Cyclotomic polynomial
Cyclotomic polynomial is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
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 i or -i 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.
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
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)
x = cyclotomic(d)
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
c = cyclotomic(i)
for coef in filter(x -> -10.0001 < x < 10.0001, coeffs(c))
x = Int(round(abs(coef)))
if 0 < x < 11 && dig[x] == 0
dig[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