Safe primes and unsafe primes
You are encouraged to solve this task according to the task description, using any language you may know.
- Definitions
-
- A safe prime is a prime p and where (p-1)/2 is also prime.
- The corresponding prime (p-1)/2 is known as a Sophie Germain prime.
- An unsafe prime is a prime p and where (p-1)/2 isn't a prime.
- An unsafe prime is a prime that isn't a safe prime.
- Task
-
- Find and display (on one line) the first 35 safe primes.
- Find and display the count of the safe primes below 1,000,000.
- Find and display the count of the safe primes below 10,000,000.
- Find and display (on one line) the first 40 unsafe primes.
- Find and display the count of the unsafe primes below 1,000,000.
- Find and display the count of the unsafe primes below 10,000,000.
- (Optional) display the counts and "below numbers" with commas.
Show all output here.
- Related Task
- Also see
-
- The OEIS article: safe primes.
- The OEIS article: unsafe primes.
ALGOL 68
<lang algol68># find and count safe and unsafe primes #
- safe primes are primes p such that ( p - 1 ) / 2 is also prime #
- unsafe primes are primes that are not safe #
PR heap=128M PR # set heap memory size for Algol 68G #
- returns a string representation of n with commas #
PROC commatise = ( INT n )STRING:
BEGIN
STRING result := "";
STRING unformatted = whole( n, 0 );
INT ch count := 0;
FOR c FROM UPB unformatted BY -1 TO LWB unformatted DO
IF ch count <= 2 THEN ch count +:= 1
ELSE ch count := 1; "," +=: result
FI;
unformatted[ c ] +=: result
OD;
result
END # commatise # ;
- sieve values #
CHAR prime = "P"; # unclassified prime # CHAR safe = "S"; # safe prime # CHAR unsafe = "U"; # unsafe prime # CHAR composite = "C"; # non-prime #
- sieve of Eratosthenes: sets s[i] to prime if i is a prime, #
- composite otherwise #
PROC sieve = ( REF[]CHAR s )VOID:
BEGIN
# start with everything flagged as prime #
FOR i TO UPB s DO s[ i ] := prime OD;
# sieve out the non-primes #
s[ 1 ] := composite;
FOR i FROM 2 TO ENTIER sqrt( UPB s ) DO
IF s[ i ] = prime THEN FOR p FROM i * i BY i TO UPB s DO s[ p ] := composite OD FI
OD
END # sieve # ;
INT max number = 10 000 000;
- construct a sieve of primes up to the maximum number #
[ 1 : max number ]CHAR primes; sieve( primes );
- classify the primes #
- ( p - 1 ) OVER 2 is non-zero for p >= 3, thus we know 2 is unsafe #
primes[ 2 ] := unsafe; FOR p FROM 3 TO UPB primes DO
IF primes[ p ] = prime THEN
primes[ p ] := IF primes[ ( p - 1 ) OVER 2 ] = composite THEN unsafe ELSE safe FI
FI
OD;
- count the primes of each type #
INT safe1 := 0, safe10 := 0; INT unsafe1 := 0, unsafe10 := 0; FOR p FROM LWB primes TO UPB primes DO
IF primes[ p ] = safe THEN
safe10 +:= 1;
IF p < 1 000 000 THEN safe1 +:= 1 FI
ELIF primes[ p ] = unsafe THEN
unsafe10 +:= 1;
IF p < 1 000 000 THEN unsafe1 +:= 1 FI
FI
OD; INT safe count := 0; print( ( "first 35 safe primes:", newline ) ); FOR p WHILE safe count < 35 DO IF primes[ p ] = safe THEN print( ( " ", whole( p, 0 ) ) ); safe count +:= 1 FI OD; print( ( newline ) ); print( ( "safe primes below 1,000,000: ", commatise( safe1 ), newline ) ); print( ( "safe primes below 10,000,000: ", commatise( safe10 ), newline ) ); print( ( "first 40 unsafe primes:", newline ) ); INT unsafe count := 0; FOR p WHILE unsafe count < 40 DO IF primes[ p ] = unsafe THEN print( ( " ", whole( p, 0 ) ) ); unsafe count +:= 1 FI OD; print( ( newline ) ); print( ( "unsafe primes below 1,000,000: ", commatise( unsafe1 ), newline ) ); print( ( "unsafe primes below 10,000,000: ", commatise( unsafe10 ), newline ) )</lang>
- Output:
first 35 safe primes: 5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619 safe primes below 1,000,000: 4,324 safe primes below 10,000,000: 30,657 first 40 unsafe primes: 2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233 unsafe primes below 1,000,000: 74,174 unsafe primes below 10,000,000: 633,922
AppleScript
<lang applescript>-- Heavy-duty Sieve of Eratosthenes handler. -- Returns a list containing either just the primes up to a given limit ('crossingsOut' = false) or, as in this task, -- both the primes and 'missing values' representing the "crossed out" non-primes ('crossingsOut' = true). on sieveForPrimes given limit:limit, crossingsOut:keepingZaps
if (limit < 1) then return {}
-- Build a list initially containing only 'missing values'. For speed, and to reduce the likelihood of hanging,
-- do this by building sublists of at most 5000 items and concatenating them afterwards.
script o
property sublists : {}
property numberList : {}
end script
set sublistSize to 5000
set mv to missing value -- Use a single 'missing value' instance for economy.
repeat sublistSize times
set end of o's numberList to mv
end repeat
-- Start with a possible < 5000-item sublist.
if (limit mod sublistSize > 0) then set end of o's sublists to items 1 thru (limit mod sublistSize) of o's numberList
-- Then any 5000-item sublists needed.
if (limit ≥ sublistSize) then
set end of o's sublists to o's numberList
repeat (limit div sublistSize - 1) times
set end of o's sublists to o's numberList's items
end repeat
end if
-- Concatenate them more-or-less evenly.
set subListCount to (count o's sublists)
repeat until (subListCount is 1)
set o's numberList to {}
repeat with i from 2 to subListCount by 2
set end of o's numberList to (item (i - 1) of o's sublists) & (item i of o's sublists)
end repeat
if (i < subListCount) then set last item of o's numberList to (end of o's numberList) & (end of o's sublists)
set o's sublists to o's numberList
set subListCount to subListCount div 2
end repeat
set o's numberList to beginning of o's sublists
-- Set the relevant list positions to 2, 3, 5, and numbers which aren't multiples of them.
if (limit > 1) then set item 2 of o's numberList to 2
if (limit > 2) then set item 3 of o's numberList to 3
if (limit > 4) then set item 5 of o's numberList to 5
if (limit < 36) then
set n to -23
else
repeat with n from 7 to (limit - 29) by 30
set item n of o's numberList to n
tell (n + 4) to set item it of o's numberList to it
tell (n + 6) to set item it of o's numberList to it
tell (n + 10) to set item it of o's numberList to it
tell (n + 12) to set item it of o's numberList to it
tell (n + 16) to set item it of o's numberList to it
tell (n + 22) to set item it of o's numberList to it
tell (n + 24) to set item it of o's numberList to it
end repeat
end if
repeat with n from (n + 30) to limit
if ((n mod 2 > 0) and (n mod 3 > 0) and (n mod 5 > 0)) then set item n of o's numberList to n
end repeat
-- "Cross out" inserted numbers which are multiples of others.
set inx to {0, 4, 6, 10, 12, 16, 22, 24}
repeat with n from 7 to ((limit ^ 0.5) div 1) by 30
repeat with inc in inx
tell (n + inc)
if (item it of o's numberList is it) then
repeat with multiple from (it * it) to limit by it
set item multiple of o's numberList to mv
end repeat
end if
end tell
end repeat
end repeat
if (keepingZaps) then return o's numberList
return o's numberList's numbers
end sieveForPrimes
-- Task code: on doTask()
set {safeQuantity, unsafeQuantity, max1, max2} to {35, 40, 1000000 - 1, 10000000 - 1}
set {safePrimes, unsafePrimes, safeCount1, safeCount2, unsafeCount1, unsafeCount2} to {{}, {}, 0, 0, 0, 0}
-- Get a list of 9,999,999 primes and "crossed out" non-primes! Also one with just the primes.
script o
property primesAndZaps : sieveForPrimes with crossingsOut given limit:max2
property primesOnly : my primesAndZaps's numbers
end script
-- Work through the primes-only list, using the other as an indexable look-up to check the related numbers.
set SophieGermainLimit to (max2 - 1) div 2
repeat with n in o's primesOnly
set n to n's contents
if (n ≤ SophieGermainLimit) then
tell (n * 2 + 1)
if (item it of o's primesAndZaps is it) then
if (safeCount2 < safeQuantity) then set end of safePrimes to it
if (it < max1) then set safeCount1 to safeCount1 + 1
set safeCount2 to safeCount2 + 1
end if
end tell
end if
if ((n is 2) or (item ((n - 1) div 2) of o's primesAndZaps is missing value)) then
if (unsafeCount2 < unsafeQuantity) then set end of unsafePrimes to n
if (n < max1) then set unsafeCount1 to unsafeCount1 + 1
set unsafeCount2 to unsafeCount2 + 1
end if
end repeat
-- Format and output the results.
set output to {}
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to ", "
set end of output to "First 35 safe primes:"
set end of output to safePrimes as text
set end of output to "There are " & safeCount1 & " safe primes < 1,000,000 and " & safeCount2 & " < 10,000,000."
set end of output to ""
set end of output to "First 40 unsafe primes:"
set end of output to unsafePrimes as text
set end of output to "There are " & unsafeCount1 & " unsafe primes < 1,000,000 and " & unsafeCount2 & " < 10,000,000."
set AppleScript's text item delimiters to linefeed
set output to output as text
set AppleScript's text item delimiters to astid
return output
end doTask
return doTask()</lang>
- Output:
<lang applescript>"First 35 safe primes: 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619 There are 4324 safe primes < 1,000,000 and 30657 < 10,000,000.
First 40 unsafe primes: 2, 3, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233 There are 74174 unsafe primes < 1,000,000 and 633922 < 10,000,000."</lang>
C
<lang c>#include <stdbool.h>
- include <stdio.h>
int primes[] = {
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331
};
- define PCOUNT (sizeof(primes) / sizeof(int))
bool isPrime(int n) {
int i;
if (n < 2) {
return false;
}
for (i = 0; i < PCOUNT; i++) {
if (n == primes[i]) {
return true;
}
if (n % primes[i] == 0) {
return false;
}
if (n < primes[i] * primes[i]) {
return true;
}
}
for (i = primes[PCOUNT - 1] + 2; i * i <= n; i += 2) {
if (n % i == 0) {
return false;
}
}
return true;
}
int main() {
int beg, end; int i, count;
// safe primes
///////////////////////////////////////////
beg = 2;
end = 1000000;
count = 0;
printf("First 35 safe primes:\n");
for (i = beg; i < end; i++) {
if (isPrime(i) && isPrime((i - 1) / 2)) {
if (count < 35) {
printf("%d ", i);
}
count++;
}
}
printf("\nThere are %d safe primes below %d\n", count, end);
beg = end;
end = end * 10;
for (i = beg; i < end; i++) {
if (isPrime(i) && isPrime((i - 1) / 2)) {
count++;
}
}
printf("There are %d safe primes below %d\n", count, end);
// unsafe primes
///////////////////////////////////////////
beg = 2;
end = 1000000;
count = 0;
printf("\nFirst 40 unsafe primes:\n");
for (i = beg; i < end; i++) {
if (isPrime(i) && !isPrime((i - 1) / 2)) {
if (count < 40) {
printf("%d ", i);
}
count++;
}
}
printf("\nThere are %d unsafe primes below %d\n", count, end);
beg = end;
end = end * 10;
for (i = beg; i < end; i++) {
if (isPrime(i) && !isPrime((i - 1) / 2)) {
count++;
}
}
printf("There are %d unsafe primes below %d\n", count, end);
return 0;
}</lang>
- Output:
First 35 safe primes: 5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619 There are 4324 safe primes below 1000000 There are 30657 safe primes below 10000000 First 40 unsafe primes: 2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233 There are 74174 unsafe primes below 1000000 There are 633922 unsafe primes below 10000000
C#
<lang csharp>using static System.Console; using System; using System.Collections; using System.Collections.Generic; using System.Linq;
public static class SafePrimes {
public static void Main() {
HashSet<int> primes = Primes(10_000_000).ToHashSet();
WriteLine("First 35 safe primes:");
WriteLine(string.Join(" ", primes.Where(IsSafe).Take(35)));
WriteLine($"There are {primes.TakeWhile(p => p < 1_000_000).Count(IsSafe):n0} safe primes below {1_000_000:n0}");
WriteLine($"There are {primes.TakeWhile(p => p < 10_000_000).Count(IsSafe):n0} safe primes below {10_000_000:n0}");
WriteLine("First 40 unsafe primes:");
WriteLine(string.Join(" ", primes.Where(IsUnsafe).Take(40)));
WriteLine($"There are {primes.TakeWhile(p => p < 1_000_000).Count(IsUnsafe):n0} unsafe primes below {1_000_000:n0}");
WriteLine($"There are {primes.TakeWhile(p => p < 10_000_000).Count(IsUnsafe):n0} unsafe primes below {10_000_000:n0}");
bool IsSafe(int prime) => primes.Contains(prime / 2);
bool IsUnsafe(int prime) => !primes.Contains(prime / 2);
}
//Method from maths library
static IEnumerable<int> Primes(int bound) {
if (bound < 2) yield break;
yield return 2;
BitArray composite = new BitArray((bound - 1) / 2);
int limit = ((int)(Math.Sqrt(bound)) - 1) / 2;
for (int i = 0; i < limit; i++) {
if (composite[i]) continue;
int prime = 2 * i + 3;
yield return prime;
for (int j = (prime * prime - 2) / 2; j < composite.Count; j += prime) composite[j] = true;
}
for (int i = limit; i < composite.Count; i++) {
if (!composite[i]) yield return 2 * i + 3;
}
}
}</lang>
- Output:
First 35 safe primes: 5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619 There are 4,324 safe primes below 1,000,000 There are 30,657 safe primes below 10,000,000 First 40 unsafe primes: 2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233 There are 74,174 unsafe primes below 1,000,000 There are 633,922 unsafe primes below 10,000,000
C++
<lang cpp>#include <iostream>
- include <iomanip>
- include <locale>
- include <sstream>
- include "prime_sieve.hpp"
const int limit1 = 1000000; const int limit2 = 10000000;
class prime_info { public:
explicit prime_info(int max) : max_print(max) {}
void add_prime(int prime) {
++count2;
if (prime < limit1)
++count1;
if (count2 <= max_print) {
if (count2 > 1)
out << ' ';
out << prime;
}
}
void print(std::ostream& os, const char* name) const {
os << "First " << max_print << " " << name << " primes: " << out.str() << '\n';
os << "Number of " << name << " primes below " << limit1 << ": " << count1 << '\n';
os << "Number of " << name << " primes below " << limit2 << ": " << count2 << '\n';
}
private:
int max_print; int count1 = 0; int count2 = 0; std::ostringstream out;
};
int main() {
// find the prime numbers up to limit2 prime_sieve sieve(limit2);
// write numbers with groups of digits separated according to the system default locale
std::cout.imbue(std::locale(""));
// count and print safe/unsafe prime numbers
prime_info safe_primes(35);
prime_info unsafe_primes(40);
for (int p = 2; p < limit2; ++p) {
if (sieve.is_prime(p)) {
if (sieve.is_prime((p - 1)/2))
safe_primes.add_prime(p);
else
unsafe_primes.add_prime(p);
}
}
safe_primes.print(std::cout, "safe");
unsafe_primes.print(std::cout, "unsafe");
return 0;
}</lang>
Contents of prime_sieve.hpp: <lang cpp>#ifndef PRIME_SIEVE_HPP
- define PRIME_SIEVE_HPP
- include <algorithm>
- include <vector>
/**
* A simple implementation of the Sieve of Eratosthenes. * See https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes. */
class prime_sieve { public:
explicit prime_sieve(size_t); bool is_prime(size_t) const;
private:
std::vector<bool> is_prime_;
};
/**
* Constructs a sieve with the given limit. * * @param limit the maximum integer that can be tested for primality */
inline prime_sieve::prime_sieve(size_t limit) {
limit = std::max(size_t(3), limit);
is_prime_.resize(limit/2, true);
for (size_t p = 3; p * p <= limit; p += 2) {
if (is_prime_[p/2 - 1]) {
size_t inc = 2 * p;
for (size_t q = p * p; q <= limit; q += inc)
is_prime_[q/2 - 1] = false;
}
}
}
/**
* Returns true if the given integer is a prime number. The integer * must be less than or equal to the limit passed to the constructor. * * @param n an integer less than or equal to the limit passed to the * constructor * @return true if the integer is prime */
inline bool prime_sieve::is_prime(size_t n) const {
if (n == 2)
return true;
if (n < 2 || n % 2 == 0)
return false;
return is_prime_.at(n/2 - 1);
}
- endif</lang>
- Output:
First 35 safe primes: 5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619 Number of safe primes below 1,000,000: 4,324 Number of safe primes below 10,000,000: 30,657 First 40 unsafe primes: 2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233 Number of unsafe primes below 1,000,000: 74,174 Number of unsafe primes below 10,000,000: 633,922
D
<lang d>import std.stdio;
immutable PRIMES = [
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331
];
bool isPrime(const int n) {
if (n < 2) {
return false;
}
foreach (p; PRIMES) {
if (n == p) {
return true;
}
if (n % p == 0) {
return false;
}
if (n < p * p) {
return true;
}
}
int i = (PRIMES[$ - 1] / 6) * 6 - 1;
while (i * i <= n) {
if (n % i == 0) {
return false;
}
i += 2;
if (n % i == 0) {
return false;
}
i += 4;
}
return true;
}
void main() {
int beg = 2; int end = 1_000_000; int count = 0;
// safe primes ///////////////////////////////////////////
writeln("First 35 safe primes:");
foreach (i; beg..end) {
if (isPrime(i) && isPrime((i - 1) / 2)) {
if (count < 35) {
write(i, ' ');
}
count++;
}
}
writefln("\nThere are %5d safe primes below %8d", count, end);
beg = end;
end *= 10;
foreach (i; beg..end) {
if (isPrime(i) && isPrime((i - 1) / 2)) {
count++;
}
}
writefln("There are %5d safe primes below %8d", count, end);
// unsafe primes ///////////////////////////////////////////
beg = 2;
end = 1_000_000;
count = 0;
writeln("\nFirst 40 unsafe primes:");
foreach (i; beg..end) {
if (isPrime(i) && !isPrime((i - 1) / 2)) {
if (count < 40) {
write(i, ' ');
}
count++;
}
}
writefln("\nThere are %6d unsafe primes below %9d", count, end);
beg = end;
end *= 10;
foreach (i; beg..end) {
if (isPrime(i) && !isPrime((i - 1) / 2)) {
count++;
}
}
writefln("There are %6d unsafe primes below %9d", count, end);
}</lang>
- Output:
First 35 safe primes: 5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619 There are 4324 safe primes below 1000000 There are 30657 safe primes below 10000000 First 40 unsafe primes: 2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233 There are 74174 unsafe primes below 1000000 There are 633922 unsafe primes below 10000000
F#
This task uses Extensible Prime Generator (F#)<lang fsharp> pCache |> Seq.filter(fun n->isPrime((n-1)/2)) |> Seq.take 35 |> Seq.iter (printf "%d ") </lang>
- Output:
5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619
<lang fsharp> printfn "There are %d safe primes less than 1000000" (pCache |> Seq.takeWhile(fun n->n<1000000) |> Seq.filter(fun n->isPrime((n-1)/2)) |> Seq.length) </lang>
- Output:
There are 4324 safe primes less than 10000000
<lang fsharp> printfn "There are %d safe primes less than 10000000" (pCache |> Seq.takeWhile(fun n->n<10000000) |> Seq.filter(fun n->isPrime((n-1)/2)) |> Seq.length) </lang>
- Output:
There are 30657 safe primes less than 10000000
<lang fsharp> pCache |> Seq.filter(fun n->not (isPrime((n-1)/2))) |> Seq.take 40 |> Seq.iter (printf "%d ") </lang>
- Output:
2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233
<lang fsharp> printfn "There are %d unsafe primes less than 1000000" (pCache |> Seq.takeWhile(fun n->n<1000000) |> Seq.filter(fun n->not (isPrime((n-1)/2))) |> Seq.length);; </lang>
- Output:
There are 74174 unsafe primes less than 1000000
<lang fsharp> printfn "There are %d unsafe primes less than 10000000" (pCache |> Seq.takeWhile(fun n->n<10000000) |> Seq.filter(fun n->not (isPrime((n-1)/2))) |> Seq.length);; </lang>
- Output:
There are 633922 unsafe primes less than 10000000
Factor
Much like the Raku example, this program uses an in-built primes generator to efficiently obtain the first ten million primes. If memory is a concern, it wouldn't be unreasonable to perform primality tests on the (odd) numbers below ten million, however. <lang factor>USING: fry interpolate kernel literals math math.primes sequences tools.memory.private ; IN: rosetta-code.safe-primes
CONSTANT: primes $[ 10,000,000 primes-upto ]
- safe/unsafe ( -- safe unsafe )
primes [ 1 - 2/ prime? ] partition ;
- count< ( seq n -- str ) '[ _ < ] count commas ;
- seq>commas ( seq -- str ) [ commas ] map " " join ;
- stats ( seq n -- head count1 count2 )
'[ _ head seq>commas ] [ 1e6 count< ] [ 1e7 count< ] tri ;
safe/unsafe [ 35 ] [ 40 ] bi* [ stats ] 2bi@
[I First 35 safe primes: ${5} Safe prime count below 1,000,000: ${4} Safe prime count below 10,000,000: ${3}
First 40 unsafe primes: ${2} Unsafe prime count below 1,000,000: ${1} Unsafe prime count below 10,000,000: ${} I]</lang>
- Output:
First 35 safe primes: 5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1,019 1,187 1,283 1,307 1,319 1,367 1,439 1,487 1,523 1,619 Safe prime count below 1,000,000: 4,324 Safe prime count below 10,000,000: 30,657 First 40 unsafe primes: 2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233 Unsafe prime count below 1,000,000: 74,174 Unsafe prime count below 10,000,000: 633,922
FreeBASIC
<lang freebasic>' version 19-01-2019 ' compile with: fbc -s console
Const As UInteger max = 10000000 Dim As UInteger i, j, sc1, usc1, sc2, usc2 Dim As String safeprimes, unsafeprimes Dim As UByte sieve()
ReDim sieve(max) ' 0 = prime, 1 = no prime sieve(0) = 1 : sieve(1) = 1
For i = 4 To max Step 2
sieve(i) = 1
Next For i = 3 To Sqr(max) +1 Step 2
If sieve(i) = 0 Then
For j = i * i To max Step i * 2
sieve(j) = 1
Next
End If
Next
usc1 = 1 : unsafeprimes = "2" For i = 3 To 3001 Step 2
If sieve(i) = 0 Then
If sieve(i \ 2) = 0 Then
sc1 += 1
If sc1 <= 35 Then
safeprimes += " " + Str(i)
End If
Else
usc1 += 1
If usc1 <= 40 Then
unsafeprimes += " " + Str(i)
End If
End If
End If
Next
For i = 3003 To max \ 10 Step 2
If sieve(i) = 0 Then
If sieve(i \ 2) = 0 Then
sc1 += 1
Else
usc1 += 1
End If
End If
Next
sc2 = sc1 : usc2 = usc1 For i = max \ 10 +1 To max Step 2
If sieve(i) = 0 Then
If sieve(i \ 2) = 0 Then
sc2 += 1
Else
usc2 += 1
End If
End If
Next
Print "the first 35 Safeprimes are: "; safeprimes Print Print "the first 40 Unsafeprimes are: "; unsafeprimes Print Print " Safeprimes Unsafeprimes" Print " Below ---------------------------" Print Using "##########, "; max \ 10; sc1; usc1 Print Using "##########, "; max ; sc2; usc2
' empty keyboard buffer While Inkey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>
- Output:
the first 35 Safeprimes are: 5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619
the first 40 Unsafeprimes are: 2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233
Safeprimes Unsafeprimes
Below ---------------------------
1,000,000 4,324 74,174
10,000,000 30,657 633,922
Frink
<lang frink> safePrimes[end=undef] := select[primes[5,end], {|p| isPrime[(p-1)/2] }] unsafePrimes[end=undef] := select[primes[2,end], {|p| p<5 or isPrime[(p-1)/2] }]
println["First 35 safe primes: " + first[safePrimes[], 35]] println["Safe primes below 1,000,000: " + length[safePrimes[1_000_000]]] println["Safe primes below 10,000,000: " + length[safePrimes[10_000_000]]]
println["First 40 unsafe primes: " + first[unsafePrimes[], 40]] println["Unsafe primes below 1,000,000: " + length[unsafePrimes[1_000_000]]] println["Unsafe primes below 10,000,000: " + length[unsafePrimes[10_000_000]]] </lang>
- Output:
First 35 safe primes: [5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619] Safe primes below 1,000,000: 4324 Safe primes below 10,000,000: 30657 First 40 unsafe primes: [2, 3, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233] Unsafe primes below 1,000,000: 74174 Unsafe primes below 10,000,000: 633922
Go
<lang go>package main
import "fmt"
func sieve(limit uint64) []bool {
limit++
// True denotes composite, false denotes prime.
c := make([]bool, limit) // all false by default
c[0] = true
c[1] = true
// apart from 2 all even numbers are of course composite
for i := uint64(4); i < limit; i += 2 {
c[i] = true
}
p := uint64(3) // Start from 3.
for {
p2 := p * p
if p2 >= limit {
break
}
for i := p2; i < limit; i += 2 * p {
c[i] = true
}
for {
p += 2
if !c[p] {
break
}
}
}
return c
}
func commatize(n int) string {
s := fmt.Sprintf("%d", n)
if n < 0 {
s = s[1:]
}
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
if n >= 0 {
return s
}
return "-" + s
}
func main() {
// sieve up to 10 million
sieved := sieve(1e7)
var safe = make([]int, 35)
count := 0
for i := 3; count < 35; i += 2 {
if !sieved[i] && !sieved[(i-1)/2] {
safe[count] = i
count++
}
}
fmt.Println("The first 35 safe primes are:\n", safe, "\n")
count = 0
for i := 3; i < 1e6; i += 2 {
if !sieved[i] && !sieved[(i-1)/2] {
count++
}
}
fmt.Println("The number of safe primes below 1,000,000 is", commatize(count), "\n")
for i := 1000001; i < 1e7; i += 2 {
if !sieved[i] && !sieved[(i-1)/2] {
count++
}
}
fmt.Println("The number of safe primes below 10,000,000 is", commatize(count), "\n")
unsafe := make([]int, 40)
unsafe[0] = 2 // since (2 - 1)/2 is not prime
count = 1
for i := 3; count < 40; i += 2 {
if !sieved[i] && sieved[(i-1)/2] {
unsafe[count] = i
count++
}
}
fmt.Println("The first 40 unsafe primes are:\n", unsafe, "\n")
count = 1
for i := 3; i < 1e6; i += 2 {
if !sieved[i] && sieved[(i-1)/2] {
count++
}
}
fmt.Println("The number of unsafe primes below 1,000,000 is", commatize(count), "\n")
for i := 1000001; i < 1e7; i += 2 {
if !sieved[i] && sieved[(i-1)/2] {
count++
}
}
fmt.Println("The number of unsafe primes below 10,000,000 is", commatize(count), "\n")
}</lang>
- Output:
The first 35 safe primes are: [5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619] The number of safe primes below 1,000,000 is 4,324 The number of safe primes below 10,000,000 is 30,657 The first 40 unsafe primes are: [2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233] The number of unsafe primes below 1,000,000 is 74,174 The number of unsafe primes below 10,000,000 is 633,922
Haskell
Uses Numbers.Prime library: http://hackage.haskell.org/package/primes-0.2.1.0/docs/Data-Numbers-Primes.html <lang haskell> import Text.Printf (printf) import Data.Numbers.Primes (isPrime, primes)
main = do
printf "First 35 safe primes: %s\n" (show $ take 35 safe) printf "There are %d safe primes below 100,000.\n" (length $ takeWhile (<1000000) safe) printf "There are %d safe primes below 10,000,000.\n\n" (length $ takeWhile (<10000000) safe)
printf "First 40 unsafe primes: %s\n" (show $ take 40 unsafe) printf "There are %d unsafe primes below 100,000.\n" (length $ takeWhile (<1000000) unsafe) printf "There are %d unsafe primes below 10,000,000.\n\n" (length $ takeWhile (<10000000) unsafe)
where safe = filter (\n -> isPrime ((n-1) `div` 2)) primes
unsafe = filter (\n -> not (isPrime((n-1) `div` 2))) primes
</lang>
- Output:
First 35 safe primes: [5,7,11,23,47,59,83,107,167,179,227,263,347,359,383,467,479,503,563,587,719,839,863,887,983,1019,1187,1283,1307,1319,1367,1439,1487,1523,1619] There are 4324 safe primes below 100,000. There are 30657 safe primes below 10,000,000. First 40 unsafe primes: [2,3,13,17,19,29,31,37,41,43,53,61,67,71,73,79,89,97,101,103,109,113,127,131,137,139,149,151,157,163,173,181,191,193,197,199,211,223,229,233] There are 74174 unsafe primes below 100,000. There are 633922 unsafe primes below 10,000,000.
J
NB. play around a bit to get primes less than ten million
p:inv 10000000
664579
p:664579
10000019
PRIMES =: p:i.664579
10 {. PRIMES
2 3 5 7 11 13 17 19 23 29
{: PRIMES
9999991
primeQ =: 1&p:
safeQ =: primeQ@:-:@:<:
Filter =: (#~`)(`:6)
SAFE =: safeQ Filter PRIMES
NB. first thirty-five safe primes
(32+3) {. SAFE
5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619
NB. first forty unsafe primes
(33+7) {. PRIMES -. SAFE
2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233
NB. tally of safe primes less than ten million
# SAFE
30657
NB. tally of safe primes below a million
# 1000000&>Filter SAFE
4324
NB. tally of perilous primes below ten million
UNSAFE =: PRIMES -. SAFE
# UNSAFE
633922
NB. tally of these below one million
K =: 1 : 'm * 1000'
+/ UNSAFE < 1 K K
74174
Essentially we have <lang J> primeQ =: 1&p: safeQ =: primeQ@:-:@:<: Filter =: (#~`)(`:6) K =: adverb def 'm * 1000' PRIMES =: i.&.:(p:inv) 10 K K SAFE =: safeQ Filter PRIMES UNSAFE =: PRIMES -. SAFE </lang> The rest of the display is mere window dressing.
Java
<lang java>public class SafePrimes {
public static void main(String... args) {
// Use Sieve of Eratosthenes to find primes
int SIEVE_SIZE = 10_000_000;
boolean[] isComposite = new boolean[SIEVE_SIZE];
// It's really a flag indicating non-prime, but composite usually applies
isComposite[0] = true;
isComposite[1] = true;
for (int n = 2; n < SIEVE_SIZE; n++) {
if (isComposite[n]) {
continue;
}
for (int i = n * 2; i < SIEVE_SIZE; i += n) {
isComposite[i] = true;
}
}
int oldSafePrimeCount = 0;
int oldUnsafePrimeCount = 0;
int safePrimeCount = 0;
int unsafePrimeCount = 0;
StringBuilder safePrimes = new StringBuilder();
StringBuilder unsafePrimes = new StringBuilder();
int safePrimesStrCount = 0;
int unsafePrimesStrCount = 0;
for (int n = 2; n < SIEVE_SIZE; n++) {
if (n == 1_000_000) {
oldSafePrimeCount = safePrimeCount;
oldUnsafePrimeCount = unsafePrimeCount;
}
if (isComposite[n]) {
continue;
}
boolean isUnsafe = isComposite[(n - 1) >>> 1];
if (isUnsafe) {
if (unsafePrimeCount < 40) {
if (unsafePrimeCount > 0) {
unsafePrimes.append(", ");
}
unsafePrimes.append(n);
unsafePrimesStrCount++;
}
unsafePrimeCount++;
}
else {
if (safePrimeCount < 35) {
if (safePrimeCount > 0) {
safePrimes.append(", ");
}
safePrimes.append(n);
safePrimesStrCount++;
}
safePrimeCount++;
}
}
System.out.println("First " + safePrimesStrCount + " safe primes: " + safePrimes.toString());
System.out.println("Number of safe primes below 1,000,000: " + oldSafePrimeCount);
System.out.println("Number of safe primes below 10,000,000: " + safePrimeCount);
System.out.println("First " + unsafePrimesStrCount + " unsafe primes: " + unsafePrimes.toString());
System.out.println("Number of unsafe primes below 1,000,000: " + oldUnsafePrimeCount);
System.out.println("Number of unsafe primes below 10,000,000: " + unsafePrimeCount);
return;
}
}</lang>
- Output:
First 35 safe primes: 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619 Number of safe primes below 1,000,000: 4324 Number of safe primes below 10,000,000: 30657 First 40 unsafe primes: 2, 3, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233 Number of unsafe primes below 1,000,000: 74174 Number of unsafe primes below 10,000,000: 633922
Julia
<lang julia>using Primes, Formatting
function parseprimelist()
primelist = primes(2, 10000000)
safeprimes = Vector{Int64}()
unsafeprimes = Vector{Int64}()
for p in primelist
if isprime(div(p - 1, 2))
push!(safeprimes, p)
else
push!(unsafeprimes, p)
end
end
println("The first 35 unsafe primes are: ", safeprimes[1:35])
println("There are ", format(sum(map(x -> x < 1000000, safeprimes)), commas=true), " safe primes less than 1 million.")
println("There are ", format(length(safeprimes), commas=true), " safe primes less than 10 million.")
println("The first 40 unsafe primes are: ", unsafeprimes[1:40])
println("There are ", format(sum(map(x -> x < 1000000, unsafeprimes)), commas=true), " unsafe primes less than 1 million.")
println("There are ", format(length(unsafeprimes), commas=true), " unsafe primes less than 10 million.")
end
parseprimelist()
</lang>
- Output:
The first 35 unsafe primes are: [5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619] There are 4,324 safe primes less than 1 million. There are 30,657 safe primes less than 10 million. The first 40 unsafe primes are: [2, 3, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233] There are 74,174 unsafe primes less than 1 million. There are 633,922 unsafe primes less than 10 million.
Kotlin
<lang scala>// Version 1.2.70
fun sieve(limit: Int): BooleanArray {
// True denotes composite, false denotes prime.
val c = BooleanArray(limit + 1) // all false by default
c[0] = true
c[1] = true
// apart from 2 all even numbers are of course composite
for (i in 4..limit step 2) c[i] = true
var p = 3 // start from 3
while (true) {
val p2 = p * p
if (p2 > limit) break
for (i in p2..limit step 2 * p) c[i] = true
while (true) {
p += 2
if (!c[p]) break
}
}
return c
}
fun main(args: Array<String>) {
// sieve up to 10 million
val sieved = sieve(10_000_000)
val safe = IntArray(35)
var count = 0
var i = 3
while (count < 35) {
if (!sieved[i] && !sieved[(i - 1) / 2]) safe[count++] = i
i += 2
}
println("The first 35 safe primes are:")
println(safe.joinToString(" ","[", "]\n"))
count = 0
for (j in 3 until 1_000_000 step 2) {
if (!sieved[j] && !sieved[(j - 1) / 2]) count++
}
System.out.printf("The number of safe primes below 1,000,000 is %,d\n\n", count)
for (j in 1_000_001 until 10_000_000 step 2) {
if (!sieved[j] && !sieved[(j - 1) / 2]) count++
}
System.out.printf("The number of safe primes below 10,000,000 is %,d\n\n", count)
val unsafe = IntArray(40)
unsafe[0] = 2 // since (2 - 1)/2 is not prime
count = 1
i = 3
while (count < 40) {
if (!sieved[i] && sieved[(i - 1) / 2]) unsafe[count++] = i
i += 2
}
println("The first 40 unsafe primes are:")
println(unsafe.joinToString(" ","[", "]\n"))
count = 1
for (j in 3 until 1_000_000 step 2) {
if (!sieved[j] && sieved[(j - 1) / 2]) count++
}
System.out.printf("The number of unsafe primes below 1,000,000 is %,d\n\n", count)
for (j in 1_000_001 until 10_000_000 step 2) {
if (!sieved[j] && sieved[(j - 1) / 2]) count++
}
System.out.printf("The number of unsafe primes below 10,000,000 is %,d\n\n", count)
}</lang>
- Output:
The first 35 safe primes are: [5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619] The number of safe primes below 1,000,000 is 4,324 The number of safe primes below 10,000,000 is 30,657 The first 40 unsafe primes are: [2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233] The number of unsafe primes below 1,000,000 is 74,174 The number of unsafe primes below 10,000,000 is 633,922
Lua
<lang lua>-- FUNCS: local function T(t) return setmetatable(t, {__index=table}) end table.filter = function(t,f) local s=T{} for _,v in ipairs(t) do if f(v) then s[#s+1]=v end end return s end table.map = function(t,f,...) local s=T{} for _,v in ipairs(t) do s[#s+1]=f(v,...) end return s end table.firstn = function(t,n) local s=T{} n=n>#t and #t or n for i = 1,n do s[i]=t[i] end return s end
-- SIEVE: local sieve, safe, unsafe, floor, N = {}, T{}, T{}, math.floor, 10000000 for i = 2,N do sieve[i]=true end for i = 2,N do if sieve[i] then for j=i*i,N,i do sieve[j]=nil end end end for i = 2,N do if sieve[i] then local t=sieve[floor((i-1)/2)] and safe or unsafe t[#t+1]=i end end
-- TASKS: local function commafy(i) return tostring(i):reverse():gsub("(%d%d%d)","%1,"):reverse():gsub("^,","") end print("First 35 safe primes : " .. safe:firstn(35):map(commafy):concat(" ")) print("# safe primes < 1,000,000 : " .. commafy(#safe:filter(function(v) return v<1e6 end))) print("# safe primes < 10,000,000 : " .. commafy(#safe)) print("First 40 unsafe primes : " .. unsafe:firstn(40):map(commafy):concat(" ")) print("# unsafe primes < 1,000,000 : " .. commafy(#unsafe:filter(function(v) return v<1e6 end))) print("# unsafe primes < 10,000,000: " .. commafy(#unsafe))</lang>
- Output:
First 35 safe primes : 5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1,019 1,187 1,283 1,307 1,319 1,367 1,439 1,487 1,523 1,619 # safe primes < 1,000,000 : 4,324 # safe primes < 10,000,000 : 30,657 First 40 unsafe primes : 2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233 # unsafe primes < 1,000,000 : 74,174 # unsafe primes < 10,000,000: 633,922
Maple
<lang Maple>showSafePrimes := proc(n::posint) local prime_list, k; prime_list := [5]; for k to n - 1 do
prime_list := [op(prime_list), NumberTheory:-NextSafePrime(prime_list[-1])];
end do; return prime_list; end proc;
showUnsafePrimes := proc(n::posint) local prime_num, k; prime_num := [2]; for k to n-1 do
prime_num := [op(prime_num), nextprime(prime_num[-1])];
end do; return remove(x -> member(x, showSafePrimes(n)), prime_num); end proc:
countSafePrimes := proc(n::posint) local counts, prime; counts := 0; prime := 5; while prime < n do prime := NumberTheory:-NextSafePrime(prime);
counts := counts + 1;
end do; return counts; end proc;
countUnsafePrimes := proc(n::posint) local safe_counts, total; safe_counts := countSafePrimes(n); total := NumberTheory:-PrimeCounting(n); return total - safe_counts; end proc;
showSafePrimes(35); showUnsafePrimes(40); countSafePrimes(1000000); countSafePrimes(10000000); countUnsafePrimes(1000000); countUnsafePrimes(10000000);</lang>
- Output:
[5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619] [2, 3, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173] 4324 30657 74174 633922
Pascal
Using unit mp_prime of Wolfgang Erhardt ( RIP ) , of which I use two sieve, to simplify things. Generating small primes and checked by the second, which starts to run 2x ahead.Sieving of consecutive prime number is much faster than primality check. <lang pascal>program Sophie; { Find and count Sophie Germain primes } { uses unit mp_prime out of mparith of Wolfgang Ehrhardt
http://wolfgang-ehrhardt.de/mp_intro.html }
{$APPTYPE CONSOLE} uses
mp_prime,sysutils;
var
pS0,pS1:TSieve;
procedure SafeOrNoSavePrimeOut(totCnt:NativeInt;CntSafe:boolean); var
cnt,pr,pSG,testPr : NativeUint;
begin
prime_sieve_reset(pS0,1); prime_sieve_reset(pS1,1); cnt := 0;
// memorize prime of the sieve, because sometimes prime_sieve_next(pS1) is to far ahead.
testPr := prime_sieve_next(pS1);
IF CntSafe then
Begin
writeln('First ',totCnt,' safe primes');
repeat
pr := prime_sieve_next(pS0);
pSG := 2*pr+1;
while testPr< pSG do
testPr := prime_sieve_next(pS1);
if pSG = testPr then
begin
write(pSG,',');
inc(cnt);
end;
until cnt >= totCnt
end
else
Begin
writeln('First ',totCnt,' unsafe primes');
repeat
pr := prime_sieve_next(pS0);
pSG := (pr-1) DIV 2;
while testPr< pSG do
testPr := prime_sieve_next(pS1);
if pSG <> testPr then
begin
write(pr,',');
inc(cnt);
end;
until cnt >= totCnt;
end;
writeln(#8,#32);
end;
function CountSafePrimes(Limit:NativeInt):NativeUint; var
cnt,pr,pSG,testPr : NativeUint;
begin
prime_sieve_reset(pS0,1);
prime_sieve_reset(pS1,1);
cnt := 0;
testPr := 0;
repeat
pr := prime_sieve_next(pS0);
pSG := 2*pr+1;
while testPr< pSG do
testPr := prime_sieve_next(pS1);
if pSG = testPr then
inc(cnt);
until pSG >= Limit;
CountSafePrimes := cnt;
end;
procedure CountSafePrimesOut(Limit:NativeUint); Begin
writeln('there are ',CountSafePrimes(limit),' safe primes out of ',
primepi32(limit),' primes up to ',Limit);
end;
procedure CountUnSafePrimesOut(Limit:NativeUint); var
prCnt: NativeUint;
Begin
prCnt := primepi32(limit);
writeln('there are ',prCnt-CountSafePrimes(limit),' unsafe primes out of ',
prCnt,' primes up to ',Limit);
end;
var
T1,T0 : INt64;
begin
T0 :=gettickcount64; prime_sieve_init(pS0,1); prime_sieve_init(pS1,1);
//Find and display (on one line) the first 35 safe primes.
SafeOrNoSavePrimeOut(35,true);
//Find and display the count of the safe primes below 1,000,000.
CountSafePrimesOut(1000*1000);
//Find and display the count of the safe primes below 10,000,000.
CountSafePrimesOut(10*1000*1000);
//Find and display (on one line) the first 40 unsafe primes.
SafeOrNoSavePrimeOut(40,false);
//Find and display the count of the unsafe primes below 1,000,000.
CountUnSafePrimesOut(1000*1000);
//Find and display the count of the unsafe primes below 10,000,000.
CountUnSafePrimesOut(10*1000*1000);
writeln;
CountSafePrimesOut(1000*1000*1000);
T1 :=gettickcount64;
writeln('runtime ',T1-T0,' ms');
end.</lang>
- Output:
First 35 safe primes 5,7,11,23,47,59,83,107,167,179,227,263,347,359,383,467,479,503,563,587,719,839,863,887,983,1019,1187,1283,1307,1319,1367,1439,1487,1523,1619 there are 4324 safe primes out of 78498 primes up to 1000000 there are 30657 safe primes out of 664579 primes up to 10000000 First 40 unsafe primes 2,3,13,17,19,29,31,37,41,43,53,61,67,71,73,79,89,97,101,103,109,113,127,131,137,139,149,151,157,163,173,181,191,193,197,199,211,223,229,233 there are 74174 unsafe primes out of 78498 primes up to 1000000 there are 633922 unsafe primes out of 664579 primes up to 10000000 there are 1775676 safe primes out of 50847534 primes up to 1000000000 runtime 2797 ms
Perl
The module ntheory does fast prime generation and testing.
<lang perl>use ntheory qw(forprimes is_prime);
my $upto = 1e7; my %class = ( safe => [], unsafe => [2] );
forprimes {
push @{$class{ is_prime(($_-1)>>1) ? 'safe' : 'unsafe' }}, $_;
} 3, $upto;
for (['safe', 35], ['unsafe', 40]) {
my($type, $quantity) = @$_;
print "The first $quantity $type primes are:\n";
print join(" ", map { comma($class{$type}->[$_-1]) } 1..$quantity), "\n";
for my $q ($upto/10, $upto) {
my $n = scalar(grep { $_ <= $q } @{$class{$type}});
printf "The number of $type primes up to %s: %s\n", comma($q), comma($n);
}
}
sub comma {
(my $s = reverse shift) =~ s/(.{3})/$1,/g;
$s =~ s/,(-?)$/$1/;
$s = reverse $s;
}</lang>
- Output:
The first 35 safe primes are: 5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1,019 1,187 1,283 1,307 1,319 1,367 1,439 1,487 1,523 1,619 The number of safe primes up to 1,000,000: 4,324 The number of safe primes up to 10,000,000: 30,657 The first 40 unsafe primes are: 2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233 The number of unsafe primes up to 1,000,000: 74,174 The number of unsafe primes up to 10,000,000: 633,922
Phix
<lang Phix>sequence safe = {}, unsafe = {} function filter_range(integer lo, hi)
while true do
integer p = get_prime(lo)
if p>hi then return lo end if
if p>2 and is_prime((p-1)/2) then
safe &= p
else
unsafe &= p
end if
lo += 1
end while
end function integer lo = filter_range(1,1_000_000),
ls = length(safe),
lu = length(unsafe)
{} = filter_range(lo,10_000_000) printf(1,"The first 35 safe primes: %v\n",{safe[1..35]}) printf(1,"Count of safe primes below 1,000,000: %,d\n",ls) printf(1,"Count of safe primes below 10,000,000: %,d\n",length(safe)) printf(1,"The first 40 unsafe primes: %v\n",{unsafe[1..40]}) printf(1,"Count of unsafe primes below 1,000,000: %,d\n",lu) printf(1,"Count of unsafe primes below 10,000,000: %,d\n",length(unsafe))</lang>
- Output:
The first 35 safe primes: {5,7,11,23,47,59,83,107,167,179,227,263,347,359,383,467,479,503,563,587,719,839,863,887,983,1019,1187,1283,1307,1319,1367,1439,1487,1523,1619}
Count of safe primes below 1,000,000: 4,324
Count of safe primes below 10,000,000: 30,657
The first 40 unsafe primes: {2,3,13,17,19,29,31,37,41,43,53,61,67,71,73,79,89,97,101,103,109,113,127,131,137,139,149,151,157,163,173,181,191,193,197,199,211,223,229,233}
Count of unsafe primes below 1,000,000: 74,174
Count of unsafe primes below 10,000,000: 633,922
Python
<lang Python> primes =[] sp =[] usp=[] n = 10000000 if 2<n:
primes.append(2)
for i in range(3,n+1,2):
for j in primes:
if(j>i/2) or (j==primes[-1]):
primes.append(i)
if((i-1)/2) in primes:
sp.append(i)
break
else:
usp.append(i)
break
if (i%j==0):
break
print('First 35 safe primes are:\n' , sp[:35]) print('There are '+str(len(sp[:1000000]))+' safe primes below 1,000,000') print('There are '+str(len(sp))+' safe primes below 10,000,000') print('First 40 unsafe primes:\n',usp[:40]) print('There are '+str(len(usp[:1000000]))+' unsafe primes below 1,000,000') print('There are '+str(len(usp))+' safe primes below 10,000,000') </lang>
- Output:
First 35 safe primes: [5,7,11,23,47,59,83,107,167,179,227,263,347,359,383,467,479,503,563,587,719,839,863,887,983,1019,1187,1283,1307,1319,1367,1439,1487,1523,1619] There are 4,234 safe primes below 1,000,000 There are 30,657 safe primes below 10,000,000 First 40 unsafe primes: [2,3,13,17,19,29,31,37,41,43,53,61,67,71,73,79,89,97,101,103,109,113,127,131,137,139,149,151,157,163,173,181,191,193,197,199,211,223,229,233] There are 74,174 unsafe primes below 1,000,000 There are 633,922 unsafe primes below 10,000,000
Raku
(formerly Perl 6)
Raku has a built-in method .is-prime to test for prime numbers. It's great for testing individual numbers or to find/filter a few thousand numbers, but when you are looking for millions, it becomes a drag. No fear, the Raku ecosystem has a fast prime sieve module available which can produce 10 million primes in a few seconds. Once we have the primes, it is just a small matter of filtering and formatting them appropriately.
<lang perl6>sub comma { $^i.flip.comb(3).join(',').flip }
use Math::Primesieve;
my $sieve = Math::Primesieve.new;
my @primes = $sieve.primes(10_000_000);
my %filter = @primes.Set;
my $primes = @primes.classify: { %filter{($_ - 1)/2} ?? 'safe' !! 'unsafe' };
for 'safe', 35, 'unsafe', 40 -> $type, $quantity {
say "The first $quantity $type primes are:";
say $primes{$type}[^$quantity]».,
say "The number of $type primes up to {comma $_}: ",
comma $primes{$type}.first(* > $_, :k) // +$primes{$type} for 1e6, 1e7;
say ;
}</lang>
- Output:
The first 35 safe primes are: (5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1,019 1,187 1,283 1,307 1,319 1,367 1,439 1,487 1,523 1,619) The number of safe primes up to 1,000,000: 4,324 The number of safe primes up to 10,000,000: 30,657 The first 40 unsafe primes are: (2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233) The number of unsafe primes up to 1,000,000: 74,174 The number of unsafe primes up to 10,000,000: 633,922
REXX
<lang rexx>/*REXX program lists a sequence (or a count) of ──safe── or ──unsafe── primes. */ parse arg N kind _ . 1 . okind; upper kind /*obtain optional arguments from the CL*/ if N== | N=="," then N= 35 /*Not specified? Then assume default.*/ if kind== | kind=="," then kind= 'SAFE' /* " " " " " */ if _\== then call ser 'too many arguments specified.' if kind\=='SAFE' & kind\=='UNSAFE' then call ser 'invalid 2nd argument: ' okind if kind =='UNSAFE' then safe= 0; else safe= 1 /*SAFE is a binary value for function.*/ w = linesize() - 1 /*obtain the usable width of the term. */ tell= (N>0); @.=; N= abs(N) /*N is negative? Then don't display. */ !.=0; !.1=2; !.2=3; !.3=5; !.4=7; !.5=11; !.6=13; !.7=17; !.8=19; #= 8 @.=; @.2=1; @.3=1; @.5=1; @.7=1; @.11=1; @.13=1; @.17=1; @.19=1; start= # + 1 m= 0; lim=0 /*# is the number of low primes so far*/ $=; do i=1 for # while lim<=N; j= !.i /* [↓] find primes, and maybe show 'em*/
call safeUnsafe; $= strip($) /*go see if other part of a KIND prime.*/
end /*i*/ /* [↑] allows faster loop (below). */
/* [↓] N: default lists up to 35 #'s.*/
do j=!.#+2 by 2 while lim<N /*continue on with the next odd prime. */
if j // 3 == 0 then iterate /*is this integer a multiple of three? */
parse var j -1 _ /*obtain the last decimal digit of J */
if _ == 5 then iterate /*is this integer a multiple of five? */
if j // 7 == 0 then iterate /* " " " " " " seven? */
if j //11 == 0 then iterate /* " " " " " " eleven?*/
if j //13 == 0 then iterate /* " " " " " " 13 ? */
if j //17 == 0 then iterate /* " " " " " " 17 ? */
if j //19 == 0 then iterate /* " " " " " " 19 ? */
/* [↓] divide by the primes. ___ */
do k=start to # while !.k * !.k<=j /*divide J by other primes ≤ √ J */
if j // !.k ==0 then iterate j /*÷ by prev. prime? ¬prime ___ */
end /*k*/ /* [↑] only divide up to √ J */
#= # + 1 /*bump the count of number of primes. */
!.#= j; @.j= 1 /*define a prime and its index value.*/
call safeUnsafe /*go see if other part of a KIND prime.*/
end /*j*/
/* [↓] display number of primes found.*/
if $\== then say $ /*display any residual primes in $ list*/ say if tell then say commas(m)' ' kind "primes found."
else say commas(m)' ' kind "primes found below or equal to " commas(N).
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ add: m= m+1; lim= m; if \tell & j>N then do; lim= j; m= m-1; end; else call app; return 1 app: if tell then if length($ j)>w then do; say $; $ =j; end; else $= $ j; return 1 ser: say; say; say '***error***' arg(1); say; say; exit 13 /*tell error message. */ commas: parse arg _; do jc=length(_)-3 to 1 by -3; _=insert(',', _, jc); end; return _ /*──────────────────────────────────────────────────────────────────────────────────────*/ safeUnsafe: ?= (j-1) % 2 /*obtain the other part of KIND prime. */
if safe then if @.? == then return 0 /*not a safe prime.*/
else return add() /*is " " " */
else if @.? == then return add() /*is an unsafe prime.*/
else return 0 /*not " " " */</lang>
This REXX program makes use of LINESIZE REXX program (or BIF) which is used to determine the screen width (or linesize) of the terminal (console). Some REXXes don't have this BIF.
The LINESIZE.REX REXX program is included here ───► LINESIZE.REX.
- output when using the default input of: 35
Shown at 5/6 size.)
5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619 35 SAFE primes found.
- output when using the input: -1000000
4,324 SAFE primes found below or equal to 1,000,000.
- output when using the input: -10000000
30,657 SAFE primes found below or equal to 10,000,000.
- output when using the input: 40 unsafe
(Shown at 5/6 size.)
2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233 40 UNSAFE primes found.
- output when using the input: -1000000 unsafe
74,174 UNSAFE primes found below or equal to 1,000,000.
- output when using the input: -10000000
633,922 UNSAFE primes found below or equal to 10,000,000.
Ring
<lang ring> load "stdlib.ring"
see "working..." + nl
p = 1 num = 0 limit1 = 36 limit2 = 41 safe1 = 1000000 safe2 = 10000000
see "the first 35 Safeprimes are: " + nl while true
p = p + 1
p2 = (p-1)/2
if isprime(p) and isprime(p2)
num = num + 1
if num < limit1
see " " + p
else
exit
ok
ok
end
see nl + "the first 40 Unsafeprimes are: " + nl p = 1 num = 0 while true
p = p + 1
p2 = (p-1)/2
if isprime(p) and not isprime(p2)
num = num + 1
if num < limit2
see " " + p
else
exit
ok
ok
end
p = 1 num1 = 0 num2 = 0 while true
p = p + 1
p2 = (p-1)/2
if isprime(p) and isprime(p2)
if p < safe1
num1 = num1 + 1
ok
if p < safe2
num2 = num2 + 1
else
exit
ok
ok
end
see nl + "safe primes below 1,000,000: " + num1 + nl see "safe primes below 10,000,000: " + num2 + nl
p = 1 num1 = 0 num2 = 0 while true
p = p + 1
p2 = (p-1)/2
if isprime(p) and not isprime(p2)
if p < safe1
num1 = num1 + 1
ok
if p < safe2
num2 = num2 + 1
else
exit
ok
ok
end
see "unsafe primes below 1,000,000: " + num1 + nl see "unsafe primes below 10,000,000: " + num2 + nl
see "done..." + nl </lang> Output:
working... the first 35 Safeprimes are: 5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619 the first 40 Unsafeprimes are: 2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233 safe primes below 1,000,000: 4324 safe primes below 10,000,000: 30657 unsafe primes below 1,000,000: 74174 unsafe primes below 10,000,000: 633922 done...
Ruby
<lang ruby>require "prime" class Integer
def safe_prime? #assumes prime ((self-1)/2).prime? end
end
def format_parts(n)
partitions = Prime.each(n).partition(&:safe_prime?).map(&:count)
"There are %d safes and %d unsafes below #{n}."% partitions
end
puts "First 35 safe-primes:" p Prime.each.lazy.select(&:safe_prime?).take(35).to_a puts format_parts(1_000_000), "\n"
puts "First 40 unsafe-primes:" p Prime.each.lazy.reject(&:safe_prime?).take(40).to_a puts format_parts(10_000_000) </lang>
- Output:
First 35 safe-primes: [5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619] There are 4324 safes and 74174 unsafes below 1000000. First 40 unsafe-primes: [2, 3, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233] There are 30657 safes and 633922 unsafes below 10000000.
Sidef
<lang ruby>func is_safeprime(p) {
is_prime(p) && is_prime((p-1)/2)
}
func is_unsafeprime(p) {
is_prime(p) && !is_prime((p-1)/2)
}
func safeprime_count(from, to) {
from..to -> count_by(is_safeprime)
}
func unsafeprime_count(from, to) {
from..to -> count_by(is_unsafeprime)
}
say "First 35 safe-primes:" say (1..Inf -> lazy.grep(is_safeprime).first(35).join(' ')) say say "First 40 unsafe-primes:" say (1..Inf -> lazy.grep(is_unsafeprime).first(40).join(' ')) say say "There are #{safeprime_count(1, 1e6)} safe-primes bellow 10^6" say "There are #{unsafeprime_count(1, 1e6)} unsafe-primes bellow 10^6" say say "There are #{safeprime_count(1, 1e7)} safe-primes bellow 10^7" say "There are #{unsafeprime_count(1, 1e7)} unsafe-primes bellow 10^7"</lang>
- Output:
First 35 safe-primes: 5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619 First 40 unsafe-primes: 2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233 There are 4324 safe-primes bellow 10^6 There are 74174 unsafe-primes bellow 10^6 There are 30657 safe-primes bellow 10^7 There are 633922 unsafe-primes bellow 10^7
Simula
<lang simula> BEGIN
CLASS BOOLARRAY(N); INTEGER N;
BEGIN
BOOLEAN ARRAY DATA(0:N-1);
END BOOLARRAY;
CLASS INTARRAY(N); INTEGER N;
BEGIN
INTEGER ARRAY DATA(0:N-1);
END INTARRAY;
REF(BOOLARRAY) PROCEDURE SIEVE(LIMIT);
INTEGER LIMIT;
BEGIN
REF(BOOLARRAY) C;
INTEGER P, P2;
LIMIT := LIMIT+1;
COMMENT TRUE DENOTES COMPOSITE, FALSE DENOTES PRIME. ;
C :- NEW BOOLARRAY(LIMIT); COMMENT ALL FALSE BY DEFAULT ;
C.DATA(0) := TRUE;
C.DATA(1) := TRUE;
COMMENT APART FROM 2 ALL EVEN NUMBERS ARE OF COURSE COMPOSITE ;
FOR I := 4 STEP 2 UNTIL LIMIT-1 DO
C.DATA(I) := TRUE;
COMMENT START FROM 3. ;
P := 3;
WHILE TRUE DO BEGIN
P2 := P * P;
IF P2 >= LIMIT THEN BEGIN
GO TO OUTER_BREAK;
END;
I := P2;
WHILE I < LIMIT DO BEGIN
C.DATA(I) := TRUE;
I := I + 2 * P;
END;
WHILE TRUE DO BEGIN
P := P + 2;
IF NOT C.DATA(P) THEN BEGIN
GO TO INNER_BREAK;
END;
END;
INNER_BREAK:
END;
OUTER_BREAK:
SIEVE :- C;
END SIEVE;
COMMENT MAIN BLOCK ;
REF(BOOLARRAY) SIEVED; REF(INTARRAY) UNSAFE, SAFE; INTEGER I, COUNT;
COMMENT SIEVE UP TO 10 MILLION ; SIEVED :- SIEVE(10000000);
SAFE :- NEW INTARRAY(35);
COUNT := 0;
I := 3;
WHILE COUNT < 35 DO BEGIN
IF NOT SIEVED.DATA(I) AND NOT SIEVED.DATA((I-1)//2) THEN BEGIN
SAFE.DATA(COUNT) := I;
COUNT := COUNT+1;
END;
I := I+2;
END;
OUTTEXT("THE FIRST 35 SAFE PRIMES ARE:");
OUTIMAGE;
OUTCHAR('[');
FOR I := 0 STEP 1 UNTIL 35-1 DO BEGIN
IF I>0 THEN OUTCHAR(' ');
OUTINT(SAFE.DATA(I), 0);
END;
OUTCHAR(']');
OUTIMAGE;
OUTIMAGE;
COUNT := 0;
FOR I := 3 STEP 2 UNTIL 1000000 DO BEGIN
IF NOT SIEVED.DATA(I) AND NOT SIEVED.DATA((I-1)//2) THEN BEGIN
COUNT := COUNT+1;
END;
END;
OUTTEXT("THE NUMBER OF SAFE PRIMES BELOW 1,000,000 IS ");
OUTINT(COUNT, 0);
OUTIMAGE;
OUTIMAGE;
FOR I := 1000001 STEP 2 UNTIL 10000000 DO BEGIN
IF NOT SIEVED.DATA(I) AND NOT SIEVED.DATA((I-1)//2) THEN
COUNT := COUNT+1;
END;
OUTTEXT("THE NUMBER OF SAFE PRIMES BELOW 10,000,000 IS ");
OUTINT(COUNT, 0);
OUTIMAGE;
OUTIMAGE;
UNSAFE :- NEW INTARRAY(40);
UNSAFE.DATA(0) := 2; COMMENT SINCE (2 - 1)/2 IS NOT PRIME ;
COUNT := 1;
I := 3;
WHILE COUNT < 40 DO BEGIN
IF NOT SIEVED.DATA(I) AND SIEVED.DATA((I-1)//2) THEN BEGIN
UNSAFE.DATA(COUNT) := I;
COUNT := COUNT+1;
END;
I := I+2;
END;
OUTTEXT("THE FIRST 40 UNSAFE PRIMES ARE:");
OUTIMAGE;
OUTCHAR('[');
FOR I := 0 STEP 1 UNTIL 40-1 DO BEGIN
IF I>0 THEN OUTCHAR(' ');
OUTINT(UNSAFE.DATA(I), 0);
END;
OUTCHAR(']');
OUTIMAGE;
OUTIMAGE;
COUNT := 1;
FOR I := 3 STEP 2 UNTIL 1000000 DO BEGIN
IF NOT SIEVED.DATA(I) AND SIEVED.DATA((I-1)//2) THEN
COUNT := COUNT+1;
END;
OUTTEXT("THE NUMBER OF UNSAFE PRIMES BELOW 1,000,000 IS ");
OUTINT(COUNT, 0);
OUTIMAGE;
OUTIMAGE;
FOR I := 1000001 STEP 2 UNTIL 10000000 DO BEGIN
IF NOT SIEVED.DATA(I) AND SIEVED.DATA((I-1)//2) THEN
COUNT := COUNT+1;
END;
OUTTEXT("THE NUMBER OF UNSAFE PRIMES BELOW 10,000,000 IS ");
OUTINT(COUNT, 0);
OUTIMAGE;
END
</lang>
- Output:
THE FIRST 35 SAFE PRIMES ARE: [5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619] THE NUMBER OF SAFE PRIMES BELOW 1,000,000 IS 4324 THE NUMBER OF SAFE PRIMES BELOW 10,000,000 IS 30657 THE FIRST 40 UNSAFE PRIMES ARE: [2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233] THE NUMBER OF UNSAFE PRIMES BELOW 1,000,000 IS 74174 THE NUMBER OF UNSAFE PRIMES BELOW 10,000,000 IS 633922
Smalltalk
<lang smalltalk>[
| isSafePrime printFirstNElements |
isSafePrime := [:p | ((p-1)//2) isPrime].
printFirstNElements :=
[:coll :n |
(coll to:n)
do:[:p | Transcript show:p]
separatedBy:[Transcript space]
].
(Iterator on:[:b | Integer primesUpTo:10000000 do:b])
partition:isSafePrime
into:[:savePrimes :unsavePrimes |
|nSaveBelow1M nSaveBelow10M nUnsaveBelow1M nUnsaveBelow10M|
nSaveBelow1M := savePrimes count:[:p | p < 1000000].
nSaveBelow10M := savePrimes size.
nUnsaveBelow1M := unsavePrimes count:[:p | p < 1000000].
nUnsaveBelow10M := unsavePrimes size.
Transcript showCR: 'first 35 safe primes:'.
printFirstNElements value:savePrimes value:35.
Transcript cr.
Transcript show: 'safe primes below 1,000,000: '.
Transcript showCR:nSaveBelow1M printStringWithThousandsSeparator.
Transcript show: 'safe primes below 10,000,000: '.
Transcript showCR:nSaveBelow10M printStringWithThousandsSeparator.
Transcript showCR: 'first 40 unsafe primes:'.
printFirstNElements value:unsavePrimes value:40.
Transcript cr.
Transcript show: 'unsafe primes below 1,000,000: '.
Transcript showCR:nUnsaveBelow1M printStringWithThousandsSeparator.
Transcript show: 'unsafe primes below 10,000,000: '.
Transcript showCR:nUnsaveBelow10M printStringWithThousandsSeparator.
]
] benchmark:'runtime: safe primes'</lang>
- Output:
first 35 safe primes: 5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619 safe primes below 1,000,000: 4,324 safe primes below 10,000,000: 30,657 first 40 unsafe primes: 2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233 unsafe primes below 1,000,000: 74,174 unsafe primes below 10,000,000: 633,922 runtime: safe primes: 996ms
Notes:
1) partition:into: is a method in collection which is a combined select+reject.
2) instead if the Iterator, I could have also used "(Integer primesUpTo:10000000) partition...",
but that would use a few additional Mb of temporary memory for the primes collection, whereas the iterator simply computes and enumerates them (without actually collecting them).
But, who cares, these days ;-)
3) time is on a 2012 MacBook 2.5Ghz i5; interpreted not jitted. Compiled/jitted time is 738ms.
Swift
<lang swift>import Foundation
class PrimeSieve {
var composite: [Bool]
init(size: Int) {
composite = Array(repeating: false, count: size/2)
var p = 3
while p * p <= size {
if !composite[p/2 - 1] {
let inc = p * 2
var q = p * p
while q <= size {
composite[q/2 - 1] = true
q += inc
}
}
p += 2
}
}
func isPrime(number: Int) -> Bool {
if number < 2 {
return false
}
if (number & 1) == 0 {
return number == 2
}
return !composite[number/2 - 1]
}
}
func commatize(_ number: Int) -> String {
let n = NSNumber(value: number) return NumberFormatter.localizedString(from: n, number: .decimal)
}
let limit1 = 1000000 let limit2 = 10000000
class PrimeInfo {
let maxPrint: Int
var count1: Int
var count2: Int
var primes: [Int]
init(maxPrint: Int) {
self.maxPrint = maxPrint
count1 = 0
count2 = 0
primes = []
}
func addPrime(prime: Int) {
count2 += 1
if prime < limit1 {
count1 += 1
}
if count2 <= maxPrint {
primes.append(prime)
}
}
func printInfo(name: String) {
print("First \(maxPrint) \(name) primes: \(primes)")
print("Number of \(name) primes below \(commatize(limit1)): \(commatize(count1))")
print("Number of \(name) primes below \(commatize(limit2)): \(commatize(count2))")
}
}
var safePrimes = PrimeInfo(maxPrint: 35) var unsafePrimes = PrimeInfo(maxPrint: 40)
let sieve = PrimeSieve(size: limit2)
for prime in 2..<limit2 {
if sieve.isPrime(number: prime) {
if sieve.isPrime(number: (prime - 1)/2) {
safePrimes.addPrime(prime: prime)
} else {
unsafePrimes.addPrime(prime: prime)
}
}
}
safePrimes.printInfo(name: "safe") unsafePrimes.printInfo(name: "unsafe")</lang>
- Output:
First 35 safe primes: [5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619] Number of safe primes below 1,000,000: 4,324 Number of safe primes below 10,000,000: 30,657 First 40 unsafe primes: [2, 3, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233] Number of unsafe primes below 1,000,000: 74,174 Number of unsafe primes below 10,000,000: 633,922
Visual Basic .NET
Dependent on using .NET Core 2.1 or 2.0, or .NET Framework 4.7.2
<lang vbnet>Imports System.Console
Namespace safety
Module SafePrimes
Dim pri_HS As HashSet(Of Integer) = Primes(10_000_000).ToHashSet()
Sub Main()
For Each UnSafe In {False, True} : Dim n As Integer = If(UnSafe, 40, 35)
WriteLine($"The first {n} {If(UnSafe, "un", "")}safe primes are:")
WriteLine(String.Join(" ", pri_HS.Where(Function(p) UnSafe Xor
pri_HS.Contains(p \ 2)).Take(n)))
Next : Dim limit As Integer = 1_000_000 : Do
Dim part = pri_HS.TakeWhile(Function(l) l < limit),
sc As Integer = part.Count(Function(p) pri_HS.Contains(p \ 2))
WriteLine($"Of the primes below {limit:n0}: {sc:n0} are safe, and {part.Count() -
sc:n0} are unsafe.") : If limit = 1_000_000 Then limit *= 10 Else Exit Do
Loop
End Sub
Private Iterator Function Primes(ByVal bound As Integer) As IEnumerable(Of Integer)
If bound < 2 Then Return
Yield 2
Dim composite As BitArray = New BitArray((bound - 1) \ 2)
Dim limit As Integer = (CInt((Math.Sqrt(bound))) - 1) \ 2
For i As Integer = 0 To limit - 1 : If composite(i) Then Continue For
Dim prime As Integer = 2 * i + 3 : Yield prime
Dim j As Integer = (prime * prime - 2) \ 2
While j < composite.Count : composite(j) = True : j += prime : End While
Next
For i As integer = limit To composite.Count - 1 : If Not composite(i) Then Yield 2 * i + 3
Next
End Function
End Module
End Namespace</lang> If not using the latest version of the System.Linq namespace, you can implement the Enumerable.ToHashSet() method by adding <lang vbnet>Imports System.Runtime.CompilerServices</lang> to the top and this module inside the safety namespace:<lang vbnet> Module Extensions
<Extension()>
Function ToHashSet(Of T)(ByVal src As IEnumerable(Of T), ByVal Optional _
IECmp As IEqualityComparer(Of T) = Nothing) As HashSet(Of T)
Return New HashSet(Of T)(src, IECmp)
End Function
End Module</lang>
- Output:
The first 35 safe primes are: 5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1019 1187 1283 1307 1319 1367 1439 1487 1523 1619 The first 40 unsafe primes are: 2 3 13 17 19 29 31 37 41 43 53 61 67 71 73 79 89 97 101 103 109 113 127 131 137 139 149 151 157 163 173 181 191 193 197 199 211 223 229 233 Of the primes below 1,000,000: 4,324 are safe, and 74,174 are unsafe. Of the primes below 10,000,000: 30,657 are safe, and 633,922 are unsafe.
zkl
Using GMP (GNU Multiple Precision Arithmetic Library, probabilistic primes), because it is easy and fast to generate primes.
Extensible prime generator#zkl could be used instead. <lang zkl>var [const] BI=Import("zklBigNum"); // libGMP // saving 664,578 primes (vs generating them on the fly) seems a bit overkill
fcn safePrime(p){ ((p-1)/2).probablyPrime() } // p is a BigInt prime
fcn safetyList(sN,nsN){
p,safe,notSafe := BI(2),List(),List();
do{
if(safePrime(p)) safe.append(p.toInt()) else notSafe.append(p.toInt());
p.nextPrime();
}while(safe.len()<sN or notSafe.len()<nsN);
println("The first %d safe primes are: %s".fmt(sN,safe[0,sN].concat(",")));
println("The first %d unsafe primes are: %s".fmt(nsN,notSafe[0,nsN].concat(",")));
}(35,40);</lang>
- Output:
The first 35 safe primes are: 5,7,11,23,47,59,83,107,167,179,227,263,347,359,383,467,479,503,563,587,719,839,863,887,983,1019,1187,1283,1307,1319,1367,1439,1487,1523,1619 The first 40 unsafe primes are: 2,3,13,17,19,29,31,37,41,43,53,61,67,71,73,79,89,97,101,103,109,113,127,131,137,139,149,151,157,163,173,181,191,193,197,199,211,223,229,233
safetyList could also be written as: <lang zkl>println("The first %d safe primes are: %s".fmt(N:=35,
Walker(BI(1).nextPrime) // gyrate (vs Walker.filter) because p mutates
.pump(N,String,safePrime,Void.Filter,String.fp1(","))));
println("The first %d unsafe primes are: %s".fmt(N=40,
Walker(BI(1).nextPrime) // or save as List
.pump(N,List,safePrime,'==(False),Void.Filter,"toInt").concat(",")));</lang>
Time to count: <lang zkl>fcn safetyCount(N,s=0,ns=0,p=BI(2)){
do{
if(safePrime(p)) s+=1; else ns+=1;
p.nextPrime()
}while(p<N);
println("The number of safe primes below %10,d is %7,d".fmt(N,s));
println("The number of unsafe primes below %10,d is %7,d".fmt(N,ns));
return(s,ns,p);
}
s,ns,p := safetyCount(1_000_000); println(); safetyCount(10_000_000,s,ns,p);</lang>
- Output:
The number of safe primes below 1,000,000 is 4,324 The number of unsafe primes below 1,000,000 is 74,174 The number of safe primes below 10,000,000 is 30,657 The number of unsafe primes below 10,000,000 is 633,922