x^5 + 1 CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[27] = x^18 + x^9 + 1 CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1 CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient: CP[1] has coefficient with magnitude = 1 CP[105] has coefficient with magnitude = 2 CP[385] has coefficient with magnitude = 3 CP[1365] has coefficient with magnitude = 4 CP[1785] has coefficient with magnitude = 5 CP[2805] has coefficient with magnitude = 6 CP[3135] has coefficient with magnitude = 7 CP[6545] has coefficient with magnitude = 8 CP[6545] has coefficient with magnitude = 9 CP[10465] has coefficient with magnitude = 10
C#
<lang csharp>using System; using System.Collections; using System.Collections.Generic; using System.Linq; using IntMap = System.Collections.Generic.Dictionary<int, int>;
public static class CyclotomicPolynomial {
public static void Main2() {
Console.WriteLine("Task 1: Cyclotomic polynomials for n <= 30:");
for (int i = 1; i <= 30; i++) {
var p = GetCyclotomicPolynomial(i);
Console.WriteLine($"CP[{i}] = {p.ToString()}");
}
Console.WriteLine();
Console.WriteLine("Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient:");
for (int i = 1, n = 0; i <= 10; i++) {
while (true) {
n++;
var p = GetCyclotomicPolynomial(n);
if (p.Any(t => Math.Abs(t.Coefficient) == i)) {
Console.WriteLine($"CP[{n}] has coefficient with magnitude = {i}");
n--;
break;
}
}
}
}
private const int MaxFactors = 100_000;
private const int Algorithm = 2;
private static readonly Term x = new Term(1, 1);
private static readonly Dictionary<int, Polynomial> polyCache =
new Dictionary<int, Polynomial> { [1] = x - 1 };
private static readonly Dictionary<int, IntMap> factorCache =
new Dictionary<int, IntMap> { [2] = new IntMap { [2] = 1 } };
private static Polynomial GetCyclotomicPolynomial(in int n) {
if (polyCache.TryGetValue(n, out var result)) return result;
var factors = GetFactors(n);
if (factors.ContainsKey(n)) { //n is prime
result = new Polynomial(from exp in ..n select x[exp]);
} else if (factors.Count == 2 && factors.Contains(2, 1) && factors.Contains(n/2, 1)) { //n = 2p
result = new Polynomial(from i in ..(n/2) select (IsOdd(i) ? -x : x)[i]);
} else if (factors.Count == 1 && factors.TryGetValue(2, out int h)) { //n = 2^h
result = x[1<<(h-1)] + 1;
} else if (factors.Count == 1 && !factors.ContainsKey(n)) { // n = p^k
(int p, int k) = factors.First();
result = new Polynomial(from i in ..p select x[i * (int)Math.Pow(p, k-1)]);
} else if (factors.Count == 2 && factors.ContainsKey(2)) { //n = 2^h * p^k
(int p, int k) = factors.First(entry => entry.Key != 2);
int twoExp = 1 << (factors[2] - 1);
result = new Polynomial(from i in ..p select (IsOdd(i) ? -x : x)[i * twoExp * (int)Math.Pow(p, k-1)]);
} else if (factors.ContainsKey(2) && IsOdd(n/2) && n/2 > 1) { // CP(2m)[x] = CP[-m][x], n is odd > 1
Polynomial cycloDiv2 = GetCyclotomicPolynomial(n/2);
result = new Polynomial(from term in cycloDiv2 select IsOdd(term.Exponent) ? -term : term);
#pragma warning disable CS0162
} else if (Algorithm == 0) {
var divisors = GetDivisors(n);
result = x[n] - 1;
foreach (int d in divisors) result /= GetCyclotomicPolynomial(d);
} else if (Algorithm == 1) {
var divisors = GetDivisors(n).ToList();
int maxDivisor = divisors.Max();
result = (x[n] - 1) / (x[maxDivisor] - 1);
foreach (int d in divisors.Where(div => maxDivisor % div == 0)) {
result /= GetCyclotomicPolynomial(d);
}
} else if (Algorithm == 2) {
int m = 1;
result = GetCyclotomicPolynomial(m);
var primes = factors.Keys.ToList();
primes.Sort();
foreach (int prime in primes) {
var cycloM = result;
result = new Polynomial(from term in cycloM select term.Coefficient * x[term.Exponent * prime]);
result /= cycloM;
m *= prime;
}
int s = n / m;
result = new Polynomial(from term in result select term.Coefficient * x[term.Exponent * s]);
#pragma warning restore CS0162
} else {
throw new InvalidOperationException("Invalid algorithm");
}
polyCache[n] = result;
return result;
}
private static bool IsOdd(int i) => (i & 1) != 0; private static bool Contains(this IntMap map, int key, int value) => map.TryGetValue(key, out int v) && v == value; private static int GetOrZero(this IntMap map, int key) => map.TryGetValue(key, out int v) ? v : 0; private static IEnumerable<T> Select<T>(this Range r, Func<int, T> f) => Enumerable.Range(r.Start.Value, r.End.Value - r.Start.Value).Select(f);
private static IntMap GetFactors(in int n) {
if (factorCache.TryGetValue(n, out var factors)) return factors;
factors = new IntMap();
if (!IsOdd(n)) {
foreach (var entry in GetFactors(n/2)) factors.Add(entry.Key, entry.Value);
factors[2] = factors.GetOrZero(2) + 1;
return Cache(n, factors);
}
for (int i = 3; i * i <= n; i+=2) {
if (n % i == 0) {
foreach (var entry in GetFactors(n/i)) factors.Add(entry.Key, entry.Value);
factors[i] = factors.GetOrZero(i) + 1;
return Cache(n, factors);
}
}
factors[n] = 1;
return Cache(n, factors);
}
private static IntMap Cache(int n, IntMap factors) {
if (n < MaxFactors) factorCache[n] = factors;
return factors;
}
private static IEnumerable<int> GetDivisors(int n) {
for (int i = 1; i * i <= n; i++) {
if (n % i == 0) {
yield return i;
int div = n / i;
if (div != i && div != n) yield return div;
}
}
}
public sealed class Polynomial : IEnumerable<Term>
{
public Polynomial() { }
public Polynomial(params Term[] terms) : this(terms.AsEnumerable()) { }
public Polynomial(IEnumerable<Term> terms) {
Terms.AddRange(terms);
Simplify();
}
private List<Term>? terms;
private List<Term> Terms => terms ??= new List<Term>();
public int Count => terms?.Count ?? 0;
public int Degree => Count == 0 ? -1 : Terms[0].Exponent;
public int LeadingCoefficient => Count == 0 ? 0 : Terms[0].Coefficient;
public IEnumerator<Term> GetEnumerator() => Terms.GetEnumerator();
IEnumerator IEnumerable.GetEnumerator() => GetEnumerator();
public override string ToString() => Count == 0 ? "0" : string.Join(" + ", Terms).Replace("+ -", "- ");
public static Polynomial operator *(Polynomial p, Term t) => new Polynomial(from s in p select s * t);
public static Polynomial operator +(Polynomial p, Polynomial q) => new Polynomial(p.Terms.Concat(q.Terms));
public static Polynomial operator -(Polynomial p, Polynomial q) => new Polynomial(p.Terms.Concat(q.Terms.Select(t => -t)));
public static Polynomial operator *(Polynomial p, Polynomial q) => new Polynomial(from s in p from t in q select s * t);
public static Polynomial operator /(Polynomial p, Polynomial q) => p.Divide(q).quotient;
public (Polynomial quotient, Polynomial remainder) Divide(Polynomial divisor) {
if (Degree < 0) return (new Polynomial(), this);
Polynomial quotient = new Polynomial();
Polynomial remainder = this;
int lcv = divisor.LeadingCoefficient;
int dv = divisor.Degree;
while (remainder.Degree >= divisor.Degree) {
int lcr = remainder.LeadingCoefficient;
Term div = new Term(lcr / lcv, remainder.Degree - dv);
quotient.Terms.Add(div);
remainder += divisor * -div;
}
quotient.Simplify();
remainder.Simplify();
return (quotient, remainder);
}
private void Simplify() {
if (Count < 2) return;
Terms.Sort((a, b) => -a.CompareTo(b));
for (int i = Terms.Count - 1; i > 0; i--) {
Term s = Terms[i-1];
Term t = Terms[i];
if (t.Exponent == s.Exponent) {
Terms[i-1] = new Term(s.Coefficient + t.Coefficient, s.Exponent);
Terms.RemoveAt(i);
}
}
Terms.RemoveAll(t => t.IsZero);
}
}
public readonly struct Term : IEquatable<Term>, IComparable<Term>
{
public Term(int coefficient, int exponent = 0) => (Coefficient, Exponent) = (coefficient, exponent);
public Term this[int exponent] => new Term(Coefficient, exponent); //Using x[exp] because x^exp has low precedence
public int Coefficient { get; }
public int Exponent { get; }
public bool IsZero => Coefficient == 0;
public static Polynomial operator +(Term left, Term right) => new Polynomial(left, right);
public static Polynomial operator -(Term left, Term right) => new Polynomial(left, -right);
public static implicit operator Term(int coefficient) => new Term(coefficient);
public static Term operator -(Term t) => new Term(-t.Coefficient, t.Exponent);
public static Term operator *(Term left, Term right) => new Term(left.Coefficient * right.Coefficient, left.Exponent + right.Exponent);
public static bool operator ==(Term left, Term right) => left.Equals(right);
public static bool operator !=(Term left, Term right) => !left.Equals(right);
public static bool operator <(Term left, Term right) => left.CompareTo(right) < 0;
public static bool operator >(Term left, Term right) => left.CompareTo(right) > 0;
public static bool operator <=(Term left, Term right) => left.CompareTo(right) <= 0;
public static bool operator >=(Term left, Term right) => left.CompareTo(right) >= 0;
public bool Equals(Term other) => Exponent == other.Exponent && Coefficient == other.Coefficient;
public override bool Equals(object? obj) => obj is Term t && Equals(t);
public override int GetHashCode() => Coefficient.GetHashCode() * 31 + Exponent.GetHashCode();
public int CompareTo(Term other) {
int c = Exponent.CompareTo(other.Exponent);
if (c != 0) return c;
return Coefficient.CompareTo(other.Coefficient);
}
public override string ToString() => (Coefficient, Exponent) switch {
(0, _) => "0",
(_, 0) => $"{Coefficient}",
(1, 1) => "x",
(-1, 1) => "-x",
(_, 1) => $"{Coefficient}x",
(1, _) => $"x^{Exponent}",
(-1, _) => $"-x^{Exponent}",
_ => $"{Coefficient}x^{Exponent}"
};
}
}</lang>
- Output:
Task 1: Cyclotomic polynomials for n <= 30: CP[1] = x - 1 CP[2] = x + 1 CP[3] = x^2 + x + 1 CP[4] = x^2 + 1 CP[5] = x^4 + x^3 + x^2 + x + 1 CP[6] = x^2 - x + 1 CP[7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[8] = x^4 + 1 CP[9] = x^6 + x^3 + 1 CP[10] = x^4 - x^3 + x^2 - x + 1 CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[12] = x^4 - x^2 + 1 CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 CP[16] = x^8 + 1 CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[18] = x^6 - x^3 + 1 CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[20] = x^8 - x^6 + x^4 - x^2 + 1 CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[24] = x^8 - x^4 + 1 CP[25] = x^20 + x^15 + x^10 + x^5 + 1 CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[27] = x^18 + x^9 + 1 CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1 CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient: CP[1] has coefficient with magnitude = 1 CP[105] has coefficient with magnitude = 2 CP[385] has coefficient with magnitude = 3 CP[1365] has coefficient with magnitude = 4 CP[1785] has coefficient with magnitude = 5 CP[2805] has coefficient with magnitude = 6 CP[3135] has coefficient with magnitude = 7 CP[6545] has coefficient with magnitude = 8 CP[6545] has coefficient with magnitude = 9 CP[10465] has coefficient with magnitude = 10
D
<lang d>import std.algorithm; import std.exception; import std.format; import std.functional; import std.math; import std.range; import std.stdio;
immutable MAX_ALL_FACTORS = 100_000; immutable ALGORITHM = 2;
//Note: Cyclotomic Polynomials have small coefficients. Not appropriate for general polynomial usage.
struct Term {
private long m_coefficient; private long m_exponent;
public this(long c, long e) {
m_coefficient = c;
m_exponent = e;
}
public long coefficient() const {
return m_coefficient;
}
public long exponent() const {
return m_exponent;
}
public Term opUnary(string op)() const {
static if (op == "-") {
return Term(-m_coefficient, m_exponent);
} else {
assert(false, "Not implemented");
}
}
public Term opBinary(string op)(Term term) const {
static if (op == "+") {
if (exponent() != term.exponent()) {
assert(false, "Error 102: Exponents not equals.");
}
return Term(coefficient() + term.coefficient(), exponent());
} else if (op == "*") {
return Term(coefficient() * term.coefficient(), exponent() + term.exponent());
} else {
assert(false, "Not implemented: " ~ op);
}
}
public void toString(scope void delegate(const(char)[]) sink) const {
auto spec = singleSpec("%s");
if (m_coefficient == 0) {
sink("0");
} else if (m_exponent == 0) {
formatValue(sink, m_coefficient, spec);
} else if (m_coefficient == 1) {
if (m_exponent == 1) {
sink("x");
} else {
sink("x^");
formatValue(sink, m_exponent, spec);
}
} else if (m_coefficient == -1) {
if (m_exponent == 1) {
sink("-x");
} else {
sink("-x^");
formatValue(sink, m_exponent, spec);
}
} else if (m_exponent == 1) {
formatValue(sink, m_coefficient, spec);
sink("x");
} else {
formatValue(sink, m_coefficient, spec);
sink("x^");
formatValue(sink, m_exponent, spec);
}
}
}
struct Polynomial {
private Term[] terms;
public this(const Term[] ts...) {
terms = ts.dup;
terms.sort!"b.exponent < a.exponent";
}
bool hasCoefficientAbs(int coeff) const {
foreach (term; terms) {
if (abs(term.coefficient) == coeff) {
return true;
}
}
return false;
}
public long leadingCoefficient() const {
return terms[0].coefficient();
}
public long degree() const {
if (terms.empty) {
return -1;
}
return terms[0].exponent();
}
public Polynomial opBinary(string op)(Term term) const {
static if (op == "+") {
Term[] newTerms;
auto added = false;
foreach (currentTerm; terms) {
if (currentTerm.exponent() == term.exponent()) {
added = true;
if (currentTerm.coefficient() + term.coefficient() != 0) {
newTerms ~= currentTerm + term;
}
} else {
newTerms ~= currentTerm;
}
}
if (!added) {
newTerms ~= term;
}
return Polynomial(newTerms);
} else if (op == "*") {
Term[] newTerms;
foreach (currentTerm; terms) {
newTerms ~= currentTerm * term;
}
return Polynomial(newTerms);
} else {
assert(false, "Not implemented: " ~ op);
}
}
public Polynomial opBinary(string op)(Polynomial rhs) const {
static if (op == "+") {
Term[] newTerms;
auto thisCount = terms.length;
auto polyCount = rhs.terms.length;
while (thisCount > 0 || polyCount > 0) {
if (thisCount == 0) {
newTerms ~= rhs.terms[polyCount - 1];
polyCount--;
} else if (polyCount == 0) {
newTerms ~= terms[thisCount - 1];
thisCount--;
} else {
auto thisTerm = terms[thisCount - 1];
auto polyTerm = rhs.terms[polyCount - 1];
if (thisTerm.exponent() == polyTerm.exponent()) {
auto t = thisTerm + polyTerm;
if (t.coefficient() != 0) {
newTerms ~= t;
}
thisCount--;
polyCount--;
} else if (thisTerm.exponent() < polyTerm.exponent()) {
newTerms ~= thisTerm;
thisCount--;
} else {
newTerms ~= polyTerm;
polyCount--;
}
}
}
return Polynomial(newTerms);
} else if (op == "/") {
Polynomial q;
auto r = Polynomial(terms);
auto lcv = rhs.leadingCoefficient();
auto dv = rhs.degree();
while (r.degree() >= rhs.degree()) {
auto lcr = r.leadingCoefficient();
auto s = lcr / lcv;
auto term = Term(s, r.degree() - dv);
q = q + term;
r = r + rhs * -term;
}
return q;
} else {
assert(false, "Not implemented: " ~ op);
}
}
public int opApply(int delegate(Term) dg) const {
foreach (term; terms) {
auto rv = dg(term);
if (rv != 0) {
return rv;
}
}
return 0;
}
public void toString(scope void delegate(const(char)[]) sink) const {
auto spec = singleSpec("%s");
if (!terms.empty) {
formatValue(sink, terms[0], spec);
foreach (t; terms[1..$]) {
if (t.coefficient > 0) {
sink(" + ");
formatValue(sink, t, spec);
} else {
sink(" - ");
formatValue(sink, -t, spec);
}
}
}
}
}
void putAll(K, V)(ref V[K] a, V[K] b) {
foreach (k, v; b) {
a[k] = v;
}
}
void merge(K, V, F)(ref V[K] a, K k, V v, F f) {
if (k in a) {
a[k] = f(a[k], v);
} else {
a[k] = v;
}
}
int sum(int a, int b) {
return a + b;
}
int[int] getFactorsImpl(int number) {
int[int] factors;
if (number % 2 == 0) {
if (number > 2) {
auto factorsDivTwo = memoize!getFactorsImpl(number / 2);
factors.putAll(factorsDivTwo);
}
factors.merge(2, 1, &sum);
return factors;
}
auto root = sqrt(cast(real) number);
auto i = 3;
while (i <= root) {
if (number % i == 0) {
factors.putAll(memoize!getFactorsImpl(number / i));
factors.merge(i, 1, &sum);
return factors;
}
i += 2;
}
factors[number] = 1;
return factors;
} alias getFactors = memoize!getFactorsImpl;
int[] getDivisors(int number) {
int[] divisors;
auto root = cast(int)sqrt(cast(real) number);
foreach (i; 1..root) {
if (number % i == 0) {
divisors ~= i;
}
auto div = number / i;
if (div != i && div != number) {
divisors ~= div;
}
}
return divisors;
}
Polynomial cyclotomicPolynomialImpl(int n) {
if (n == 1) {
// Polynomial: x - 1
return Polynomial(Term(1, 1), Term(-1, 0));
}
auto factors = getFactors(n);
if (n in factors) {
// n prime
Term[] terms;
foreach (i; 0..n) {
terms ~= Term(1, i);
}
return Polynomial(terms);
} else if (factors.length == 2 && 2 in factors && factors[2] == 1 && (n / 2) in factors && factors[n / 2] == 1) {
// n = 2p
auto prime = n / 2;
Term[] terms;
auto coeff = -1;
foreach (i; 0..prime) {
coeff *= -1;
terms ~= Term(coeff, i);
}
return Polynomial(terms);
} else if (factors.length == 1 && 2 in factors) {
// n = 2^h
auto h = factors[2];
Term[] terms;
terms ~= Term(1, 2 ^^ (h - 1));
terms ~= Term(1, 0);
return Polynomial(terms);
} else if (factors.length == 1 && n !in factors) {
// n = p^k
auto p = 0;
auto k = 0;
foreach (prime, v; factors) {
if (prime > p) {
p = prime;
k = v;
}
}
Term[] terms;
foreach (i; 0..p) {
terms ~= Term(1, (i * p) ^^ (k - 1));
}
return Polynomial(terms);
} else if (factors.length == 2 && 2 in factors) {
// n = 2^h * p^k
auto p = 0;
auto k = 0;
foreach (prime, v; factors) {
if (prime != 2 && prime > p) {
p = prime;
k = v;
}
}
Term[] terms;
auto coeff = -1;
auto twoExp = 2 ^^ (factors[2] - 1);
foreach (i; 0..p) {
coeff *= -1;
auto exponent = i * twoExp * p ^^ (k - 1);
terms ~= Term(coeff, exponent);
}
return Polynomial(terms);
} else if (2 in factors && n / 2 % 2 == 1 && n / 2 > 1) {
// CP(2m)[x] = CP(-m)[x], n odd integer > 1
auto cycloDiv2 = memoize!cyclotomicPolynomialImpl(n / 2);
Term[] terms;
foreach (term; cycloDiv2) {
if (term.exponent() % 2 == 0) {
terms ~= term;
} else {
terms ~= -term;
}
}
return Polynomial(terms);
}
if (ALGORITHM == 0) {
// Slow - uses basic definition.
auto divisors = getDivisors(n);
// Polynomial: ( x^n - 1 )
auto cyclo = Polynomial(Term(1, n), Term(-1, 0));
foreach (i; divisors) {
auto p = memoize!cyclotomicPolynomialImpl(i);
cyclo = cyclo / p;
}
return cyclo;
}
if (ALGORITHM == 1) {
// Faster. Remove Max divisor (and all divisors of max divisor) - only one divide for all divisors of Max Divisor
auto divisors = getDivisors(n);
auto maxDivisor = int.min;
foreach (div; divisors) {
maxDivisor = max(maxDivisor, div);
}
int[] divisorsExceptMax;
foreach (div; divisors) {
if (maxDivisor % div != 0) {
divisorsExceptMax ~= div;
}
}
// Polynomial: ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor
auto cyclo = Polynomial(Term(1, n), Term(-1, 0)) / Polynomial(Term(1, maxDivisor), Term(-1, 0));
foreach (i; divisorsExceptMax) {
auto p = memoize!cyclotomicPolynomialImpl(i);
cyclo = cyclo / p;
}
return cyclo;
}
if (ALGORITHM == 2) {
// Fastest
// Let p ; q be primes such that p does not divide n, and q q divides n.
// Then CP(np)[x] = CP(n)[x^p] / CP(n)[x]
auto m = 1;
auto cyclo = memoize!cyclotomicPolynomialImpl(m);
auto primes = factors.keys;
primes.sort;
foreach (prime; primes) {
// CP(m)[x]
auto cycloM = cyclo;
// Compute CP(m)[x^p].
Term[] terms;
foreach (term; cycloM) {
terms ~= Term(term.coefficient(), term.exponent() * prime);
}
cyclo = Polynomial(terms) / cycloM;
m *= prime;
}
// Now, m is the largest square free divisor of n
auto s = n / m;
// Compute CP(n)[x] = CP(m)[x^s]
Term[] terms;
foreach (term; cyclo) {
terms ~= Term(term.coefficient(), term.exponent() * s);
}
return Polynomial(terms);
}
assert(false, "Error 103: Invalid algorithm");
} alias cyclotomicPolynomial = memoize!cyclotomicPolynomialImpl;
void main() {
writeln("Task 1: cyclotomic polynomials for n <= 30:");
foreach (i; 1 .. 31) {
auto p = cyclotomicPolynomial(i);
writefln("CP[%d] = %s", i, p);
}
writeln;
writeln("Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient:");
auto n = 0;
foreach (i; 1 .. 11) {
while (true) {
n++;
auto cyclo = cyclotomicPolynomial(n);
if (cyclo.hasCoefficientAbs(i)) {
writefln("CP[%d] has coefficient with magnitude = %d", n, i);
n--;
break;
}
}
}
}</lang>
- Output:
Task 1: cyclotomic polynomials for n <= 30: CP[1] = x - 1 CP[2] = x + 1 CP[3] = x^2 + x + 1 CP[4] = x^2 + 1 CP[5] = x^4 + x^3 + x^2 + x + 1 CP[6] = x^2 - x + 1 CP[7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[8] = x^4 + 1 CP[9] = x^6 + x^3 + 1 CP[10] = x^4 - x^3 + x^2 - x + 1 CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[12] = x^4 - x^2 + 1 CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 CP[16] = x^8 + 1 CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[18] = x^6 - x^3 + 1 CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[20] = x^8 - x^6 + x^4 - x^2 + 1 CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[24] = x^8 - x^4 + 1 CP[25] = x^20 + x^15 + x^10 + x^5 + 1 CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[27] = x^36 + x^9 + 1 CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1 CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient: CP[1] has coefficient with magnitude = 1 CP[105] has coefficient with magnitude = 2 CP[385] has coefficient with magnitude = 3 CP[1365] has coefficient with magnitude = 4 CP[1785] has coefficient with magnitude = 5 CP[2805] has coefficient with magnitude = 6 CP[3135] has coefficient with magnitude = 7 CP[6545] has coefficient with magnitude = 8 CP[6545] has coefficient with magnitude = 9 CP[10465] has coefficient with magnitude = 10
Fermat
This isn't terribly efficient if you have to calculate many cyclotomics- store them in an array rather than using recursion instead if you need to do that- but it showcases Fermat's strength at polynomial expressions. <lang fermat> &(J=x); {adjoin x as the variable in the polynomials}
Func Cyclotomic(n) =
if n=1 then x-1 fi; {first cyclotomic polynomial is x^n-1}
r:=x^n-1; {caclulate cyclotomic by division}
for d = 1 to n-1 do
if Divides(d,n) then
r:=r\Cyclotomic(d)
fi;
od;
r.; {return the polynomial}
Func Hascoef(n, k) =
p:=Cyclotomic(n);
for d = 0 to Deg(p) do
if |(Coef(p,d))|=k then Return(1) fi
od;
0.;
for d = 1 to 30 do
!!(d,' : ',Cyclotomic(d))
od;
for m = 1 to 10 do
i:=1;
while not Hascoef(i, m) do
i:+
od;
!!(m,' : ',i);
od;</lang>
Go
<lang go>package main
import (
"fmt" "log" "math" "sort" "strings"
)
const (
algo = 2 maxAllFactors = 100000
)
func iabs(i int) int {
if i < 0 {
return -i
}
return i
}
type term struct{ coef, exp int }
func (t term) mul(t2 term) term {
return term{t.coef * t2.coef, t.exp + t2.exp}
}
func (t term) add(t2 term) term {
if t.exp != t2.exp {
log.Fatal("exponents unequal in term.add method")
}
return term{t.coef + t2.coef, t.exp}
}
func (t term) negate() term { return term{-t.coef, t.exp} }
func (t term) String() string {
switch {
case t.coef == 0:
return "0"
case t.exp == 0:
return fmt.Sprintf("%d", t.coef)
case t.coef == 1:
if t.exp == 1 {
return "x"
} else {
return fmt.Sprintf("x^%d", t.exp)
}
case t.exp == 1:
return fmt.Sprintf("%dx", t.coef)
}
return fmt.Sprintf("%dx^%d", t.coef, t.exp)
}
type poly struct{ terms []term }
// pass coef, exp in pairs as parameters func newPoly(values ...int) poly {
le := len(values)
if le == 0 {
return poly{[]term{term{0, 0}}}
}
if le%2 != 0 {
log.Fatalf("odd number of parameters (%d) passed to newPoly function", le)
}
var terms []term
for i := 0; i < le; i += 2 {
terms = append(terms, term{values[i], values[i+1]})
}
p := poly{terms}.tidy()
return p
}
func (p poly) hasCoefAbs(coef int) bool {
for _, t := range p.terms {
if iabs(t.coef) == coef {
return true
}
}
return false
}
func (p poly) add(p2 poly) poly {
p3 := newPoly()
le, le2 := len(p.terms), len(p2.terms)
for le > 0 || le2 > 0 {
if le == 0 {
p3.terms = append(p3.terms, p2.terms[le2-1])
le2--
} else if le2 == 0 {
p3.terms = append(p3.terms, p.terms[le-1])
le--
} else {
t := p.terms[le-1]
t2 := p2.terms[le2-1]
if t.exp == t2.exp {
t3 := t.add(t2)
if t3.coef != 0 {
p3.terms = append(p3.terms, t3)
}
le--
le2--
} else if t.exp < t2.exp {
p3.terms = append(p3.terms, t)
le--
} else {
p3.terms = append(p3.terms, t2)
le2--
}
}
}
return p3.tidy()
}
func (p poly) addTerm(t term) poly {
q := newPoly()
added := false
for i := 0; i < len(p.terms); i++ {
ct := p.terms[i]
if ct.exp == t.exp {
added = true
if ct.coef+t.coef != 0 {
q.terms = append(q.terms, ct.add(t))
}
} else {
q.terms = append(q.terms, ct)
}
}
if !added {
q.terms = append(q.terms, t)
}
return q.tidy()
}
func (p poly) mulTerm(t term) poly {
q := newPoly()
for i := 0; i < len(p.terms); i++ {
ct := p.terms[i]
q.terms = append(q.terms, ct.mul(t))
}
return q.tidy()
}
func (p poly) div(v poly) poly {
q := newPoly()
lcv := v.leadingCoef()
dv := v.degree()
for p.degree() >= v.degree() {
lcp := p.leadingCoef()
s := lcp / lcv
t := term{s, p.degree() - dv}
q = q.addTerm(t)
p = p.add(v.mulTerm(t.negate()))
}
return q.tidy()
}
func (p poly) leadingCoef() int {
return p.terms[0].coef
}
func (p poly) degree() int {
return p.terms[0].exp
}
func (p poly) String() string {
var sb strings.Builder
first := true
for _, t := range p.terms {
if first {
sb.WriteString(t.String())
first = false
} else {
sb.WriteString(" ")
if t.coef > 0 {
sb.WriteString("+ ")
sb.WriteString(t.String())
} else {
sb.WriteString("- ")
sb.WriteString(t.negate().String())
}
}
}
return sb.String()
}
// in place descending sort by term.exp func (p poly) sortTerms() {
sort.Slice(p.terms, func(i, j int) bool {
return p.terms[i].exp > p.terms[j].exp
})
}
// sort terms and remove any unnecesary zero terms func (p poly) tidy() poly {
p.sortTerms()
if p.degree() == 0 {
return p
}
for i := len(p.terms) - 1; i >= 0; i-- {
if p.terms[i].coef == 0 {
copy(p.terms[i:], p.terms[i+1:])
p.terms[len(p.terms)-1] = term{0, 0}
p.terms = p.terms[:len(p.terms)-1]
}
}
if len(p.terms) == 0 {
p.terms = append(p.terms, term{0, 0})
}
return p
}
func getDivisors(n int) []int {
var divs []int
sqrt := int(math.Sqrt(float64(n)))
for i := 1; i <= sqrt; i++ {
if n%i == 0 {
divs = append(divs, i)
d := n / i
if d != i && d != n {
divs = append(divs, d)
}
}
}
return divs
}
var (
computed = make(map[int]poly) allFactors = make(map[int]map[int]int)
)
func init() {
f := map[int]int{2: 1}
allFactors[2] = f
}
func getFactors(n int) map[int]int {
if f, ok := allFactors[n]; ok {
return f
}
factors := make(map[int]int)
if n%2 == 0 {
factorsDivTwo := getFactors(n / 2)
for k, v := range factorsDivTwo {
factors[k] = v
}
factors[2]++
if n < maxAllFactors {
allFactors[n] = factors
}
return factors
}
prime := true
sqrt := int(math.Sqrt(float64(n)))
for i := 3; i <= sqrt; i += 2 {
if n%i == 0 {
prime = false
for k, v := range getFactors(n / i) {
factors[k] = v
}
factors[i]++
if n < maxAllFactors {
allFactors[n] = factors
}
return factors
}
}
if prime {
factors[n] = 1
if n < maxAllFactors {
allFactors[n] = factors
}
}
return factors
}
func cycloPoly(n int) poly {
if p, ok := computed[n]; ok {
return p
}
if n == 1 {
// polynomial: x - 1
p := newPoly(1, 1, -1, 0)
computed[1] = p
return p
}
factors := getFactors(n)
cyclo := newPoly()
if _, ok := factors[n]; ok {
// n is prime
for i := 0; i < n; i++ {
cyclo.terms = append(cyclo.terms, term{1, i})
}
} else if len(factors) == 2 && factors[2] == 1 && factors[n/2] == 1 {
// n == 2p
prime := n / 2
coef := -1
for i := 0; i < prime; i++ {
coef *= -1
cyclo.terms = append(cyclo.terms, term{coef, i})
}
} else if len(factors) == 1 {
if h, ok := factors[2]; ok {
// n == 2^h
cyclo.terms = append(cyclo.terms, term{1, 1 << (h - 1)}, term{1, 0})
} else if _, ok := factors[n]; !ok {
// n == p ^ k
p := 0
for prime := range factors {
p = prime
}
k := factors[p]
for i := 0; i < p; i++ {
pk := int(math.Pow(float64(p), float64(k-1)))
cyclo.terms = append(cyclo.terms, term{1, i * pk})
}
}
} else if len(factors) == 2 && factors[2] != 0 {
// n = 2^h * p^k
p := 0
for prime := range factors {
if prime != 2 {
p = prime
}
}
coef := -1
twoExp := 1 << (factors[2] - 1)
k := factors[p]
for i := 0; i < p; i++ {
coef *= -1
pk := int(math.Pow(float64(p), float64(k-1)))
cyclo.terms = append(cyclo.terms, term{coef, i * twoExp * pk})
}
} else if factors[2] != 0 && ((n/2)%2 == 1) && (n/2) > 1 {
// CP(2m)[x] == CP(-m)[x], n odd integer > 1
cycloDiv2 := cycloPoly(n / 2)
for _, t := range cycloDiv2.terms {
t2 := t
if t.exp%2 != 0 {
t2 = t.negate()
}
cyclo.terms = append(cyclo.terms, t2)
}
} else if algo == 0 {
// slow - uses basic definition
divs := getDivisors(n)
// polynomial: x^n - 1
cyclo = newPoly(1, n, -1, 0)
for _, i := range divs {
p := cycloPoly(i)
cyclo = cyclo.div(p)
}
} else if algo == 1 {
// faster - remove max divisor (and all divisors of max divisor)
// only one divide for all divisors of max divisor
divs := getDivisors(n)
maxDiv := math.MinInt32
for _, d := range divs {
if d > maxDiv {
maxDiv = d
}
}
var divsExceptMax []int
for _, d := range divs {
if maxDiv%d != 0 {
divsExceptMax = append(divsExceptMax, d)
}
}
// polynomial: ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor
cyclo = newPoly(1, n, -1, 0)
cyclo = cyclo.div(newPoly(1, maxDiv, -1, 0))
for _, i := range divsExceptMax {
p := cycloPoly(i)
cyclo = cyclo.div(p)
}
} else if algo == 2 {
// fastest
// let p, q be primes such that p does not divide n, and q divides n
// then CP(np)[x] = CP(n)[x^p] / CP(n)[x]
m := 1
cyclo = cycloPoly(m)
var primes []int
for prime := range factors {
primes = append(primes, prime)
}
sort.Ints(primes)
for _, prime := range primes {
// CP(m)[x]
cycloM := cyclo
// compute CP(m)[x^p]
var terms []term
for _, t := range cycloM.terms {
terms = append(terms, term{t.coef, t.exp * prime})
}
cyclo = newPoly()
cyclo.terms = append(cyclo.terms, terms...)
cyclo = cyclo.tidy()
cyclo = cyclo.div(cycloM)
m *= prime
}
// now, m is the largest square free divisor of n
s := n / m
// Compute CP(n)[x] = CP(m)[x^s]
var terms []term
for _, t := range cyclo.terms {
terms = append(terms, term{t.coef, t.exp * s})
}
cyclo = newPoly()
cyclo.terms = append(cyclo.terms, terms...)
} else {
log.Fatal("invalid algorithm")
}
cyclo = cyclo.tidy()
computed[n] = cyclo
return cyclo
}
func main() {
fmt.Println("Task 1: cyclotomic polynomials for n <= 30:")
for i := 1; i <= 30; i++ {
p := cycloPoly(i)
fmt.Printf("CP[%2d] = %s\n", i, p)
}
fmt.Println("\nTask 2: Smallest cyclotomic polynomial with n or -n as a coefficient:")
n := 0
for i := 1; i <= 10; i++ {
for {
n++
cyclo := cycloPoly(n)
if cyclo.hasCoefAbs(i) {
fmt.Printf("CP[%d] has coefficient with magnitude = %d\n", n, i)
n--
break
}
}
}
}</lang>
- Output:
Task 1: cyclotomic polynomials for n <= 30: CP[ 1] = x - 1 CP[ 2] = x + 1 CP[ 3] = x^2 + x + 1 CP[ 4] = x^2 + 1 CP[ 5] = x^4 + x^3 + x^2 + x + 1 CP[ 6] = x^2 - x + 1 CP[ 7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[ 8] = x^4 + 1 CP[ 9] = x^6 + x^3 + 1 CP[10] = x^4 - x^3 + x^2 - x + 1 CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[12] = x^4 - x^2 + 1 CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 CP[16] = x^8 + 1 CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[18] = x^6 - x^3 + 1 CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[20] = x^8 - x^6 + x^4 - x^2 + 1 CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[24] = x^8 - x^4 + 1 CP[25] = x^20 + x^15 + x^10 + x^5 + 1 CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[27] = x^18 + x^9 + 1 CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1 CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient: CP[1] has coefficient with magnitude = 1 CP[105] has coefficient with magnitude = 2 CP[385] has coefficient with magnitude = 3 CP[1365] has coefficient with magnitude = 4 CP[1785] has coefficient with magnitude = 5 CP[2805] has coefficient with magnitude = 6 CP[3135] has coefficient with magnitude = 7 CP[6545] has coefficient with magnitude = 8 CP[6545] has coefficient with magnitude = 9 CP[10465] has coefficient with magnitude = 10
Haskell
Uses synthetic polynomial division and simple memoization.
<lang haskell>import Data.List import Data.Numbers.Primes (primeFactors)
negateVar p = zipWith (*) p $ reverse $ take (length p) $ cycle [1,-1]
lift p 1 = p lift p n = intercalate (replicate (n-1) 0) (pure <$> p)
shortDiv :: [Integer] -> [Integer] -> [Integer] shortDiv p1 (_:p2) = unfoldr go (length p1 - length p2, p1)
where go (0, _) = Nothing go (i, h:t) = Just (h, (i-1, zipWith (+) (map (h *) ker) t)) ker = negate <$> p2 ++ repeat 0
primePowerFactors = sortOn fst . map (\x-> (head x, length x)) . group . primeFactors
-- simple memoization cyclotomics :: Integer cyclotomics = cyclotomic <$> [0..]
cyclotomic :: Int -> [Integer] cyclotomic 0 = [0] cyclotomic 1 = [1, -1] cyclotomic 2 = [1, 1] cyclotomic n = case primePowerFactors n of
-- for n = 2^k
[(2,h)] -> 1 : replicate (2 ^ (h-1) - 1) 0 ++ [1]
-- for prime n
[(p,1)] -> replicate n 1
-- for power of prime n
[(p,m)] -> lift (cyclotomics !! p) (p^(m-1))
-- for n = 2*p and prime p
[(2,1),(p,1)] -> take (n `div` 2) $ cycle [1,-1]
-- for n = 2*m and odd m
(2,1):_ -> negateVar $ cyclotomics !! (n `div` 2)
-- general case
(p, m):ps -> let cm = cyclotomics !! (n `div` (p ^ m))
in lift (lift cm p `shortDiv` cm) (p^(m-1))</lang>
Simple examples
λ> cyclotomic 7 [1,1,1,1,1,1,1] λ> cyclotomic 9 [1,0,0,1,0,0,1] λ> cyclotomic 16 [1,0,0,0,0,0,0,0,1]
The task solution
<lang haskell>showPoly [] = "0" showPoly p = foldl1 (\r -> (r ++) . term) $
dropWhile null $
foldMap (\(c, n) -> [show c ++ expt n]) $
zip (reverse p) [0..]
where
expt = \case 0 -> ""
1 -> "*x"
n -> "*x^" ++ show n
term = \case [] -> ""
'0':'*':t -> ""
'-':'1':'*':t -> " - " ++ t
'1':'*':t -> " + " ++ t
'-':t -> " - " ++ t
t -> " + " ++ t
main = do
mapM_ (print . showPoly . cyclotomic) [1..30]
putStrLn $ replicate 40 '-'
mapM_ showLine $ take 4 task2
where
showLine (j, i, l) = putStrLn $ concat [ show j
, " appears in CM(", show i
, ") of length ", show l ]
-- in order to make computations faster we leave only each 5-th polynomial
task2 = (1,1,2) : tail (search 1 $ zip [0,5..] $ skipBy 5 cyclotomics)
where
search i ((k, p):ps) = if i `notElem` (abs <$> p)
then search i ps
else (i, k, length p) : search (i+1) ((k, p):ps)
skipBy n [] = [] skipBy n lst = let (x:_, b) = splitAt n lst in x:skipBy n b</lang>
Result
"-1 + x^1" "1 + x^1" "1 + x^1 + x^2" "1 + x^2" "1 + x^1 + x^2 + x^3 + x^4" "1 - x^1 + x^2" "1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6" "1 + x^4" "1 + x^3 + x^6" "1 - x^1 + x^2 - x^3 + x^4" "1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10" "1 - x^2 + x^4" "1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12" "1 - x^1 + x^2 - x^3 + x^4 - x^5 + x^6" "1 - x^1 + x^3 - x^4 + x^5 - x^7 + x^8" "1 + x^8" "1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16" "1 - x^3 + x^6" "1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18" "1 - x^2 + x^4 - x^6 + x^8" "1 - x^1 + x^3 - x^4 + x^6 - x^8 + x^9 - x^11 + x^12" "1 - x^1 + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10" "1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18 + x^19 + x^20 + x^21 + x^22" "1 - x^4 + x^8" "1 + x^5 + x^10 + x^15 + x^20" "1 - x^1 + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10 - x^11 + x^12" "1 + x^9 + x^18" "1 - x^2 + x^4 - x^6 + x^8 - x^10 + x^12" "1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18 + x^19 + x^20 + x^21 + x^22 + x^23 + x^24 + x^25 + x^26 + x^27 + x^28" "1 + x^1 - x^3 - x^4 - x^5 + x^7 + x^8" ---------------------------------------- 1 appears in CM(1) having 2 terms 2 appears in CM(105) having 49 terms 3 appears in CM(385) having 241 terms 4 appears in CM(1365) having 577 terms 5 appears in CM(1785) having 769 terms 6 appears in CM(2805) having 1281 terms 7 appears in CM(3135) having 1441 terms 8 appears in CP(6545) having 3841 terms 9 appears in CP(6545) having 3841 terms 10 appears in CP(10465) having 6337 terms
Computations take a while...
J
For values up to 70, we can find cyclotomic polynomials by finding a polynomial with roots of unity relatively prime to the order of the polynomial:
<lang J>cyclo=: {{<.-:1+(++) p. 1;^0j2p1* y%~1+I.1=y+.1+i.y}}</lang>
This approach suggests that cyclotomic polynomial zero should be f0(x)= 1
Implementation of routine to find nth cyclotomic polynomial:
<lang J>{{ if.0>nc<'cache' do.cache=:y end.}} (,1);_1 1
cyclotomic=: {{
if.y<#cache do.
if.#c=. y{::cache do.
c return.
end.
end.
c=. unpad cyclotomic000 y
if. y>:#cache do. cache=:(100+y){.cache end.
cache=: (<c) y} cache
c
}}
cyclotomic000=: {{ assert.0<y
'q p'=. __ q: y
if. 1=#q do.
,(y%*/q) {."0 q#1
elseif.2={.q do.
,(y%*/q) {."0 (* 1 _1 $~ #) cyclotomic */}.q
elseif. 1 e. 1 < p do.
,(y%*/q) {."0 cyclotomic */q
else.
(_1,(-y){.1) pDiv ;+//.@(*/)each/ cyclotomic each}:*/@>,{1,each q
end.
}}
NB. discard high order zero coefficients in representation of polynomial
unpad=: {.~ 1+0 i:~0=]
NB. polynomial division, optimized for somewhat sparse polynomials pDiv=: Template:Q=. $j=. 2 + x -&</lang>
If you take all the divisors of a number. (For example, for 12, the divisors are: 1, 2, 3, 4, 6 and 12) and find the product of their cyclotomic polynomials (for example, for 12, x-1, x+1, x2+x+1, x2+1, x2-x+1, and x4-x2+1) you get xn-1 (for 12, that would of course be x12-1).
Notes:
- the coefficients of a cyclotomic polynomials after 1 form a palindrome (that's the q#1 phrase in the implementation).
- the coefficients of the cyclotomic polynomial for a prime number has as many terms as that number, and the coefficients are all 1.
- powers of primes add zero coefficients to the polynomial (that's the ,(y%*/q) {."0 ... phrase in the implementation). This means that we can mostly ignore powers of prime numbers -- they're just going to correspond to zeros we add to the base polynomial.
- an even base cyclotomic polynomial is the same as the corresponding odd base cyclotomic polynomial except with x replaced by negative x. (that's the (* 1 _1 $~ #) phrase in the implementation.
- To deal with the general case, we use polynomial division, xn-1 divided by the polynomial product the cyclotomic polynomials of the proper divisors of number we're looking for.
- +//.@(*/) is polynomial product in J.
Task examples:
<lang J>taskfmt=: {{
c=. ":each j=.cyclotomic y
raw=. rplc&'_-' ;:inv}.,'+';"0|.(*|j)#c('(',[,],')'"_)each '*x^',&":L:0 <"0 i.#c
txt=. raw rplc'(1*x^0)';'1';'(-1*x^0)';'(-1)';'*x^1)';'*x)'
LF,~'CP[',y,&":']= ',rplc&('(x)';'x';'+ (-1)';'- 1')txt rplc'(1*';'(';'(-1*';'(-'
}}
taskorder=: {{
r=.$k=.0
while.y>#r do.k=.k+1
if.(1+#r) e.|cyclotomic k do.
r=. r,k
k=. k-1
end.
end.r
}}
;taskfmt each 1+i.30
CP[1]= x - 1 CP[2]= x + 1 CP[3]= (x^2) + x + 1 CP[4]= (x^2) + 1 CP[5]= (x^4) + (x^3) + (x^2) + x + 1 CP[6]= (x^2) + (-x) + 1 CP[7]= (x^6) + (x^5) + (x^4) + (x^3) + (x^2) + x + 1 CP[8]= (x^4) + 1 CP[9]= (x^6) + (x^3) + 1 CP[10]= (x^4) + (-x^3) + (x^2) + (-x) + 1 CP[11]= (x^10) + (x^9) + (x^8) + (x^7) + (x^6) + (x^5) + (x^4) + (x^3) + (x^2) + x + 1 CP[12]= (x^4) + (-x^2) + 1 CP[13]= (x^12) + (x^11) + (x^10) + (x^9) + (x^8) + (x^7) + (x^6) + (x^5) + (x^4) + (x^3) + (x^2) + x + 1 CP[14]= (x^6) + (-x^5) + (x^4) + (-x^3) + (x^2) + (-x) + 1 CP[15]= (x^8) + (-x^7) + (x^5) + (-x^4) + (x^3) + (-x) + 1 CP[16]= (x^8) + 1 CP[17]= (x^16) + (x^15) + (x^14) + (x^13) + (x^12) + (x^11) + (x^10) + (x^9) + (x^8) + (x^7) + (x^6) + (x^5) + (x^4) + (x^3) + (x^2) + x + 1 CP[18]= (x^6) + (-x^3) + 1 CP[19]= (x^18) + (x^17) + (x^16) + (x^15) + (x^14) + (x^13) + (x^12) + (x^11) + (x^10) + (x^9) + (x^8) + (x^7) + (x^6) + (x^5) + (x^4) + (x^3) + (x^2) + x + 1 CP[20]= (x^8) + (-x^6) + (x^4) + (-x^2) + 1 CP[21]= (x^12) + (-x^11) + (x^9) + (-x^8) + (x^6) + (-x^4) + (x^3) + (-x) + 1 CP[22]= (x^10) + (-x^9) + (x^8) + (-x^7) + (x^6) + (-x^5) + (x^4) + (-x^3) + (x^2) + (-x) + 1 CP[23]= (x^22) + (x^21) + (x^20) + (x^19) + (x^18) + (x^17) + (x^16) + (x^15) + (x^14) + (x^13) + (x^12) + (x^11) + (x^10) + (x^9) + (x^8) + (x^7) + (x^6) + (x^5) + (x^4) + (x^3) + (x^2) + x + 1 CP[24]= (x^8) + (-x^4) + 1 CP[25]= (x^20) + (x^15) + (x^10) + (x^5) + 1 CP[26]= (x^12) + (-x^11) + (x^10) + (-x^9) + (x^8) + (-x^7) + (x^6) + (-x^5) + (x^4) + (-x^3) + (x^2) + (-x) + 1 CP[27]= (x^18) + (x^9) + 1 CP[28]= (x^12) + (-x^10) + (x^8) + (-x^6) + (x^4) + (-x^2) + 1 CP[29]= (x^28) + (x^27) + (x^26) + (x^25) + (x^24) + (x^23) + (x^22) + (x^21) + (x^20) + (x^19) + (x^18) + (x^17) + (x^16) + (x^15) + (x^14) + (x^13) + (x^12) + (x^11) + (x^10) + (x^9) + (x^8) + (x^7) + (x^6) + (x^5) + (x^4) + (x^3) + (x^2) + x + 1 CP[30]= (x^8) + (x^7) + (-x^5) + (-x^4) + (-x^3) + x + 1
(,.~#\) taskorder 10 1 1 2 105 3 385 4 1365 5 1785 6 2805 7 3135 8 6545 9 6545
10 10465</lang>
Another approach
As noted in the J programming forum, we can improve the big-O character of this algorithm by using the Fast Fourier transform for polynomial multiplication and division.
<lang J>NB. install'math/fftw' require'math/fftw'
cyclotomic000=: {{ assert.0<y
if. y = 1 do. _1 1 return. end.
'q p'=. __ q: y
if. 1=#q do.
,(y%*/q) {."0 q#1
elseif.2={.q do.
,(y%*/q) {."0 (* 1 _1 $~ #) cyclotomic */}.q
elseif. 1 e. 1 < p do.
,(y%*/q) {."0 cyclotomic */q
else.
lgl=. {:$ ctlist=. cyclotomic "0 }:*/@>,{1,each q NB. ctlist is 2-d table of polynomial divisors
lgd=. # dividend=. _1,(-y){.1 NB. (x^n) - 1, and its size
lg=. >.&.(2&^.) lgl >. lgd NB. required lengths of all polynomials for fft transforms
NB. really, "divisor" is the fft of the divisor!
divisor=. */ fftw"1 lg{."1 ctlist NB. FFT article doesn't deal with lists of multiplicands
unpad roundreal ifftw"1 divisor %~ fftw lg{.dividend NB. similar to article's multiplication
end.
}}
roundreal =: [: <. 0.5 + 9&o.</lang>
This variation for polynomial division is only valid when there's no remainder to be concerned with (which is the case, here).
This approach gave slightly over a 16x speedup for taskorder 10, from a 2 element cache, with an approximately 50% increased memory footprint. (Remember, of course, that benchmarks and benchmark ratios have dependencies on computer architecture and language implementation, and the host environment.)
Java
<lang java> import java.util.ArrayList; import java.util.Collections; import java.util.Comparator; import java.util.HashMap; import java.util.List; import java.util.Map; import java.util.TreeMap;
public class CyclotomicPolynomial {
@SuppressWarnings("unused")
private static int divisions = 0;
private static int algorithm = 2;
public static void main(String[] args) throws Exception {
System.out.println("Task 1: cyclotomic polynomials for n <= 30:");
for ( int i = 1 ; i <= 30 ; i++ ) {
Polynomial p = cyclotomicPolynomial(i);
System.out.printf("CP[%d] = %s%n", i, p);
}
System.out.println("Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient:");
int n = 0;
for ( int i = 1 ; i <= 10 ; i++ ) {
while ( true ) {
n++;
Polynomial cyclo = cyclotomicPolynomial(n);
if ( cyclo.hasCoefficientAbs(i) ) {
System.out.printf("CP[%d] has coefficient with magnitude = %d%n", n, i);
n--;
break;
}
}
}
}
private static final Map<Integer, Polynomial> COMPUTED = new HashMap<>();
private static Polynomial cyclotomicPolynomial(int n) {
if ( COMPUTED.containsKey(n) ) {
return COMPUTED.get(n);
}
//System.out.println("COMPUTE: n = " + n);
if ( n == 1 ) {
// Polynomial: x - 1
Polynomial p = new Polynomial(1, 1, -1, 0);
COMPUTED.put(1, p);
return p;
}
Map<Integer,Integer> factors = getFactors(n);
if ( factors.containsKey(n) ) {
// n prime
List<Term> termList = new ArrayList<>();
for ( int index = 0 ; index < n ; index++ ) {
termList.add(new Term(1, index));
}
Polynomial cyclo = new Polynomial(termList);
COMPUTED.put(n, cyclo);
return cyclo;
}
else if ( factors.size() == 2 && factors.containsKey(2) && factors.get(2) == 1 && factors.containsKey(n/2) && factors.get(n/2) == 1 ) {
// n = 2p
int prime = n/2;
List<Term> termList = new ArrayList<>();
int coeff = -1;
for ( int index = 0 ; index < prime ; index++ ) {
coeff *= -1;
termList.add(new Term(coeff, index));
}
Polynomial cyclo = new Polynomial(termList);
COMPUTED.put(n, cyclo);
return cyclo;
}
else if ( factors.size() == 1 && factors.containsKey(2) ) {
// n = 2^h
int h = factors.get(2);
List<Term> termList = new ArrayList<>();
termList.add(new Term(1, (int) Math.pow(2, h-1)));
termList.add(new Term(1, 0));
Polynomial cyclo = new Polynomial(termList);
COMPUTED.put(n, cyclo);
return cyclo;
}
else if ( factors.size() == 1 && ! factors.containsKey(n) ) {
// n = p^k
int p = 0;
for ( int prime : factors.keySet() ) {
p = prime;
}
int k = factors.get(p);
List<Term> termList = new ArrayList<>();
for ( int index = 0 ; index < p ; index++ ) {
termList.add(new Term(1, index * (int) Math.pow(p, k-1)));
}
Polynomial cyclo = new Polynomial(termList);
COMPUTED.put(n, cyclo);
return cyclo;
}
else if ( factors.size() == 2 && factors.containsKey(2) ) {
// n = 2^h * p^k
int p = 0;
for ( int prime : factors.keySet() ) {
if ( prime != 2 ) {
p = prime;
}
}
List<Term> termList = new ArrayList<>();
int coeff = -1;
int twoExp = (int) Math.pow(2, factors.get(2)-1);
int k = factors.get(p);
for ( int index = 0 ; index < p ; index++ ) {
coeff *= -1;
termList.add(new Term(coeff, index * twoExp * (int) Math.pow(p, k-1)));
}
Polynomial cyclo = new Polynomial(termList);
COMPUTED.put(n, cyclo);
return cyclo;
}
else if ( factors.containsKey(2) && ((n/2) % 2 == 1) && (n/2) > 1 ) {
// CP(2m)[x] = CP(-m)[x], n odd integer > 1
Polynomial cycloDiv2 = cyclotomicPolynomial(n/2);
List<Term> termList = new ArrayList<>();
for ( Term term : cycloDiv2.polynomialTerms ) {
termList.add(term.exponent % 2 == 0 ? term : term.negate());
}
Polynomial cyclo = new Polynomial(termList);
COMPUTED.put(n, cyclo);
return cyclo;
}
// General Case
if ( algorithm == 0 ) {
// Slow - uses basic definition.
List<Integer> divisors = getDivisors(n);
// Polynomial: ( x^n - 1 )
Polynomial cyclo = new Polynomial(1, n, -1, 0);
for ( int i : divisors ) {
Polynomial p = cyclotomicPolynomial(i);
cyclo = cyclo.divide(p);
}
COMPUTED.put(n, cyclo);
return cyclo;
}
else if ( algorithm == 1 ) {
// Faster. Remove Max divisor (and all divisors of max divisor) - only one divide for all divisors of Max Divisor
List<Integer> divisors = getDivisors(n);
int maxDivisor = Integer.MIN_VALUE;
for ( int div : divisors ) {
maxDivisor = Math.max(maxDivisor, div);
}
List<Integer> divisorsExceptMax = new ArrayList<Integer>();
for ( int div : divisors ) {
if ( maxDivisor % div != 0 ) {
divisorsExceptMax.add(div);
}
}
// Polynomial: ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor
Polynomial cyclo = new Polynomial(1, n, -1, 0).divide(new Polynomial(1, maxDivisor, -1, 0));
for ( int i : divisorsExceptMax ) {
Polynomial p = cyclotomicPolynomial(i);
cyclo = cyclo.divide(p);
}
COMPUTED.put(n, cyclo);
return cyclo;
}
else if ( algorithm == 2 ) {
// Fastest
// Let p ; q be primes such that p does not divide n, and q q divides n.
// Then CP(np)[x] = CP(n)[x^p] / CP(n)[x]
int m = 1;
Polynomial cyclo = cyclotomicPolynomial(m);
List<Integer> primes = new ArrayList<>(factors.keySet());
Collections.sort(primes);
for ( int prime : primes ) {
// CP(m)[x]
Polynomial cycloM = cyclo;
// Compute CP(m)[x^p].
List<Term> termList = new ArrayList<>();
for ( Term t : cycloM.polynomialTerms ) {
termList.add(new Term(t.coefficient, t.exponent * prime));
}
cyclo = new Polynomial(termList).divide(cycloM);
m = m * prime;
}
// Now, m is the largest square free divisor of n
int s = n / m;
// Compute CP(n)[x] = CP(m)[x^s]
List<Term> termList = new ArrayList<>();
for ( Term t : cyclo.polynomialTerms ) {
termList.add(new Term(t.coefficient, t.exponent * s));
}
cyclo = new Polynomial(termList);
COMPUTED.put(n, cyclo);
return cyclo;
}
else {
throw new RuntimeException("ERROR 103: Invalid algorithm.");
}
}
private static final List<Integer> getDivisors(int number) {
List<Integer> divisors = new ArrayList<Integer>();
long sqrt = (long) Math.sqrt(number);
for ( int i = 1 ; i <= sqrt ; i++ ) {
if ( number % i == 0 ) {
divisors.add(i);
int div = number / i;
if ( div != i && div != number ) {
divisors.add(div);
}
}
}
return divisors;
}
private static final Map<Integer,Map<Integer,Integer>> allFactors = new TreeMap<Integer,Map<Integer,Integer>>();
static {
Map<Integer,Integer> factors = new TreeMap<Integer,Integer>();
factors.put(2, 1);
allFactors.put(2, factors);
}
public static Integer MAX_ALL_FACTORS = 100000;
public static final Map<Integer,Integer> getFactors(Integer number) {
if ( allFactors.containsKey(number) ) {
return allFactors.get(number);
}
Map<Integer,Integer> factors = new TreeMap<Integer,Integer>();
if ( number % 2 == 0 ) {
Map<Integer,Integer> factorsdDivTwo = getFactors(number/2);
factors.putAll(factorsdDivTwo);
factors.merge(2, 1, (v1, v2) -> v1 + v2);
if ( number < MAX_ALL_FACTORS )
allFactors.put(number, factors);
return factors;
}
boolean prime = true;
long sqrt = (long) Math.sqrt(number);
for ( int i = 3 ; i <= sqrt ; i += 2 ) {
if ( number % i == 0 ) {
prime = false;
factors.putAll(getFactors(number/i));
factors.merge(i, 1, (v1, v2) -> v1 + v2);
if ( number < MAX_ALL_FACTORS )
allFactors.put(number, factors);
return factors;
}
}
if ( prime ) {
factors.put(number, 1);
if ( number < MAX_ALL_FACTORS )
allFactors.put(number, factors);
}
return factors;
}
private static final class Polynomial {
private List<Term> polynomialTerms;
// Format - coeff, exp, coeff, exp, (repeating in pairs) . . .
public Polynomial(int ... values) {
if ( values.length % 2 != 0 ) {
throw new IllegalArgumentException("ERROR 102: Polynomial constructor. Length must be even. Length = " + values.length);
}
polynomialTerms = new ArrayList<>();
for ( int i = 0 ; i < values.length ; i += 2 ) {
Term t = new Term(values[i], values[i+1]);
polynomialTerms.add(t);
}
Collections.sort(polynomialTerms, new TermSorter());
}
public Polynomial() {
// zero
polynomialTerms = new ArrayList<>();
polynomialTerms.add(new Term(0,0));
}
private boolean hasCoefficientAbs(int coeff) {
for ( Term term : polynomialTerms ) {
if ( Math.abs(term.coefficient) == coeff ) {
return true;
}
}
return false;
}
private Polynomial(List<Term> termList) {
if ( termList.size() == 0 ) {
// zero
termList.add(new Term(0,0));
}
else {
// Remove zero terms if needed
for ( int i = 0 ; i < termList.size() ; i++ ) {
if ( termList.get(i).coefficient == 0 ) {
termList.remove(i);
}
}
}
if ( termList.size() == 0 ) {
// zero
termList.add(new Term(0,0));
}
polynomialTerms = termList;
Collections.sort(polynomialTerms, new TermSorter());
}
public Polynomial divide(Polynomial v) {
//long start = System.currentTimeMillis();
divisions++;
Polynomial q = new Polynomial();
Polynomial r = this;
long lcv = v.leadingCoefficient();
long dv = v.degree();
while ( r.degree() >= v.degree() ) {
long lcr = r.leadingCoefficient();
long s = lcr / lcv; // Integer division
Term term = new Term(s, r.degree() - dv);
q = q.add(term);
r = r.add(v.multiply(term.negate()));
}
//long end = System.currentTimeMillis();
//System.out.printf("Divide: Elapsed = %d, Degree 1 = %d, Degree 2 = %d%n", (end-start), this.degree(), v.degree());
return q;
}
public Polynomial add(Polynomial polynomial) {
List<Term> termList = new ArrayList<>();
int thisCount = polynomialTerms.size();
int polyCount = polynomial.polynomialTerms.size();
while ( thisCount > 0 || polyCount > 0 ) {
Term thisTerm = thisCount == 0 ? null : polynomialTerms.get(thisCount-1);
Term polyTerm = polyCount == 0 ? null : polynomial.polynomialTerms.get(polyCount-1);
if ( thisTerm == null ) {
termList.add(polyTerm.clone());
polyCount--;
}
else if (polyTerm == null ) {
termList.add(thisTerm.clone());
thisCount--;
}
else if ( thisTerm.degree() == polyTerm.degree() ) {
Term t = thisTerm.add(polyTerm);
if ( t.coefficient != 0 ) {
termList.add(t);
}
thisCount--;
polyCount--;
}
else if ( thisTerm.degree() < polyTerm.degree() ) {
termList.add(thisTerm.clone());
thisCount--;
}
else {
termList.add(polyTerm.clone());
polyCount--;
}
}
return new Polynomial(termList);
}
public Polynomial add(Term term) {
List<Term> termList = new ArrayList<>();
boolean added = false;
for ( int index = 0 ; index < polynomialTerms.size() ; index++ ) {
Term currentTerm = polynomialTerms.get(index);
if ( currentTerm.exponent == term.exponent ) {
added = true;
if ( currentTerm.coefficient + term.coefficient != 0 ) {
termList.add(currentTerm.add(term));
}
}
else {
termList.add(currentTerm.clone());
}
}
if ( ! added ) {
termList.add(term.clone());
}
return new Polynomial(termList);
}
public Polynomial multiply(Term term) {
List<Term> termList = new ArrayList<>();
for ( int index = 0 ; index < polynomialTerms.size() ; index++ ) {
Term currentTerm = polynomialTerms.get(index);
termList.add(currentTerm.clone().multiply(term));
}
return new Polynomial(termList);
}
public Polynomial clone() {
List<Term> clone = new ArrayList<>();
for ( Term t : polynomialTerms ) {
clone.add(new Term(t.coefficient, t.exponent));
}
return new Polynomial(clone);
}
public long leadingCoefficient() {
// long coefficient = 0; // long degree = Integer.MIN_VALUE; // for ( Term t : polynomialTerms ) { // if ( t.degree() > degree ) { // coefficient = t.coefficient; // degree = t.degree(); // } // }
return polynomialTerms.get(0).coefficient;
}
public long degree() {
// long degree = Integer.MIN_VALUE; // for ( Term t : polynomialTerms ) { // if ( t.degree() > degree ) { // degree = t.degree(); // } // }
return polynomialTerms.get(0).exponent;
}
@Override
public String toString() {
StringBuilder sb = new StringBuilder();
boolean first = true;
for ( Term term : polynomialTerms ) {
if ( first ) {
sb.append(term);
first = false;
}
else {
sb.append(" ");
if ( term.coefficient > 0 ) {
sb.append("+ ");
sb.append(term);
}
else {
sb.append("- ");
sb.append(term.negate());
}
}
}
return sb.toString();
}
}
private static final class TermSorter implements Comparator<Term> {
@Override
public int compare(Term o1, Term o2) {
return (int) (o2.exponent - o1.exponent);
}
}
// Note: Cyclotomic Polynomials have small coefficients. Not appropriate for general polynomial usage.
private static final class Term {
long coefficient;
long exponent;
public Term(long c, long e) {
coefficient = c;
exponent = e;
}
public Term clone() {
return new Term(coefficient, exponent);
}
public Term multiply(Term term) {
return new Term(coefficient * term.coefficient, exponent + term.exponent);
}
public Term add(Term term) {
if ( exponent != term.exponent ) {
throw new RuntimeException("ERROR 102: Exponents not equal.");
}
return new Term(coefficient + term.coefficient, exponent);
}
public Term negate() {
return new Term(-coefficient, exponent);
}
public long degree() {
return exponent;
}
@Override
public String toString() {
if ( coefficient == 0 ) {
return "0";
}
if ( exponent == 0 ) {
return "" + coefficient;
}
if ( coefficient == 1 ) {
if ( exponent == 1 ) {
return "x";
}
else {
return "x^" + exponent;
}
}
if ( exponent == 1 ) {
return coefficient + "x";
}
return coefficient + "x^" + exponent;
}
}
} </lang>
- Output:
Task 1: cyclotomic polynomials for n <= 30: CP[1] = x - 1 CP[2] = x + 1 CP[3] = x^2 + x + 1 CP[4] = x^2 + 1 CP[5] = x^4 + x^3 + x^2 + x + 1 CP[6] = x^2 - x + 1 CP[7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[8] = x^4 + 1 CP[9] = x^6 + x^3 + 1 CP[10] = x^4 - x^3 + x^2 - x + 1 CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[12] = x^4 - x^2 + 1 CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 CP[16] = x^8 + 1 CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[18] = x^6 - x^3 + 1 CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[20] = x^8 - x^6 + x^4 - x^2 + 1 CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[24] = x^8 - x^4 + 1 CP[25] = x^20 + x^15 + x^10 + x^5 + 1 CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[27] = x^18 + x^9 + 1 CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1 CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient: CP[1] has coefficient with magnitude = 1 CP[105] has coefficient with magnitude = 2 CP[385] has coefficient with magnitude = 3 CP[1365] has coefficient with magnitude = 4 CP[1785] has coefficient with magnitude = 5 CP[2805] has coefficient with magnitude = 6 CP[3135] has coefficient with magnitude = 7 CP[6545] has coefficient with magnitude = 8 CP[6545] has coefficient with magnitude = 9 CP[10465] has coefficient with magnitude = 10
Julia
<lang julia>using Primes, Polynomials
- memoize cache for recursive calls
const cyclotomics = Dict([1 => Poly([big"-1", big"1"]), 2 => Poly([big"1", big"1"])])
- get all integer divisors of an integer, except itself
function divisors(n::Integer)
f = [one(n)]
for (p, e) in factor(n)
f = reduce(vcat, [f * p^j for j in 1:e], init=f)
end
return resize!(f, length(f) - 1)
end
"""
cyclotomic(n::Integer)
Calculate the n -th cyclotomic polynomial. See wikipedia article at bottom of section /wiki/Cyclotomic_polynomial#Fundamental_tools The algorithm is reliable but slow for large n > 1000. """ function cyclotomic(n::Integer)
if haskey(cyclotomics, n)
c = cyclotomics[n]
elseif isprime(n)
c = Poly(ones(BigInt, n))
cyclotomics[n] = c
else # recursive formula seen in wikipedia article
c = Poly([big"-1"; zeros(BigInt, n - 1); big"1"])
for d in divisors(n)
c ÷= cyclotomic(d)
end
cyclotomics[n] = c
end
return c
end
println("First 30 cyclotomic polynomials:") for i in 1:30
println(rpad("$i: ", 5), cyclotomic(BigInt(i)))
end
const dig = zeros(BigInt, 10) for i in 1:1000000
if all(x -> x != 0, dig)
break
end
for coef in coeffs(cyclotomic(i))
x = abs(coef)
if 0 < x < 11 && dig[Int(x)] == 0
dig[Int(x)] = coef < 0 ? -i : i
end
end
end for (i, n) in enumerate(dig)
println("The cyclotomic polynomial Φ(", abs(n), ") has a coefficient that is ", n < 0 ? -i : i)
end
</lang>
- Output:
First 30 cyclotomic polynomials: 1: Poly(-1 + x) 2: Poly(1 + x) 3: Poly(1 + x + x^2) 4: Poly(1.0 + 1.0*x^2) 5: Poly(1 + x + x^2 + x^3 + x^4) 6: Poly(1.0 - 1.0*x + 1.0*x^2) 7: Poly(1 + x + x^2 + x^3 + x^4 + x^5 + x^6) 8: Poly(1.0 + 1.0*x^4) 9: Poly(1.0 + 1.0*x^3 + 1.0*x^6) 10: Poly(1.0 - 1.0*x + 1.0*x^2 - 1.0*x^3 + 1.0*x^4) 11: Poly(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10) 12: Poly(1.0 - 1.0*x^2 + 1.0*x^4) 13: Poly(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12) 14: Poly(1.0 - 1.0*x + 1.0*x^2 - 1.0*x^3 + 1.0*x^4 - 1.0*x^5 + 1.0*x^6) 15: Poly(1.0 - 1.0*x + 1.0*x^3 - 1.0*x^4 + 1.0*x^5 - 1.0*x^7 + 1.0*x^8) 16: Poly(1.0 + 1.0*x^8) 17: Poly(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16) 18: Poly(1.0 - 1.0*x^3 + 1.0*x^6) 19: Poly(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18) 20: Poly(1.0 - 1.0*x^2 + 1.0*x^4 - 1.0*x^6 + 1.0*x^8) 21: Poly(1.0 - 1.0*x + 1.0*x^3 - 1.0*x^4 + 1.0*x^6 - 1.0*x^8 + 1.0*x^9 - 1.0*x^11 + 1.0*x^12) 22: Poly(1.0 - 1.0*x + 1.0*x^2 - 1.0*x^3 + 1.0*x^4 - 1.0*x^5 + 1.0*x^6 - 1.0*x^7 + 1.0*x^8 - 1.0*x^9 + 1.0*x^10) 23: Poly(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18 + x^19 + x^20 + x^21 + x^22) 24: Poly(1.0 - 1.0*x^4 + 1.0*x^8) 25: Poly(1.0 + 1.0*x^5 + 1.0*x^10 + 1.0*x^15 + 1.0*x^20) 26: Poly(1.0 - 1.0*x + 1.0*x^2 - 1.0*x^3 + 1.0*x^4 - 1.0*x^5 + 1.0*x^6 - 1.0*x^7 + 1.0*x^8 - 1.0*x^9 + 1.0*x^10 - 1.0*x^11 + 1.0*x^12) 27: Poly(1.0 + 1.0*x^9 + 1.0*x^18) 28: Poly(1.0 - 1.0*x^2 + 1.0*x^4 - 1.0*x^6 + 1.0*x^8 - 1.0*x^10 + 1.0*x^12) 29: Poly(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18 + x^19 + x^20 + x^21 + x^22 + x^23 + x^24 + x^25 + x^26 + x^27 + x^28) 30: Poly(1.0 + 1.0*x - 1.0*x^3 - 1.0*x^4 - 1.0*x^5 + 1.0*x^7 + 1.0*x^8) The cyclotomic polynomial Φ(1) has a coefficient that is -1 The cyclotomic polynomial Φ(105) has a coefficient that is -2 The cyclotomic polynomial Φ(385) has a coefficient that is -3 The cyclotomic polynomial Φ(1365) has a coefficient that is -4 The cyclotomic polynomial Φ(1785) has a coefficient that is 5 The cyclotomic polynomial Φ(2805) has a coefficient that is -6 The cyclotomic polynomial Φ(3135) has a coefficient that is 7 The cyclotomic polynomial Φ(6545) has a coefficient that is -8 The cyclotomic polynomial Φ(6545) has a coefficient that is 9 The cyclotomic polynomial Φ(10465) has a coefficient that is 10
Kotlin
<lang scala>import java.util.TreeMap import kotlin.math.abs import kotlin.math.pow import kotlin.math.sqrt
private const val algorithm = 2
fun main() {
println("Task 1: cyclotomic polynomials for n <= 30:")
for (i in 1..30) {
val p = cyclotomicPolynomial(i)
println("CP[$i] = $p")
}
println()
println("Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient:")
var n = 0
for (i in 1..10) {
while (true) {
n++
val cyclo = cyclotomicPolynomial(n)
if (cyclo!!.hasCoefficientAbs(i)) {
println("CP[$n] has coefficient with magnitude = $i")
n--
break
}
}
}
}
private val COMPUTED: MutableMap<Int, Polynomial> = HashMap() private fun cyclotomicPolynomial(n: Int): Polynomial? {
if (COMPUTED.containsKey(n)) {
return COMPUTED[n]
}
if (n == 1) {
// Polynomial: x - 1
val p = Polynomial(1, 1, -1, 0)
COMPUTED[1] = p
return p
}
val factors = getFactors(n)
if (factors.containsKey(n)) {
// n prime
val termList: MutableList<Term> = ArrayList()
for (index in 0 until n) {
termList.add(Term(1, index.toLong()))
}
val cyclo = Polynomial(termList)
COMPUTED[n] = cyclo
return cyclo
} else if (factors.size == 2 && factors.containsKey(2) && factors[2] == 1 && factors.containsKey(n / 2) && factors[n / 2] == 1) {
// n = 2p
val prime = n / 2
val termList: MutableList<Term> = ArrayList()
var coeff = -1
for (index in 0 until prime) {
coeff *= -1
termList.add(Term(coeff.toLong(), index.toLong()))
}
val cyclo = Polynomial(termList)
COMPUTED[n] = cyclo
return cyclo
} else if (factors.size == 1 && factors.containsKey(2)) {
// n = 2^h
val h = factors[2]!!
val termList: MutableList<Term> = ArrayList()
termList.add(Term(1, 2.0.pow((h - 1).toDouble()).toLong()))
termList.add(Term(1, 0))
val cyclo = Polynomial(termList)
COMPUTED[n] = cyclo
return cyclo
} else if (factors.size == 1 && !factors.containsKey(n)) {
// n = p^k
var p = 0
for (prime in factors.keys) {
p = prime
}
val k = factors[p]!!
val termList: MutableList<Term> = ArrayList()
for (index in 0 until p) {
termList.add(Term(1, (index * p.toDouble().pow(k - 1.toDouble()).toInt()).toLong()))
}
val cyclo = Polynomial(termList)
COMPUTED[n] = cyclo
return cyclo
} else if (factors.size == 2 && factors.containsKey(2)) {
// n = 2^h * p^k
var p = 0
for (prime in factors.keys) {
if (prime != 2) {
p = prime
}
}
val termList: MutableList<Term> = ArrayList()
var coeff = -1
val twoExp = 2.0.pow((factors[2]!!) - 1.toDouble()).toInt()
val k = factors[p]!!
for (index in 0 until p) {
coeff *= -1
termList.add(Term(coeff.toLong(), (index * twoExp * p.toDouble().pow(k - 1.toDouble()).toInt()).toLong()))
}
val cyclo = Polynomial(termList)
COMPUTED[n] = cyclo
return cyclo
} else if (factors.containsKey(2) && n / 2 % 2 == 1 && n / 2 > 1) {
// CP(2m)[x] = CP(-m)[x], n odd integer > 1
val cycloDiv2 = cyclotomicPolynomial(n / 2)
val termList: MutableList<Term> = ArrayList()
for (term in cycloDiv2!!.polynomialTerms) {
termList.add(if (term.exponent % 2 == 0L) term else term.negate())
}
val cyclo = Polynomial(termList)
COMPUTED[n] = cyclo
return cyclo
}
// General Case
return when (algorithm) {
0 -> {
// Slow - uses basic definition.
val divisors = getDivisors(n)
// Polynomial: ( x^n - 1 )
var cyclo = Polynomial(1, n, -1, 0)
for (i in divisors) {
val p = cyclotomicPolynomial(i)
cyclo = cyclo.divide(p)
}
COMPUTED[n] = cyclo
cyclo
}
1 -> {
// Faster. Remove Max divisor (and all divisors of max divisor) - only one divide for all divisors of Max Divisor
val divisors = getDivisors(n)
var maxDivisor = Int.MIN_VALUE
for (div in divisors) {
maxDivisor = maxDivisor.coerceAtLeast(div)
}
val divisorsExceptMax: MutableList<Int> = ArrayList()
for (div in divisors) {
if (maxDivisor % div != 0) {
divisorsExceptMax.add(div)
}
}
// Polynomial: ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor
var cyclo = Polynomial(1, n, -1, 0).divide(Polynomial(1, maxDivisor, -1, 0))
for (i in divisorsExceptMax) {
val p = cyclotomicPolynomial(i)
cyclo = cyclo.divide(p)
}
COMPUTED[n] = cyclo
cyclo
}
2 -> {
// Fastest
// Let p ; q be primes such that p does not divide n, and q q divides n.
// Then CP(np)[x] = CP(n)[x^p] / CP(n)[x]
var m = 1
var cyclo = cyclotomicPolynomial(m)
val primes = factors.keys.toMutableList()
primes.sort()
for (prime in primes) {
// CP(m)[x]
val cycloM = cyclo
// Compute CP(m)[x^p].
val termList: MutableList<Term> = ArrayList()
for (t in cycloM!!.polynomialTerms) {
termList.add(Term(t.coefficient, t.exponent * prime))
}
cyclo = Polynomial(termList).divide(cycloM)
m *= prime
}
// Now, m is the largest square free divisor of n
val s = n / m
// Compute CP(n)[x] = CP(m)[x^s]
val termList: MutableList<Term> = ArrayList()
for (t in cyclo!!.polynomialTerms) {
termList.add(Term(t.coefficient, t.exponent * s))
}
cyclo = Polynomial(termList)
COMPUTED[n] = cyclo
cyclo
}
else -> {
throw RuntimeException("ERROR 103: Invalid algorithm.")
}
}
}
private fun getDivisors(number: Int): List<Int> {
val divisors: MutableList<Int> = ArrayList()
val sqrt = sqrt(number.toDouble()).toLong()
for (i in 1..sqrt) {
if (number % i == 0L) {
divisors.add(i.toInt())
val div = (number / i).toInt()
if (div.toLong() != i && div != number) {
divisors.add(div)
}
}
}
return divisors
}
private fun crutch(): MutableMap<Int, Map<Int, Int>> {
val allFactors: MutableMap<Int, Map<Int, Int>> = TreeMap()
val factors: MutableMap<Int, Int> = TreeMap() factors[2] = 1
allFactors[2] = factors return allFactors
}
private val allFactors = crutch()
var MAX_ALL_FACTORS = 100000
fun getFactors(number: Int): Map<Int, Int> {
if (allFactors.containsKey(number)) {
return allFactors[number]!!
}
val factors: MutableMap<Int, Int> = TreeMap()
if (number % 2 == 0) {
val factorsDivTwo = getFactors(number / 2)
factors.putAll(factorsDivTwo)
factors.merge(2, 1) { a: Int?, b: Int? -> Integer.sum(a!!, b!!) }
if (number < MAX_ALL_FACTORS) allFactors[number] = factors
return factors
}
val sqrt = sqrt(number.toDouble()).toLong()
var i = 3
while (i <= sqrt) {
if (number % i == 0) {
factors.putAll(getFactors(number / i))
factors.merge(i, 1) { a: Int?, b: Int? -> Integer.sum(a!!, b!!) }
if (number < MAX_ALL_FACTORS) {
allFactors[number] = factors
}
return factors
}
i += 2
}
factors[number] = 1
if (number < MAX_ALL_FACTORS) {
allFactors[number] = factors
}
return factors
}
private class Polynomial {
val polynomialTerms: MutableList<Term>
// Format - coeff, exp, coeff, exp, (repeating in pairs) . . .
constructor(vararg values: Int) {
require(values.size % 2 == 0) { "ERROR 102: Polynomial constructor. Length must be even. Length = " + values.size }
polynomialTerms = mutableListOf()
var i = 0
while (i < values.size) {
val t = Term(values[i].toLong(), values[i + 1].toLong())
polynomialTerms.add(t)
i += 2
}
polynomialTerms.sortWith(TermSorter())
}
constructor() {
// zero
polynomialTerms = ArrayList()
polynomialTerms.add(Term(0, 0))
}
fun hasCoefficientAbs(coeff: Int): Boolean {
for (term in polynomialTerms) {
if (abs(term.coefficient) == coeff.toLong()) {
return true
}
}
return false
}
constructor(termList: MutableList<Term>) {
if (termList.isEmpty()) {
// zero
termList.add(Term(0, 0))
} else {
// Remove zero terms if needed
termList.removeIf { t -> t.coefficient == 0L }
}
if (termList.size == 0) {
// zero
termList.add(Term(0, 0))
}
polynomialTerms = termList
polynomialTerms.sortWith(TermSorter())
}
fun divide(v: Polynomial?): Polynomial {
var q = Polynomial()
var r = this
val lcv = v!!.leadingCoefficient()
val dv = v.degree()
while (r.degree() >= v.degree()) {
val lcr = r.leadingCoefficient()
val s = lcr / lcv // Integer division
val term = Term(s, r.degree() - dv)
q = q.add(term)
r = r.add(v.multiply(term.negate()))
}
return q
}
fun add(polynomial: Polynomial): Polynomial {
val termList: MutableList<Term> = ArrayList()
var thisCount = polynomialTerms.size
var polyCount = polynomial.polynomialTerms.size
while (thisCount > 0 || polyCount > 0) {
val thisTerm = if (thisCount == 0) null else polynomialTerms[thisCount - 1]
val polyTerm = if (polyCount == 0) null else polynomial.polynomialTerms[polyCount - 1]
when {
thisTerm == null -> {
termList.add(polyTerm!!.clone())
polyCount--
}
polyTerm == null -> {
termList.add(thisTerm.clone())
thisCount--
}
thisTerm.degree() == polyTerm.degree() -> {
val t = thisTerm.add(polyTerm)
if (t.coefficient != 0L) {
termList.add(t)
}
thisCount--
polyCount--
}
thisTerm.degree() < polyTerm.degree() -> {
termList.add(thisTerm.clone())
thisCount--
}
else -> {
termList.add(polyTerm.clone())
polyCount--
}
}
}
return Polynomial(termList)
}
fun add(term: Term): Polynomial {
val termList: MutableList<Term> = ArrayList()
var added = false
for (currentTerm in polynomialTerms) {
if (currentTerm.exponent == term.exponent) {
added = true
if (currentTerm.coefficient + term.coefficient != 0L) {
termList.add(currentTerm.add(term))
}
} else {
termList.add(currentTerm.clone())
}
}
if (!added) {
termList.add(term.clone())
}
return Polynomial(termList)
}
fun multiply(term: Term): Polynomial {
val termList: MutableList<Term> = ArrayList()
for (currentTerm in polynomialTerms) {
termList.add(currentTerm.clone().multiply(term))
}
return Polynomial(termList)
}
fun leadingCoefficient(): Long {
return polynomialTerms[0].coefficient
}
fun degree(): Long {
return polynomialTerms[0].exponent
}
override fun toString(): String {
val sb = StringBuilder()
var first = true
for (term in polynomialTerms) {
if (first) {
sb.append(term)
first = false
} else {
sb.append(" ")
if (term.coefficient > 0) {
sb.append("+ ")
sb.append(term)
} else {
sb.append("- ")
sb.append(term.negate())
}
}
}
return sb.toString()
}
}
private class TermSorter : Comparator<Term> {
override fun compare(o1: Term, o2: Term): Int {
return (o2.exponent - o1.exponent).toInt()
}
}
// Note: Cyclotomic Polynomials have small coefficients. Not appropriate for general polynomial usage. private class Term(var coefficient: Long, var exponent: Long) {
fun clone(): Term {
return Term(coefficient, exponent)
}
fun multiply(term: Term): Term {
return Term(coefficient * term.coefficient, exponent + term.exponent)
}
fun add(term: Term): Term {
if (exponent != term.exponent) {
throw RuntimeException("ERROR 102: Exponents not equal.")
}
return Term(coefficient + term.coefficient, exponent)
}
fun negate(): Term {
return Term(-coefficient, exponent)
}
fun degree(): Long {
return exponent
}
override fun toString(): String {
if (coefficient == 0L) {
return "0"
}
if (exponent == 0L) {
return "" + coefficient
}
if (coefficient == 1L) {
return if (exponent == 1L) {
"x"
} else {
"x^$exponent"
}
}
return if (exponent == 1L) {
coefficient.toString() + "x"
} else coefficient.toString() + "x^" + exponent
}
}</lang>
- Output:
Task 1: cyclotomic polynomials for n <= 30: CP[1] = x - 1 CP[2] = x + 1 CP[3] = x^2 + x + 1 CP[4] = x^2 + 1 CP[5] = x^4 + x^3 + x^2 + x + 1 CP[6] = x^2 - x + 1 CP[7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[8] = x^4 + 1 CP[9] = x^6 + x^3 + 1 CP[10] = x^4 - x^3 + x^2 - x + 1 CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[12] = x^4 - x^2 + 1 CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 CP[16] = x^8 + 1 CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[18] = x^6 - x^3 + 1 CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[20] = x^8 - x^6 + x^4 - x^2 + 1 CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[24] = x^8 - x^4 + 1 CP[25] = x^20 + x^15 + x^10 + x^5 + 1 CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[27] = x^18 + x^9 + 1 CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1 CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient: CP[1] has coefficient with magnitude = 1 CP[105] has coefficient with magnitude = 2 CP[385] has coefficient with magnitude = 3 CP[1365] has coefficient with magnitude = 4 CP[1785] has coefficient with magnitude = 5 CP[2805] has coefficient with magnitude = 6 CP[3135] has coefficient with magnitude = 7 CP[6545] has coefficient with magnitude = 8 CP[6545] has coefficient with magnitude = 9 CP[10465] has coefficient with magnitude = 10
Maple
<lang maple>with(NumberTheory): for n to 30 do lprint(Phi(n,x)) od:
x-1 x+1 x^2+x+1 x^2+1 x^4+x^3+x^2+x+1 x^2-x+1 x^6+x^5+x^4+x^3+x^2+x+1 x^4+1 x^6+x^3+1 x^4-x^3+x^2-x+1 x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 x^4-x^2+1 x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 x^6-x^5+x^4-x^3+x^2-x+1 x^8-x^7+x^5-x^4+x^3-x+1 x^8+1 x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 x^6-x^3+1 x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 x^8-x^6+x^4-x^2+1 x^12-x^11+x^9-x^8+x^6-x^4+x^3-x+1 x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1 x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 x^8-x^4+1 x^20+x^15+x^10+x^5+1 x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1 x^18+x^9+1 x^12-x^10+x^8-x^6+x^4-x^2+1 x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 x^8+x^7-x^5-x^4-x^3+x+1
PhiSet:=[seq(map(abs,{coeffs(Phi(k,x),x)}),k=1..15000)]: [seq(ListTools:-SelectFirst(s->member(n,s),PhiSet,output=indices),n=1..20)];
- [1, 105, 385, 1365, 1785, 2805, 3135, 6545, 6545, 10465, 10465,
- 10465, 10465, 10465, 11305, 11305, 11305, 11305, 11305, 11305]</lang>
Mathematica / Wolfram Language
<lang Mathematica>Cyclotomic[#, x] & /@ Range[30] // Column i = 1; n = 10; PrintTemporary[Dynamic[{magnitudes, i}]]; magnitudes = ConstantArray[True, n]; While[Or @@ magnitudes,
coeff = Abs[CoefficientList[Cyclotomic[i, x], x]];
coeff = Select[coeff, Between[{1, n}]];
coeff = DeleteDuplicates[coeff];
If[Or @@ magnitudescoeff,
Do[
If[magnitudesc == True,
Print["CyclotomicPolynomial(", i,
") has coefficient with magnitude ", c]
]
,
{c, coeff}
];
magnitudescoeff = False;
];
i++;
]</lang>
- Output:
-1+x 1+x 1+x+x^2 1+x^2 1+x+x^2+x^3+x^4 1-x+x^2 1+x+x^2+x^3+x^4+x^5+x^6 1+x^4 1+x^3+x^6 1-x+x^2-x^3+x^4 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10 1-x^2+x^4 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12 1-x+x^2-x^3+x^4-x^5+x^6 1-x+x^3-x^4+x^5-x^7+x^8 1+x^8 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16 1-x^3+x^6 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18 1-x^2+x^4-x^6+x^8 1-x+x^3-x^4+x^6-x^8+x^9-x^11+x^12 1-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8-x^9+x^10 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18+x^19+x^20+x^21+x^22 1-x^4+x^8 1+x^5+x^10+x^15+x^20 1-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8-x^9+x^10-x^11+x^12 1+x^9+x^18 1-x^2+x^4-x^6+x^8-x^10+x^12 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18+x^19+x^20+x^21+x^22+x^23+x^24+x^25+x^26+x^27+x^28 1+x-x^3-x^4-x^5+x^7+x^8 CyclotomicPolynomial(1) has coefficient with magnitude 1 CyclotomicPolynomial(105) has coefficient with magnitude 2 CyclotomicPolynomial(385) has coefficient with magnitude 3 CyclotomicPolynomial(1365) has coefficient with magnitude 4 CyclotomicPolynomial(1785) has coefficient with magnitude 5 CyclotomicPolynomial(2805) has coefficient with magnitude 6 CyclotomicPolynomial(3135) has coefficient with magnitude 7 CyclotomicPolynomial(6545) has coefficient with magnitude 8 CyclotomicPolynomial(6545) has coefficient with magnitude 9 CyclotomicPolynomial(10465) has coefficient with magnitude 10
Nim
We use Java algorithm with ideas from C#, D, Go and Kotlin. We have kept only algorithm number 2 as other algorithms are much less efficient. We have also done some Nim specific improvements in order to get better performances.
<lang Nim>import algorithm, math, sequtils, strformat, tables
type
Term = tuple[coeff: int; exp: Natural] Polynomial = seq[Term]
# Table used to represent the list of factors of a number. # If, for a number "n", "k" is present in the table "f" of its factors, # "f[k]" contains the exponent of "k" in the prime factor decomposition. Factors = Table[int, int]
- Miscellaneous.
- Parity tests.
template isOdd(n: int): bool = (n and 1) != 0 template isEven(n: int): bool = (n and 1) == 0
- ---------------------------------------------------------------------------------------------------
proc sort(poly: var Polynomial) {.inline.} =
## Sort procedure for the terms of a polynomial (high degree first). algorithm.sort(poly, proc(x, y: Term): int = cmp(x.exp, y.exp), Descending)
- Superscripts.
const Superscripts: array['0'..'9', string] = ["⁰", "¹", "²", "³", "⁴", "⁵", "⁶", "⁷", "⁸", "⁹"]
func superscript(n: Natural): string =
## Return the Unicode string to use to represent an exponent. if n == 1: return "" for d in $n: result.add(Superscripts[d])
- Term operations.
func term(coeff, exp: int): Term =
## Create a term. if exp < 0: raise newException(ValueError, "term exponent cannot be negative") (coeff, Natural exp)
- ---------------------------------------------------------------------------------------------------
func `*`(a, b: Term): Term =
## Multiply two terms. (a.coeff * b.coeff, Natural a.exp + b.exp)
- ---------------------------------------------------------------------------------------------------
func `+`(a, b: Term): Term =
## Add two terms.
if a.exp != b.exp: raise newException(ValueError, "addition of terms with unequal exponents") (a.coeff + b.coeff, a.exp)
- ---------------------------------------------------------------------------------------------------
func `-`(a: Term): Term =
## Return the opposite of a term. (-a.coeff, a.exp)
- ---------------------------------------------------------------------------------------------------
func `$`(a: Term): string =
## Return the string representation of a term. if a.coeff == 0: "0" elif a.exp == 0: $a.coeff elif a.coeff == 1: 'x' & superscript(a.exp) elif a.coeff == -1: "-x" & superscript(a.exp) else: $a.coeff & 'x' & superscript(a.exp)
- Polynomial.
func polynomial(terms: varargs[Term]): Polynomial =
## Create a polynomial described by its terms.
for t in terms:
if t.coeff != 0:
result.add(t)
if result.len == 0:
return @[term(0, 0)]
sort(result)
- ---------------------------------------------------------------------------------------------------
func hasCoeffAbs(poly: Polynomial; coeff: int): bool =
## Return true if the polynomial contains a given coefficient.
for t in poly:
if abs(t.coeff) == coeff:
return true
- ---------------------------------------------------------------------------------------------------
func leadingCoeff(poly: Polynomial): int {.inline.} =
## Return the coefficient of the term with the highest degree. poly[0].coeff
- ---------------------------------------------------------------------------------------------------
func degree(poly: Polynomial): int {.inline.} =
## Return the degree of the polynomial. if poly.len == 0: -1 else: poly[0].exp
- ---------------------------------------------------------------------------------------------------
func `+`(poly: Polynomial; someTerm: Term): Polynomial =
## Add a term to a polynomial.
var added = false
for currTerm in poly:
if currterm.exp == someTerm.exp:
added = true
if currTerm.coeff + someTerm.coeff != 0:
result.add(currTerm + someTerm)
else:
result.add(currTerm)
if not added: result.add(someTerm)
- ---------------------------------------------------------------------------------------------------
func `+`(a, b: Polynomial): Polynomial =
## Add two polynomials.
var aIndex = a.high var bIndex = b.high
while aIndex >= 0 or bIndex >= 0:
if aIndex < 0:
result &= b[bIndex]
dec bIndex
elif bIndex < 0:
result &= a[aIndex]
dec aIndex
else:
let t1 = a[aIndex]
let t2 = b[bIndex]
if t1.exp == t2.exp:
let t3 = t1 + t2
if t3.coeff != 0:
result.add(t3)
dec aIndex
dec bIndex
elif t1.exp < t2.exp:
result.add(t1)
dec aIndex
else:
result.add(t2)
dec bIndex
sort(result)
- ---------------------------------------------------------------------------------------------------
func `*`(poly: Polynomial; someTerm: Term): Polynomial =
## Multiply a polynomial by a term. for currTerm in poly: result.add(currTerm * someTerm)
- ---------------------------------------------------------------------------------------------------
func `/`(a, b: Polynomial): Polynomial =
## Divide a polynomial by another polynomial.
var a = a let lcb = b.leadingCoeff let db = b.degree while a.degree >= b.degree: let lca = a.leadingCoeff let s = lca div lcb let t = term(s, a.degree - db) result = result + t a = a + b * -t
- ---------------------------------------------------------------------------------------------------
func `$`(poly: Polynomial): string =
## Return the string representation of a polynomial.
for t in poly:
if result.len == 0:
result.add($t)
else:
if t.coeff > 0:
result.add('+')
result.add($t)
else:
result.add('-')
result.add($(-t))
- Cyclotomic polynomial.
var
# Cache of list of factors.
factorCache: Table[int, Factors] = {2: {2: 1}.toTable}.toTable
# Cache of cyclotomic polynomials. Initialized with 1 -> x - 1.
polyCache: Table[int, Polynomial] = {1: polynomial(term(1, 1), term(-1, 0))}.toTable
- ---------------------------------------------------------------------------------------------------
proc getFactors(n: int): Factors =
## Return the list of factors of a number.
if n in factorCache: return factorCache[n]
if n.isEven: result = getFactors(n div 2) result[2] = result.getOrDefault(2) + 1 factorCache[n] = result return
var i = 3
while i * i <= n:
if n mod i == 0:
result = getFactors( n div i)
result[i] = result.getOrDefault(i) + 1
factorCache[n] = result
return
inc i, 2
result[n] = 1 factorCache[n] = result
- ---------------------------------------------------------------------------------------------------
proc cycloPoly(n: int): Polynomial =
## Find the nth cyclotomic polynomial.
if n in polyCache: return polyCache[n]
let factors = getFactors(n)
if n in factors:
# n is prime.
for i in countdown(n - 1, 0): # Add the terms by decreasing degrees.
result.add(term(1, i))
elif factors.len == 2 and factors.getOrDefault(2) == 1 and factors.getOrDefault(n div 2) == 1:
# n = 2 x prime.
let prime = n div 2
var coeff = -1
for i in countdown(prime - 1, 0): # Add the terms by decreasing degrees.
coeff *= -1
result.add(term(coeff, i))
elif factors.len == 1 and 2 in factors: # n = 2 ^ h. let h = factors[2] result.add([term(1, 1 shl (h - 1)), term(1, 0)])
elif factors.len == 1 and n notin factors:
# n = prime ^ k.
var p, k = 0
for prime, v in factors.pairs:
if prime > p:
p = prime
k = v
for i in countdown(p - 1, 0): # Add the terms by decreasing degrees.
result.add(term(1, i * p^(k-1)))
elif factors.len == 2 and 2 in factors:
# n = 2 ^ h x prime ^ k.
var p, k = 0
for prime, v in factors.pairs:
if prime != 2 and prime > p:
p = prime
k = v
var coeff = -1
let twoExp = 1 shl (factors[2] - 1)
for i in countdown(p - 1, 0): # Add the terms by decreasing degrees.
coeff *= -1
result.add(term(coeff, i * twoExp * p^(k-1)))
elif 2 in factors and isOdd(n div 2) and n div 2 > 1:
# CP(2m)[x] = CP(-m)[x], n odd integer > 1.
let cycloDiv2 = cycloPoly(n div 2)
for t in cycloDiv2:
result.add(if t.exp.isEven: t else: -t)
else:
# Let p, q be primes such that p does not divide n, and q divides n.
# Then CP(np)[x] = CP(n)[x^p] / CP(n)[x].
var m = 1
var cyclo = cycloPoly(m)
let primes = sorted(toSeq(factors.keys))
for prime in primes:
# Compute CP(m)[x^p].
var terms: Polynomial
for t in cyclo:
terms.add(term(t.coeff, t.exp * prime))
cyclo = terms / cyclo
m *= prime
# Now, m is the largest square free divisor of n.
let s = n div m
# Compute CP(n)[x] = CP(m)[x^s].
for t in cyclo:
result.add(term(t.coeff, t.exp * s))
polyCache[n] = result
- ———————————————————————————————————————————————————————————————————————————————————————————————————
echo "Cyclotomic polynomials for n ⩽ 30:" for i in 1..30:
echo &"Φ{'(' & $i & ')':4} = {cycloPoly(i)}"
echo "" echo "Smallest cyclotomic polynomial with n or -n as a coefficient:" var n = 0 for i in 1..10:
while true:
inc n
if cycloPoly(n).hasCoeffAbs(i):
echo &"Φ{'(' & $n & ')':7} has coefficient with magnitude = {i}"
dec n
break</lang>
- Output:
The program runs in 41 seconds on our reasonably performing laptop.
Cyclotomic polynomials for n ⩽ 30: Φ(1) = x-1 Φ(2) = x+1 Φ(3) = x²+x+1 Φ(4) = x²+1 Φ(5) = x⁴+x³+x²+x+1 Φ(6) = x²-x+1 Φ(7) = x⁶+x⁵+x⁴+x³+x²+x+1 Φ(8) = x⁴+1 Φ(9) = x⁶+x³+1 Φ(10) = x⁴-x³+x²-x+1 Φ(11) = x¹⁰+x⁹+x⁸+x⁷+x⁶+x⁵+x⁴+x³+x²+x+1 Φ(12) = x⁴-x²+1 Φ(13) = x¹²+x¹¹+x¹⁰+x⁹+x⁸+x⁷+x⁶+x⁵+x⁴+x³+x²+x+1 Φ(14) = x⁶-x⁵+x⁴-x³+x²-x+1 Φ(15) = x⁸-x⁷+x⁵-x⁴+x³-x+1 Φ(16) = x⁸+1 Φ(17) = x¹⁶+x¹⁵+x¹⁴+x¹³+x¹²+x¹¹+x¹⁰+x⁹+x⁸+x⁷+x⁶+x⁵+x⁴+x³+x²+x+1 Φ(18) = x⁶-x³+1 Φ(19) = x¹⁸+x¹⁷+x¹⁶+x¹⁵+x¹⁴+x¹³+x¹²+x¹¹+x¹⁰+x⁹+x⁸+x⁷+x⁶+x⁵+x⁴+x³+x²+x+1 Φ(20) = x⁸-x⁶+x⁴-x²+1 Φ(21) = x¹²-x¹¹+x⁹-x⁸+x⁶-x⁴+x³-x+1 Φ(22) = x¹⁰-x⁹+x⁸-x⁷+x⁶-x⁵+x⁴-x³+x²-x+1 Φ(23) = x²²+x²¹+x²⁰+x¹⁹+x¹⁸+x¹⁷+x¹⁶+x¹⁵+x¹⁴+x¹³+x¹²+x¹¹+x¹⁰+x⁹+x⁸+x⁷+x⁶+x⁵+x⁴+x³+x²+x+1 Φ(24) = x⁸-x⁴+1 Φ(25) = x²⁰+x¹⁵+x¹⁰+x⁵+1 Φ(26) = x¹²-x¹¹+x¹⁰-x⁹+x⁸-x⁷+x⁶-x⁵+x⁴-x³+x²-x+1 Φ(27) = x¹⁸+x⁹+1 Φ(28) = x¹²-x¹⁰+x⁸-x⁶+x⁴-x²+1 Φ(29) = x²⁸+x²⁷+x²⁶+x²⁵+x²⁴+x²³+x²²+x²¹+x²⁰+x¹⁹+x¹⁸+x¹⁷+x¹⁶+x¹⁵+x¹⁴+x¹³+x¹²+x¹¹+x¹⁰+x⁹+x⁸+x⁷+x⁶+x⁵+x⁴+x³+x²+x+1 Φ(30) = x⁸+x⁷-x⁵-x⁴-x³+x+1 Smallest cyclotomic polynomial with n or -n as a coefficient: Φ(1) has coefficient with magnitude = 1 Φ(105) has coefficient with magnitude = 2 Φ(385) has coefficient with magnitude = 3 Φ(1365) has coefficient with magnitude = 4 Φ(1785) has coefficient with magnitude = 5 Φ(2805) has coefficient with magnitude = 6 Φ(3135) has coefficient with magnitude = 7 Φ(6545) has coefficient with magnitude = 8 Φ(6545) has coefficient with magnitude = 9 Φ(10465) has coefficient with magnitude = 10
PARI/GP
Cyclotomic polynomials are a built-in function. <lang parigp> for(n=1,30,print(n," : ",polcyclo(n)))
contains_coeff(n, d) = p=polcyclo(n);for(k=0,poldegree(p),if(abs(polcoef(p,k))==d,return(1)));return(0)
for(d=1,10,i=1; while(contains_coeff(i,d)==0,i=i+1);print(d," : ",i)) </lang>
- Output:
1 : x - 1 2 : x + 1 3 : x^2 + x + 1 4 : x^2 + 1 5 : x^4 + x^3 + x^2 + x + 1 6 : x^2 - x + 1 7 : x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 8 : x^4 + 1 9 : x^6 + x^3 + 1 10 : x^4 - x^3 + x^2 - x + 1 11 : x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 12 : x^4 - x^2 + 1 13 : x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 14 : x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 15 : x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 16 : x^8 + 1 17 : x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 18 : x^6 - x^3 + 1 19 : x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 20 : x^8 - x^6 + x^4 - x^2 + 1 21 : x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 22 : x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 23 : x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 24 : x^8 - x^4 + 1 25 : x^20 + x^15 + x^10 + x^5 + 1 26 : x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 27 : x^18 + x^9 + 1 28 : x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1 29 : x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 30 : x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 1 : 1 2 : 105 3 : 385 4 : 1365 5 : 1785 6 : 2805 7 : 3135 8 : 6545 9 : 6545 10 : 10465
Perl
Conveniently, the module Math::Polynomial::Cyclotomic exists to do all the work. An exponent too large error prevents reaching the 10th step of the 2nd part of the task.
<lang perl>use feature 'say';
use List::Util qw(first);
use Math::Polynomial::Cyclotomic qw(cyclo_poly_iterate);
say 'First 30 cyclotomic polynomials:'; my $it = cyclo_poly_iterate(1); say "$_: " . $it->() for 1 .. 30;
say "\nSmallest cyclotomic polynomial with n or -n as a coefficient:"; $it = cyclo_poly_iterate(1);
for (my ($n, $k) = (1, 1) ; $n <= 10 ; ++$k) {
my $poly = $it->();
while (my $c = first { abs($_) == $n } $poly->coeff) {
say "CP $k has coefficient with magnitude = $n";
$n++;
}
}</lang>
- Output:
First 30 cyclotomic polynomials: 1: (x - 1) 2: (x + 1) 3: (x^2 + x + 1) 4: (x^2 + 1) 5: (x^4 + x^3 + x^2 + x + 1) 6: (x^2 - x + 1) 7: (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) 8: (x^4 + 1) 9: (x^6 + x^3 + 1) 10: (x^4 - x^3 + x^2 - x + 1) 11: (x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) 12: (x^4 - x^2 + 1) 13: (x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) 14: (x^6 - x^5 + x^4 - x^3 + x^2 - x + 1) 15: (x^8 - x^7 + x^5 - x^4 + x^3 - x + 1) 16: (x^8 + 1) 17: (x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) 18: (x^6 - x^3 + 1) 19: (x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) 20: (x^8 - x^6 + x^4 - x^2 + 1) 21: (x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1) 22: (x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1) 23: (x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) 24: (x^8 - x^4 + 1) 25: (x^20 + x^15 + x^10 + x^5 + 1) 26: (x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1) 27: (x^18 + x^9 + 1) 28: (x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1) 29: (x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) 30: (x^8 + x^7 - x^5 - x^4 - x^3 + x + 1) Smallest cyclotomic polynomial with n or -n as a coefficient: CP 1 has coefficient with magnitude = 1 CP 105 has coefficient with magnitude = 2 CP 385 has coefficient with magnitude = 3 CP 1365 has coefficient with magnitude = 4 CP 1785 has coefficient with magnitude = 5 CP 2805 has coefficient with magnitude = 6 CP 3135 has coefficient with magnitude = 7 CP 6545 has coefficient with magnitude = 8 CP 6545 has coefficient with magnitude = 9
Phix
Uses several routines from Polynomial_long_division#Phix, tweaked slightly to check remainder is zero and trim the quotient.
-- demo\rosetta\Cyclotomic_Polynomial.exw with javascript_semantics function degree(sequence p) for i=length(p) to 1 by -1 do if p[i]!=0 then return i end if end for return -1 end function function poly_div(sequence n, d) while length(d)<length(n) do d &=0 end while integer dn = degree(n), dd = degree(d) if dd<0 then throw("divide by zero") end if sequence quot = repeat(0,dn) while dn>=dd do integer k = dn-dd integer qk = n[dn]/d[dd] quot[k+1] = qk sequence d2 = d[1..length(d)-k] for i=1 to length(d2) do integer mi = -i n[mi] -= d2[mi]*qk end for dn = degree(n) end while -- return {quot,n} -- (n is now the remainder) if n!=repeat(0,length(n)) then ?9/0 end if while quot[$]=0 do quot = quot[1..$-1] end while return quot end function function poly(sequence si) -- display helper string r = "" for t=length(si) to 1 by -1 do integer sit = si[t] if sit!=0 then if sit=1 and t>1 then r &= iff(r=""? "":" + ") elsif sit=-1 and t>1 then r &= iff(r=""?"-":" - ") else if r!="" then r &= iff(sit<0?" - ":" + ") sit = abs(sit) end if r &= sprintf("%d",sit) end if r &= iff(t>1?"x"&iff(t>2?sprintf("^%d",t-1):""):"") end if end for if r="" then r="0" end if return r end function --</Polynomial_long_division.exw> --# memoize cache for recursive calls constant cyclotomics = new_dict({{1,{-1,1}},{2,{1,1}}}) function cyclotomic(integer n) -- -- Calculate the nth cyclotomic polynomial. -- See wikipedia article at bottom of section /wiki/Cyclotomic_polynomial#Fundamental_tools -- The algorithm is reliable but slow for large n > 1000. -- sequence c if getd_index(n,cyclotomics)!=NULL then c = getd(n,cyclotomics) else if is_prime(n) then c = repeat(1,n) else -- recursive formula seen in wikipedia article c = -1&repeat(0,n-1)&1 sequence f = factors(n,-1) for i=1 to length(f) do c = poly_div(c,deep_copy(cyclotomic(f[i]))) end for end if setd(n,c,cyclotomics) end if return c end function for i=1 to 30 do sequence z = cyclotomic(i) string s = poly(z) printf(1,"cp(%2d) = %s\n",{i,s}) if i>1 and z!=reverse(z) then ?9/0 end if -- sanity check end for integer found = 0, n = 1, cheat = 0 sequence fn = repeat(false,10), nxt = {105,385,1365,1785,2805,3135,6545,6545,10465,10465} atom t1 = time()+1 puts(1,"\n") while found<iff(platform()=JS?5:10) do sequence z = cyclotomic(n) for i=1 to length(z) do atom azi = abs(z[i]) if azi>=1 and azi<=10 and fn[azi]=0 then printf(1,"cp(%d) has a coefficient with magnitude %d\n",{n,azi}) cheat = azi -- (comment this out to prevent cheating!) found += 1 fn[azi] = true t1 = time()+1 end if end for if cheat then {n,cheat} = {nxt[cheat],0} else n += iff(n=1?4:10) end if if time()>t1 and platform()!=JS then printf(1,"working (%d) ...\r",n) t1 = time()+1 end if end while
- Output:
If you disable the cheating, and if in a particularly self harming mood replace it with n+=1, you will get exactly the same output, eventually.
(The distributed version contains simple instrumentation showing cp(1260) executes the line in the heart of poly_div() that subtracts a multiple of qk over 15 million times.)
cp( 1) = x - 1 cp( 2) = x + 1 cp( 3) = x^2 + x + 1 cp( 4) = x^2 + 1 cp( 5) = x^4 + x^3 + x^2 + x + 1 cp( 6) = x^2 - x + 1 cp( 7) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 cp( 8) = x^4 + 1 cp( 9) = x^6 + x^3 + 1 cp(10) = x^4 - x^3 + x^2 - x + 1 cp(11) = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 cp(12) = x^4 - x^2 + 1 cp(13) = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 cp(14) = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 cp(15) = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 cp(16) = x^8 + 1 cp(17) = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 cp(18) = x^6 - x^3 + 1 cp(19) = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 cp(20) = x^8 - x^6 + x^4 - x^2 + 1 cp(21) = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 cp(22) = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 cp(23) = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 cp(24) = x^8 - x^4 + 1 cp(25) = x^20 + x^15 + x^10 + x^5 + 1 cp(26) = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 cp(27) = x^18 + x^9 + 1 cp(28) = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1 cp(29) = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 cp(30) = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 cp(1) has a coefficient with magnitude 1 cp(105) has a coefficient with magnitude 2 cp(385) has a coefficient with magnitude 3 cp(1365) has a coefficient with magnitude 4 cp(1785) has a coefficient with magnitude 5 cp(2805) has a coefficient with magnitude 6 cp(3135) has a coefficient with magnitude 7 cp(6545) has a coefficient with magnitude 8 cp(6545) has a coefficient with magnitude 9 cp(10465) has a coefficient with magnitude 10
Python
<lang python>from itertools import count, chain from collections import deque
def primes(_cache=[2, 3]):
yield from _cache
for n in count(_cache[-1]+2, 2):
if isprime(n):
_cache.append(n)
yield n
def isprime(n):
for p in primes():
if n%p == 0:
return False
if p*p > n:
return True
def factors(n):
for p in primes():
# prime factoring is such a non-issue for small numbers that, for
# this example, we might even just say
# for p in count(2):
if p*p > n:
if n > 1:
yield(n, 1, 1)
break
if n%p == 0:
cnt = 0
while True:
n, cnt = n//p, cnt+1
if n%p != 0: break
yield p, cnt, n
- ^^ not the most sophisticated prime number routines, because no need
- Returns (list1, list2) representing the division between
- two polinomials. A list p of integers means the product
- (x^p[0] - 1) * (x^p[1] - 1) * ...
def cyclotomic(n):
def poly_div(num, den):
return (num[0] + den[1], num[1] + den[0])
def elevate(poly, n): # replace poly p(x) with p(x**n)
powerup = lambda p, n: [a*n for a in p]
return poly if n == 1 else (powerup(poly[0], n), powerup(poly[1], n))
if n == 0:
return ([], [])
if n == 1:
return ([1], [])
p, m, r = next(factors(n)) poly = cyclotomic(r) return elevate(poly_div(elevate(poly, p), poly), p**(m-1))
def to_text(poly):
def getx(c, e):
if e == 0:
return '1'
elif e == 1:
return 'x'
return 'x' + (.join('⁰¹²³⁴⁵⁶⁷⁸⁹'[i] for i in map(int, str(e))))
parts = []
for (c,e) in (poly):
if c < 0:
coef = ' - ' if c == -1 else f' - {-c} '
else:
coef = (parts and ' + ' or ) if c == 1 else f' + {c}'
parts.append(coef + getx(c,e))
return .join(parts)
def terms(poly):
# convert above representation of division to (coef, power) pairs
def merge(a, b):
# a, b should be deques. They may change during the course.
while a or b:
l = a[0] if a else (0, -1) # sentinel value
r = b[0] if b else (0, -1)
if l[1] > r[1]:
a.popleft()
elif l[1] < r[1]:
b.popleft()
l = r
else:
a.popleft()
b.popleft()
l = (l[0] + r[0], l[1])
yield l
def mul(poly, p): # p means polynomial x^p - 1
poly = list(poly)
return merge(deque((c, e+p) for c,e in poly),
deque((-c, e) for c,e in poly))
def div(poly, p): # p means polynomial x^p - 1
q = deque()
for c,e in merge(deque(poly), q):
if c:
q.append((c, e - p))
yield (c, e - p)
if e == p: break
p = [(1, 0)] # 1*x^0, i.e. 1
for x in poly[0]: # numerator
p = mul(p, x)
for x in sorted(poly[1], reverse=True): # denominator
p = div(p, x)
return p
for n in chain(range(11), [2]):
print(f'{n}: {to_text(terms(cyclotomic(n)))}')
want = 1 for n in count():
c = [c for c,_ in terms(cyclotomic(n))]
while want in c or -want in c:
print(f'C[{want}]: {n}')
want += 1</lang>
- Output:
Only showing first 10 polynomials to avoid clutter.
0: 1 1: x - 1 2: x + 1 3: x² + x + 1 4: x² + 1 5: x⁴ + x³ + x² + x + 1 6: x² - x + 1 7: x⁶ + x⁵ + x⁴ + x³ + x² + x + 1 8: x⁴ + 1 9: x⁶ + x³ + 1 10: x⁴ - x³ + x² - x + 1 105: x⁴⁸ + x⁴⁷ + x⁴⁶ - x⁴³ - x⁴² - 2 x⁴¹ - x⁴⁰ - x³⁹ + x³⁶ + x³⁵ + x³⁴ + x³³ + x³² + x³¹ - x²⁸ - x²⁶ - x²⁴ - x²² - x²⁰ + x¹⁷ + x¹⁶ + x¹⁵ + x¹⁴ + x¹³ + x¹² - x⁹ - x⁸ - 2 x⁷ - x⁶ - x⁵ + x² + x + 1 C[1]: 0 C[2]: 105 C[3]: 385 C[4]: 1365 C[5]: 1785 C[6]: 2805 C[7]: 3135 C[8]: 6545 C[9]: 6545 C[10]: 10465 C[11]: 10465 C[12]: 10465 C[13]: 10465 C[14]: 10465 C[15]: 11305 C[16]: 11305 C[17]: 11305 C[18]: 11305 C[19]: 11305 C[20]: 11305 C[21]: 11305 C[22]: 15015 C[23]: 15015
Raku
(formerly Perl 6)
Uses the same library as Perl, so comes with the same caveats. <lang perl6>use Math::Polynomial::Cyclotomic:from<Perl5> <cyclo_poly_iterate cyclo_poly>;
say 'First 30 cyclotomic polynomials:'; my $iterator = cyclo_poly_iterate(1); say "Φ($_) = " ~ super $iterator().Str for 1..30;
say "\nSmallest cyclotomic polynomial with |n| as a coefficient:"; say "Φ(1) has a coefficient magnitude: 1";
my $index = 0; for 2..9 -> $coefficient {
loop {
$index += 5;
my \Φ = cyclo_poly($index);
next unless Φ ~~ / $coefficient\* /;
say "Φ($index) has a coefficient magnitude: $coefficient";
$index -= 5;
last;
}
}
sub super ($str) {
$str.subst( / '^' (\d+) /, { $0.trans([<0123456789>.comb] => [<⁰¹²³⁴⁵⁶⁷⁸⁹>.comb]) }, :g)
}</lang>
First 30 cyclotomic polynomials: Φ(1) = (x - 1) Φ(2) = (x + 1) Φ(3) = (x² + x + 1) Φ(4) = (x² + 1) Φ(5) = (x⁴ + x³ + x² + x + 1) Φ(6) = (x² - x + 1) Φ(7) = (x⁶ + x⁵ + x⁴ + x³ + x² + x + 1) Φ(8) = (x⁴ + 1) Φ(9) = (x⁶ + x³ + 1) Φ(10) = (x⁴ - x³ + x² - x + 1) Φ(11) = (x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1) Φ(12) = (x⁴ - x² + 1) Φ(13) = (x¹² + x¹¹ + x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1) Φ(14) = (x⁶ - x⁵ + x⁴ - x³ + x² - x + 1) Φ(15) = (x⁸ - x⁷ + x⁵ - x⁴ + x³ - x + 1) Φ(16) = (x⁸ + 1) Φ(17) = (x¹⁶ + x¹⁵ + x¹⁴ + x¹³ + x¹² + x¹¹ + x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1) Φ(18) = (x⁶ - x³ + 1) Φ(19) = (x¹⁸ + x¹⁷ + x¹⁶ + x¹⁵ + x¹⁴ + x¹³ + x¹² + x¹¹ + x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1) Φ(20) = (x⁸ - x⁶ + x⁴ - x² + 1) Φ(21) = (x¹² - x¹¹ + x⁹ - x⁸ + x⁶ - x⁴ + x³ - x + 1) Φ(22) = (x¹⁰ - x⁹ + x⁸ - x⁷ + x⁶ - x⁵ + x⁴ - x³ + x² - x + 1) Φ(23) = (x²² + x²¹ + x²⁰ + x¹⁹ + x¹⁸ + x¹⁷ + x¹⁶ + x¹⁵ + x¹⁴ + x¹³ + x¹² + x¹¹ + x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1) Φ(24) = (x⁸ - x⁴ + 1) Φ(25) = (x²⁰ + x¹⁵ + x¹⁰ + x⁵ + 1) Φ(26) = (x¹² - x¹¹ + x¹⁰ - x⁹ + x⁸ - x⁷ + x⁶ - x⁵ + x⁴ - x³ + x² - x + 1) Φ(27) = (x¹⁸ + x⁹ + 1) Φ(28) = (x¹² - x¹⁰ + x⁸ - x⁶ + x⁴ - x² + 1) Φ(29) = (x²⁸ + x²⁷ + x²⁶ + x²⁵ + x²⁴ + x²³ + x²² + x²¹ + x²⁰ + x¹⁹ + x¹⁸ + x¹⁷ + x¹⁶ + x¹⁵ + x¹⁴ + x¹³ + x¹² + x¹¹ + x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1) Φ(30) = (x⁸ + x⁷ - x⁵ - x⁴ - x³ + x + 1) Smallest cyclotomic polynomial with |n| as a coefficient: Φ(1) has a coefficient magnitude: 1 Φ(105) has a coefficient magnitude: 2 Φ(385) has a coefficient magnitude: 3 Φ(1365) has a coefficient magnitude: 4 Φ(1785) has a coefficient magnitude: 5 Φ(2805) has a coefficient magnitude: 6 Φ(3135) has a coefficient magnitude: 7 Φ(6545) has a coefficient magnitude: 8 Φ(6545) has a coefficient magnitude: 9
Sidef
Solution based on polynomial interpolation (slow). <lang ruby>var Poly = require('Math::Polynomial') Poly.string_config(Hash(fold_sign => true, prefix => "", suffix => ""))
func poly_interpolation(v) {
v.len.of {|n| v.len.of {|k| n**k } }.msolve(v)
}
say "First 30 cyclotomic polynomials:" for k in (1..30) {
var a = (k+1).of { cyclotomic(k, _) }
var Φ = poly_interpolation(a)
say ("Φ(#{k}) = ", Poly.new(Φ...))
}
say "\nSmallest cyclotomic polynomial with n or -n as a coefficient:" for n in (1..10) { # very slow
var k = (1..Inf -> first {|k|
poly_interpolation((k+1).of { cyclotomic(k, _) }).first { .abs == n }
})
say "Φ(#{k}) has coefficient with magnitude #{n}"
}</lang>
Slightly faster solution, using the Math::Polynomial::Cyclotomic Perl module. <lang ruby>var Poly = require('Math::Polynomial')
require('Math::Polynomial::Cyclotomic')
Poly.string_config(Hash(fold_sign => true, prefix => "", suffix => ""))
say "First 30 cyclotomic polynomials:" for k in (1..30) {
say ("Φ(#{k}) = ", Poly.new.cyclotomic(k))
}
say "\nSmallest cyclotomic polynomial with n or -n as a coefficient:" for n in (1..10) {
var p = Poly.new
var k = (1..Inf -> first {|k|
[p.cyclotomic(k).coeff].first { .abs == n }
})
say "Φ(#{k}) has coefficient with magnitude = #{n}"
}</lang>
- Output:
First 30 cyclotomic polynomials: Φ(1) = x - 1 Φ(2) = x + 1 Φ(3) = x^2 + x + 1 Φ(4) = x^2 + 1 Φ(5) = x^4 + x^3 + x^2 + x + 1 Φ(6) = x^2 - x + 1 Φ(7) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 Φ(8) = x^4 + 1 Φ(9) = x^6 + x^3 + 1 Φ(10) = x^4 - x^3 + x^2 - x + 1 Φ(11) = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 Φ(12) = x^4 - x^2 + 1 Φ(13) = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 Φ(14) = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 Φ(15) = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 Φ(16) = x^8 + 1 Φ(17) = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 Φ(18) = x^6 - x^3 + 1 Φ(19) = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 Φ(20) = x^8 - x^6 + x^4 - x^2 + 1 Φ(21) = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 Φ(22) = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 Φ(23) = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 Φ(24) = x^8 - x^4 + 1 Φ(25) = x^20 + x^15 + x^10 + x^5 + 1 Φ(26) = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 Φ(27) = x^18 + x^9 + 1 Φ(28) = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1 Φ(29) = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 Φ(30) = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 Smallest cyclotomic polynomial with n or -n as a coefficient: Φ(1) has coefficient with magnitude = 1 Φ(105) has coefficient with magnitude = 2 Φ(385) has coefficient with magnitude = 3 Φ(1365) has coefficient with magnitude = 4 Φ(1785) has coefficient with magnitude = 5 Φ(2805) has coefficient with magnitude = 6 Φ(3135) has coefficient with magnitude = 7 ^C
Visual Basic .NET
<lang vbnet>Imports System.Text
Module Module1
Private ReadOnly MAX_ALL_FACTORS As Integer = 100_000
- Const ALGORITHM = 2
Class Term
Implements IComparable(Of Term)
Public ReadOnly Property Coefficient As Long
Public ReadOnly Property Exponent As Long
Public Sub New(c As Long, Optional e As Long = 0)
Coefficient = c
Exponent = e
End Sub
Public Shared Operator -(t As Term) As Term
Return New Term(-t.Coefficient, t.Exponent)
End Operator
Public Shared Operator +(lhs As Term, rhs As Term) As Term
If lhs.Exponent <> rhs.Exponent Then
Throw New ArgumentException("Exponents not equal")
End If
Return New Term(lhs.Coefficient + rhs.Coefficient, lhs.Exponent)
End Operator
Public Shared Operator *(lhs As Term, rhs As Term) As Term
Return New Term(lhs.Coefficient * rhs.Coefficient, lhs.Exponent + rhs.Exponent)
End Operator
Public Function CompareTo(other As Term) As Integer Implements IComparable(Of Term).CompareTo
Return -Exponent.CompareTo(other.Exponent)
End Function
Public Overrides Function ToString() As String
If Coefficient = 0 Then
Return "0"
End If
If Exponent = 0 Then
Return Coefficient.ToString
End If
If Coefficient = 1 Then
If Exponent = 1 Then
Return "x"
End If
Return String.Format("x^{0}", Exponent)
End If
If Coefficient = -1 Then
If Exponent = 1 Then
Return "-x"
End If
Return String.Format("-x^{0}", Exponent)
End If
If Exponent = 1 Then
Return String.Format("{0}x", Coefficient)
End If
Return String.Format("{0}x^{1}", Coefficient, Exponent)
End Function
End Class
Class Polynomial
Implements IEnumerable(Of Term)
Private ReadOnly polyTerms As New List(Of Term)
Public Sub New()
polyTerms.Add(New Term(0))
End Sub
Public Sub New(ParamArray values() As Term)
If values.Length = 0 Then
polyTerms.Add(New Term(0))
Else
polyTerms.AddRange(values)
End If
Normalize()
End Sub
Public Sub New(values As IEnumerable(Of Term))
polyTerms.AddRange(values)
If polyTerms.Count = 0 Then
polyTerms.Add(New Term(0))
End If
Normalize()
End Sub
Public Function LeadingCoeficient() As Long
Return polyTerms(0).Coefficient
End Function
Public Function Degree() As Long
Return polyTerms(0).Exponent
End Function
Public Function HasCoefficentAbs(coeff As Long) As Boolean
For Each t In polyTerms
If Math.Abs(t.Coefficient) = coeff Then
Return True
End If
Next
Return False
End Function
Public Function GetEnumerator() As IEnumerator(Of Term) Implements IEnumerable(Of Term).GetEnumerator
Return polyTerms.GetEnumerator
End Function
Private Function IEnumerable_GetEnumerator() As IEnumerator Implements IEnumerable.GetEnumerator
Return polyTerms.GetEnumerator
End Function
Private Sub Normalize()
polyTerms.Sort(Function(a As Term, b As Term) a.CompareTo(b))
End Sub
Public Shared Operator +(lhs As Polynomial, rhs As Term) As Polynomial
Dim terms As New List(Of Term)
Dim added = False
For Each ct In lhs
If ct.Exponent = rhs.Exponent Then
added = True
If ct.Coefficient + rhs.Coefficient <> 0 Then
terms.Add(ct + rhs)
End If
Else
terms.Add(ct)
End If
Next
If Not added Then
terms.Add(rhs)
End If
Return New Polynomial(terms)
End Operator
Public Shared Operator *(lhs As Polynomial, rhs As Term) As Polynomial
Dim terms As New List(Of Term)
For Each ct In lhs
terms.Add(ct * rhs)
Next
Return New Polynomial(terms)
End Operator
Public Shared Operator +(lhs As Polynomial, rhs As Polynomial) As Polynomial
Dim terms As New List(Of Term)
Dim thisCount = lhs.polyTerms.Count
Dim polyCount = rhs.polyTerms.Count
While thisCount > 0 OrElse polyCount > 0
If thisCount = 0 Then
Dim polyTerm = rhs.polyTerms(polyCount - 1)
terms.Add(polyTerm)
polyCount -= 1
ElseIf polyCount = 0 Then
Dim thisTerm = lhs.polyTerms(thisCount - 1)
terms.Add(thisTerm)
thisCount -= 1
Else
Dim polyTerm = rhs.polyTerms(polyCount - 1)
Dim thisTerm = lhs.polyTerms(thisCount - 1)
If thisTerm.Exponent = polyTerm.Exponent Then
Dim t = thisTerm + polyTerm
If t.Coefficient <> 0 Then
terms.Add(t)
End If
thisCount -= 1
polyCount -= 1
ElseIf thisTerm.Exponent < polyTerm.Exponent Then
terms.Add(thisTerm)
thisCount -= 1
Else
terms.Add(polyTerm)
polyCount -= 1
End If
End If
End While
Return New Polynomial(terms)
End Operator
Public Shared Operator *(lhs As Polynomial, rhs As Polynomial) As Polynomial
Throw New Exception("Not implemented")
End Operator
Public Shared Operator /(lhs As Polynomial, rhs As Polynomial) As Polynomial
Dim q As New Polynomial
Dim r = lhs
Dim lcv = rhs.LeadingCoeficient
Dim dv = rhs.Degree
While r.Degree >= rhs.Degree
Dim lcr = r.LeadingCoeficient
Dim s = lcr \ lcv
Dim t As New Term(s, r.Degree() - dv)
q += t
r += rhs * -t
End While
Return q
End Operator
Public Overrides Function ToString() As String
Dim builder As New StringBuilder
Dim it = polyTerms.GetEnumerator()
If it.MoveNext Then
builder.Append(it.Current)
End If
While it.MoveNext
If it.Current.Coefficient < 0 Then
builder.Append(" - ")
builder.Append(-it.Current)
Else
builder.Append(" + ")
builder.Append(it.Current)
End If
End While
Return builder.ToString
End Function
End Class
Function GetDivisors(number As Integer) As List(Of Integer)
Dim divisors As New List(Of Integer)
Dim root = CType(Math.Sqrt(number), Long)
For i = 1 To root
If number Mod i = 0 Then
divisors.Add(i)
Dim div = number \ i
If div <> i AndAlso div <> number Then
divisors.Add(div)
End If
End If
Next
Return divisors
End Function
Private ReadOnly allFactors As New Dictionary(Of Integer, Dictionary(Of Integer, Integer)) From {{2, New Dictionary(Of Integer, Integer) From Template:2, 1}}
Function GetFactors(number As Integer) As Dictionary(Of Integer, Integer)
If allFactors.ContainsKey(number) Then
Return allFactors(number)
End If
Dim factors As New Dictionary(Of Integer, Integer)
If number Mod 2 = 0 Then
Dim factorsDivTwo = GetFactors(number \ 2)
For Each pair In factorsDivTwo
If Not factors.ContainsKey(pair.Key) Then
factors.Add(pair.Key, pair.Value)
End If
Next
If factors.ContainsKey(2) Then
factors(2) += 1
Else
factors.Add(2, 1)
End If
If number < MAX_ALL_FACTORS Then
allFactors.Add(number, factors)
End If
Return factors
End If
Dim root = CType(Math.Sqrt(number), Long)
Dim i = 3L
While i <= root
If number Mod i = 0 Then
Dim factorsDivI = GetFactors(number \ i)
For Each pair In factorsDivI
If Not factors.ContainsKey(pair.Key) Then
factors.Add(pair.Key, pair.Value)
End If
Next
If factors.ContainsKey(i) Then
factors(i) += 1
Else
factors.Add(i, 1)
End If
If number < MAX_ALL_FACTORS Then
allFactors.Add(number, factors)
End If
Return factors
End If
i += 2
End While
factors.Add(number, 1)
If number < MAX_ALL_FACTORS Then
allFactors.Add(number, factors)
End If
Return factors
End Function
Private ReadOnly computedPolynomials As New Dictionary(Of Integer, Polynomial)
Function CyclotomicPolynomial(n As Integer) As Polynomial
If computedPolynomials.ContainsKey(n) Then
Return computedPolynomials(n)
End If
If n = 1 Then
REM polynomial: x - 1
Dim p As New Polynomial(New Term(1, 1), New Term(-1))
computedPolynomials.Add(n, p)
Return p
End If
Dim factors = GetFactors(n)
Dim terms As New List(Of Term)
Dim cyclo As Polynomial
If factors.ContainsKey(n) Then
REM n prime
For index = 1 To n
terms.Add(New Term(1, index - 1))
Next
cyclo = New Polynomial(terms)
computedPolynomials.Add(n, cyclo)
Return cyclo
ElseIf factors.Count = 2 AndAlso factors.ContainsKey(2) AndAlso factors(2) = 1 AndAlso factors.ContainsKey(n / 2) AndAlso factors(n / 2) = 1 Then
REM n = 2p
Dim prime = n \ 2
Dim coeff = -1
For index = 1 To prime
coeff *= -1
terms.Add(New Term(coeff, index - 1))
Next
cyclo = New Polynomial(terms)
computedPolynomials.Add(n, cyclo)
Return cyclo
ElseIf factors.Count = 1 AndAlso factors.ContainsKey(2) Then
REM n = 2^h
Dim h = factors(2)
terms = New List(Of Term) From {
New Term(1, Math.Pow(2, h - 1)),
New Term(1)
}
cyclo = New Polynomial(terms)
computedPolynomials.Add(n, cyclo)
Return cyclo
ElseIf factors.Count = 1 AndAlso factors.ContainsKey(n) Then
REM n = p^k
Dim p = 0
Dim k = 0
For Each it In factors
p = it.Key
k = it.Value
Next
For index = 1 To p
terms.Add(New Term(1, (index - 1) * Math.Pow(p, k - 1)))
Next
cyclo = New Polynomial(terms)
computedPolynomials.Add(n, cyclo)
Return cyclo
ElseIf factors.Count = 2 AndAlso factors.ContainsKey(2) Then
REM n = 2^h * p^k
Dim p = 0
For Each it In factors
If it.Key <> 2 Then
p = it.Key
End If
Next
Dim coeff = -1
Dim twoExp = CType(Math.Pow(2, factors(2) - 1), Long)
Dim k = factors(p)
For index = 1 To p
coeff *= -1
terms.Add(New Term(coeff, (index - 1) * twoExp * Math.Pow(p, k - 1)))
Next
cyclo = New Polynomial(terms)
computedPolynomials.Add(n, cyclo)
Return cyclo
ElseIf factors.ContainsKey(2) AndAlso (n / 2) Mod 2 = 1 AndAlso n / 2 > 1 Then
REM CP(2m)[x] = CP(-m)[x], n odd integer > 1
Dim cycloDiv2 = CyclotomicPolynomial(n \ 2)
For Each t In cycloDiv2
If t.Exponent Mod 2 = 0 Then
terms.Add(t)
Else
terms.Add(-t)
End If
Next
cyclo = New Polynomial(terms)
computedPolynomials.Add(n, cyclo)
Return cyclo
End If
- If ALGORITHM = 0 Then
REM slow - uses basic definition
Dim divisors = GetDivisors(n)
REM Polynomial: (x^n - 1)
cyclo = New Polynomial(New Term(1, n), New Term(-1))
For Each i In divisors
Dim p = CyclotomicPolynomial(i)
cyclo /= p
Next
computedPolynomials.Add(n, cyclo)
Return cyclo
- ElseIf ALGORITHM = 1 Then
REM Faster. Remove Max divisor (and all divisors of max divisor) - only one divide for all divisors of Max Divisor
Dim divisors = GetDivisors(n)
Dim maxDivisor = Integer.MinValue
For Each div In divisors
maxDivisor = Math.Max(maxDivisor, div)
Next
Dim divisorExceptMax As New List(Of Integer)
For Each div In divisors
If maxDivisor Mod div <> 0 Then
divisorExceptMax.Add(div)
End If
Next
REM Polynomial: ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor
cyclo = New Polynomial(New Term(1, n), New Term(-1)) / New Polynomial(New Term(1, maxDivisor), New Term(-1))
For Each i In divisorExceptMax
Dim p = CyclotomicPolynomial(i)
cyclo /= p
Next
computedPolynomials.Add(n, cyclo)
Return cyclo
- ElseIf ALGORITHM = 2 Then
REM Fastest
REM Let p ; q be primes such that p does not divide n, and q divides n
REM Then Cp(np)[x] = CP(n)[x^p] / CP(n)[x]
Dim m = 1
cyclo = CyclotomicPolynomial(m)
Dim primes As New List(Of Integer)
For Each it In factors
primes.Add(it.Key)
Next
primes.Sort()
For Each prime In primes
REM CP(m)[x]
Dim cycloM = cyclo
REM Compute CP(m)[x^p]
terms = New List(Of Term)
For Each t In cyclo
terms.Add(New Term(t.Coefficient, t.Exponent * prime))
Next
cyclo = New Polynomial(terms) / cycloM
m *= prime
Next
REM Now, m is the largest square free divisor of n
Dim s = n \ m
REM Compute CP(n)[x] = CP(m)[x^s]
terms = New List(Of Term)
For Each t In cyclo
terms.Add(New Term(t.Coefficient, t.Exponent * s))
Next
cyclo = New Polynomial(terms)
computedPolynomials.Add(n, cyclo)
Return cyclo
- Else
Throw New Exception("Invalid algorithm")
- End If
End Function
Sub Main()
Console.WriteLine("Task 1: cyclotomic polynomials for n <= 30:")
For i = 1 To 30
Dim p = CyclotomicPolynomial(i)
Console.WriteLine("CP[{0}] = {1}", i, p)
Next
Console.WriteLine()
Console.WriteLine("Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient:")
Dim n = 0
For i = 1 To 10
While True
n += 1
Dim cyclo = CyclotomicPolynomial(n)
If cyclo.HasCoefficentAbs(i) Then
Console.WriteLine("CP[{0}] has coefficient with magnitude = {1}", n, i)
n -= 1
Exit While
End If
End While
Next
End Sub
End Module</lang>
- Output:
Task 1: cyclotomic polynomials for n <= 30: CP[1] = x - 1 CP[2] = x + 1 CP[3] = x^2 + x + 1 CP[4] = x^2 + 1 CP[5] = x^4 + x^3 + x^2 + x + 1 CP[6] = x^2 - x + 1 CP[7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[8] = x^4 + 1 CP[9] = x^6 + x^3 + 1 CP[10] = x^4 - x^3 + x^2 - x + 1 CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[12] = x^4 - x^2 + 1 CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 CP[16] = x^8 + 1 CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[18] = x^6 - x^3 + 1 CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[20] = x^8 - x^6 + x^4 - x^2 + 1 CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[24] = x^8 - x^4 + 1 CP[25] = x^20 + x^15 + x^10 + x^5 + 1 CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[27] = x^18 + x^9 + 1 CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1 CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient: CP[1] has coefficient with magnitude = 1 CP[105] has coefficient with magnitude = 2 CP[385] has coefficient with magnitude = 3 CP[1365] has coefficient with magnitude = 4 CP[1785] has coefficient with magnitude = 5 CP[2805] has coefficient with magnitude = 6 CP[3135] has coefficient with magnitude = 7 CP[6545] has coefficient with magnitude = 8 CP[6545] has coefficient with magnitude = 9 CP[10465] has coefficient with magnitude = 10
Wren
Second part is very slow. Limited to first 7 to finish in a reasonable time - 5 minutes on my machine. <lang ecmascript>import "/trait" for Stepped import "/sort" for Sort import "/math" for Int, Nums import "/fmt" for Fmt
var algo = 2 var maxAllFactors = 1e5
class Term {
construct new(coef, exp) {
_coef = coef
_exp = exp
}
coef { _coef }
exp { _exp }
*(t) { Term.new(_coef * t.coef, _exp + t.exp) }
+(t) {
if (_exp != t.exp) Fiber.abort("Exponents unequal in term '+' method.")
return Term.new(_coef + t.coef, _exp)
}
- { Term.new(-_coef, _exp) }
toString {
if (_coef == 0) return "0"
if (_exp == 0) return _coef.toString
if (_coef == 1) return (_exp == 1) ? "x" : "x^%(_exp)"
if (_exp == 1) return "%(_coef)x"
return "%(_coef)x^%(_exp)"
}
}
class Poly {
// pass coef, exp in pairs as parameters
construct new(values) {
var le = values.count
if (le == 0) {
_terms = [Term.new(0, 0)]
} else {
if (le%2 != 0) Fiber.abort("Odd number of parameters(%(le)) passed to Poly constructor.")
_terms = []
for (i in Stepped.new(0...le, 2)) _terms.add(Term.new(values[i], values[i+1]))
tidy()
}
}
terms { _terms }
hasCoefAbs(coef) { _terms.any { |t| t.coef.abs == coef } }
+(p2) {
var p3 = Poly.new([])
var le = _terms.count
var le2 = p2.terms.count
while (le > 0 || le2 > 0) {
if (le == 0) {
p3.terms.add(p2.terms[le2-1])
le2 = le2 - 1
} else if (le2 == 0) {
p3.terms.add(_terms[le-1])
le = le - 1
} else {
var t = _terms[le-1]
var t2 = p2.terms[le2-1]
if (t.exp == t2.exp) {
var t3 = t + t2
if (t3.coef != 0) p3.terms.add(t3)
le = le - 1
le2 = le2 - 1
} else if (t.exp < t2.exp) {
p3.terms.add(t)
le = le - 1
} else {
p3.terms.add(t2)
le2 = le2 - 1
}
}
}
p3.tidy()
return p3
}
addTerm(t) {
var q = Poly.new([])
var added = false
for (i in 0..._terms.count) {
var ct = _terms[i]
if (ct.exp == t.exp) {
added = true
if (ct.coef + t.coef != 0) q.terms.add(ct + t)
} else {
q.terms.add(ct)
}
}
if (!added) q.terms.add(t)
q.tidy()
return q
}
mulTerm(t) {
var q = Poly.new([])
for (i in 0..._terms.count) {
var ct = _terms[i]
q.terms.add(ct * t)
}
q.tidy()
return q
}
/(v) {
var p = this
var q = Poly.new([])
var lcv = v.leadingCoef
var dv = v.degree
while (p.degree >= v.degree) {
var lcp = p.leadingCoef
var s = (lcp/lcv).truncate
var t = Term.new(s, p.degree - dv)
q = q.addTerm(t)
p = p + v.mulTerm(-t)
}
q.tidy()
return q
}
leadingCoef { _terms[0].coef }
degree { _terms[0].exp }
toString {
var sb = ""
var first = true
for (t in _terms) {
if (first) {
sb = sb + t.toString
first = false
} else {
sb = sb + " "
if (t.coef > 0) {
sb = sb + "+ "
sb = sb + t.toString
} else {
sb = sb + "- "
sb = sb + (-t).toString
}
}
}
return sb
}
// in place descending sort by term.exp
sortTerms() {
var cmp = Fn.new { |t1, t2| (t2.exp - t1.exp).sign }
Sort.quick(_terms, 0, _terms.count-1, cmp)
}
// sort terms and remove any unnecesary zero terms
tidy() {
sortTerms()
if (degree > 0) {
for (i in _terms.count-1..0) {
if (_terms[i].coef == 0) _terms.removeAt(i)
}
if (_terms.count == 0) _terms.add(Term.new(0, 0))
}
}
}
var computed = {} var allFactors = {2: {2: 1}}
var getFactors // recursive function getFactors = Fn.new { |n|
var f = allFactors[n]
if (f) return f
var factors = {}
if (n%2 == 0) {
var factorsDivTwo = getFactors.call(n/2)
for (me in factorsDivTwo) factors[me.key] = me.value
factors[2] = factors[2] ? factors[2] + 1 : 1
if (n < maxAllFactors) allFactors[n] = factors
return factors
}
var prime = true
var sqrt = n.sqrt.floor
var i = 3
while (i <= sqrt){
if (n%i == 0) {
prime = false
for (me in getFactors.call(n/i)) factors[me.key] = me.value
factors[i] = factors[i] ? factors[i] + 1 : 1
if (n < maxAllFactors) allFactors[n] = factors
return factors
}
i = i + 2
}
if (prime) {
factors[n] = 1
if (n < maxAllFactors) allFactors[n] = factors
}
return factors
}
var cycloPoly // recursive function cycloPoly = Fn.new { |n|
var p = computed[n]
if (p) return p
if (n == 1) {
// polynomialL x - 1
p = Poly.new([1, 1, -1, 0])
computed[1] = p
return p
}
var factors = getFactors.call(n)
var cyclo = Poly.new([])
if (factors[n]) {
// n is prime
for (i in 0...n) cyclo.terms.add(Term.new(1, i))
} else if (factors.count == 2 && factors[2] == 1 && factors[n/2] == 1) {
// n == 2p
var prime = n / 2
var coef = -1
for (i in 0...prime) {
coef = coef * (-1)
cyclo.terms.add(Term.new(coef, i))
}
} else if (factors.count == 1) {
var h = factors[2]
if (h) { // n == 2^h
cyclo.terms.addAll([Term.new(1, 1 << (h-1)), Term.new(1, 0)])
} else if (!factors[n]) {
// n == p ^ k
var p = 0
for (prime in factors.keys) p = prime
var k = factors[p]
for (i in 0...p) {
var pk = p.pow(k-1).floor
cyclo.terms.add(Term.new(1, i * pk))
}
}
} else if (factors.count == 2 && factors[2]) {
// n = 2^h * p^k
var p = 0
for (prime in factors.keys) if (prime != 2) p = prime
var coef = -1
var twoExp = 1 << (factors[2] - 1)
var k = factors[p]
for (i in 0...p) {
coef = coef * (-1)
var pk = p.pow(k-1).floor
cyclo.terms.add(Term.new(coef, i * twoExp * pk))
}
} else if (factors[2] && (n/2) % 2 == 1 && (n/2) > 1) {
// CP(2m)[x] == CP(-m)[x], n odd integer > 1
var cycloDiv2 = cycloPoly.call(n/2)
for (t in cycloDiv2.terms) {
var t2 = t
if (t.exp % 2 != 0) t2 = -t
cyclo.terms.add(t2)
}
} else if (algo == 0) {
// slow - uses basic definition
var divs = Int.properDivisors(n)
// polynomial: x^n - 1
var cyclo = Poly.new([1, n, -1, 0])
for (i in divs) {
var p = cycloPoly.call(i)
cyclo = cyclo / p
}
} else if (algo == 1) {
// faster - remove max divisor (and all divisors of max divisor)
// only one divide for all divisors of max divisor
var divs = Int.properDivisors(n)
var maxDiv = Nums.max(divs)
var divsExceptMax = divs.where { |d| maxDiv % d != 0 }.toList
// polynomial: ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor
cyclo = Poly.new([1, n, -1, 0])
cyclo = cyclo / Poly.new([1, maxDiv, -1, 0])
for (i in divsExceptMax) {
var p = cycloPoly.call(i)
cyclo = cyclo / p
}
} else if (algo == 2) {
// fastest
// let p, q be primes such that p does not divide n, and q divides n
// then CP(np)[x] = CP(n)[x^p] / CP(n)[x]
var m = 1
cyclo = cycloPoly.call(m)
var primes = []
for (prime in factors.keys) primes.add(prime)
Sort.quick(primes)
for (prime in primes) {
// CP(m)[x]
var cycloM = cyclo
// compute CP(m)[x^p]
var terms = []
for (t in cycloM.terms) terms.add(Term.new(t.coef, t.exp * prime))
cyclo = Poly.new([])
cyclo.terms.addAll(terms)
cyclo.tidy()
cyclo = cyclo / cycloM
m = m * prime
}
// now, m is the largest square free divisor of n
var s = n / m
// Compute CP(n)[x] = CP(m)[x^s]
var terms = []
for (t in cyclo.terms) terms.add(Term.new(t.coef, t.exp * s))
cyclo = Poly.new([])
cyclo.terms.addAll(terms)
} else {
Fiber.abort("Invalid algorithm.")
}
cyclo.tidy()
computed[n] = cyclo
return cyclo
}
System.print("Task 1: cyclotomic polynomials for n <= 30:") for (i in 1..30) {
var p = cycloPoly.call(i)
Fmt.print("CP[$2d] = $s", i, p)
}
System.print("\nTask 2: Smallest cyclotomic polynomial with n or -n as a coefficient:") var n = 0 for (i in 1..7) {
while(true) {
n = n + 1
var cyclo = cycloPoly.call(n)
if (cyclo.hasCoefAbs(i)) {
Fmt.print("CP[$d] has coefficient with magnitude = $d", n, i)
n = n - 1
break
}
}
}</lang>
- Output:
Task 1: cyclotomic polynomials for n <= 30: CP[ 1] = x - 1 CP[ 2] = x + 1 CP[ 3] = x^2 + x + 1 CP[ 4] = x^2 + 1 CP[ 5] = x^4 + x^3 + x^2 + x + 1 CP[ 6] = x^2 - x + 1 CP[ 7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[ 8] = x^4 + 1 CP[ 9] = x^6 + x^3 + 1 CP[10] = x^4 - x^3 + x^2 - x + 1 CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[12] = x^4 - x^2 + 1 CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 CP[16] = x^8 + 1 CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[18] = x^6 - x^3 + 1 CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[20] = x^8 - x^6 + x^4 - x^2 + 1 CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[24] = x^8 - x^4 + 1 CP[25] = x^20 + x^15 + x^10 + x^5 + 1 CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 CP[27] = x^18 + x^9 + 1 CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1 CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient: CP[1] has coefficient with magnitude = 1 CP[105] has coefficient with magnitude = 2 CP[385] has coefficient with magnitude = 3 CP[1365] has coefficient with magnitude = 4 CP[1785] has coefficient with magnitude = 5 CP[2805] has coefficient with magnitude = 6 CP[3135] has coefficient with magnitude = 7