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Mersenne primes

From Rosetta Code
Mersenne primes 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.
Task

Create code that will list (preferably calculate) all of the   Mersenne primes   until some limitation is reached.


The number of   known   Mersenne primes is   51   (as of June, 2020),   and the largest known Mersenne prime contains contains   24,862,048   decimal digits.


Also see



11l[edit]

Translation of: D
F is_prime(BigInt bi)
I bi < 2 {R 0B}
I bi % 2 == 0 {R bi == 2}
I bi % 3 == 0 {R bi == 3}
 
V test = BigInt(5)
L test * test < bi
I bi % test == 0
R 0B
test += 2
I bi % test == 0
R 0B
test += 4
 
R 1B
 
V base = BigInt(2)
L(p) 1..31
I is_prime(base - 1)
print(‘2 ^ ’p‘ - 1’)
base *= 2
Output:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1

AppleScript[edit]

 
on isPrime(integ)
set isComposite to ""
if (integ / 2) = (integ / 2 div 1) then
log integ & " is composite because 2 is a factor" as string --buttons {"OK", "Cancel"} default button 1 cancel button 2
 
else
set x to 2
set sqrtOfInteg to integ ^ 0.5
repeat until x = integ ^ 0.5 + 1 as integer
if (integ / x) = integ / x div 1 then
log integ & " is composite because " & x & " & " & (integ / x div 1) & " are factors" as string --buttons {"OK", "Cancel"} default button 1 cancel button 2
set isComposite to 1
set x to x + 1
else
 
set x to x + 1
end if
 
 
 
end repeat
log integ & " is prime" as string --buttons {"OK", "Cancel"} default button 1 cancel button 2
if isComposite = 1 then
log integ & "is composite"
else
display dialog integ
end if
end if
 
end isPrime
set x to 2
repeat
isPrime(((2 ^ x) - 1) div 1)
set x to x + 1
end repeat
 

AWK[edit]

 
# syntax: GAWK --bignum -f MERSENNE_PRIMES.AWK
BEGIN {
base = 2
for (i=1; i<62; i++) {
if (is_prime(base-1)) {
printf("2 ^ %d - 1\n",i)
}
base *= 2
}
exit(0)
}
function is_prime(n, d) {
d = 5
if (n < 2) { return(0) }
if (n % 2 == 0) { return(n == 2) }
if (n % 3 == 0) { return(n == 3) }
while (d*d <= n) {
if (n % d == 0) { return(0) }
d += 2
if (n % d == 0) { return(0) }
d += 4
}
return(1)
}
 
Output:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1
2 ^ 61 - 1

C[edit]

#include <stdbool.h>
#include <stdint.h>
#include <stdio.h>
 
bool isPrime(uint64_t n) {
uint64_t test;
 
if (n < 2) return false;
if (n % 2 == 0) return n == 2;
if (n % 3 == 0) return n == 3;
 
test = 5;
while (test * test < n) {
if (n % test == 0) return false;
test += 2;
if (n % test == 0) return false;
test += 4;
}
 
return true;
}
 
int main() {
uint64_t base = 2;
int pow;
 
for (pow = 1; pow <= 32; pow++) {
if (isPrime(base - 1)) {
printf("2 ^ %d - 1\n", pow);
}
base *= 2;
}
 
return 0;
}
Output:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1

C++[edit]

Translation of: C
#include <iostream>
 
bool isPrime(uint64_t n) {
if (n < 2) return false;
if (n % 2 == 0) return n == 2;
if (n % 3 == 0) return n == 3;
 
uint64_t test = 5;
while (test * test < n) {
if (n % test == 0) return false;
test += 2;
if (n % test == 0) return false;
test += 4;
}
 
return true;
}
 
int main() {
uint64_t base = 2;
 
for (int pow = 1; pow <= 32; pow++) {
if (isPrime(base - 1)) {
std::cout << "2 ^ " << pow << " - 1\n";
}
base *= 2;
}
 
return 0;
}
Output:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1

C#[edit]

Needs a better primality checking algorithm to do really large prime numbers.

using System;
using System.Numerics;
 
namespace MersennePrimes {
class Program {
static BigInteger Sqrt(BigInteger x) {
if (x < 0) throw new ArgumentException("Negative argument.");
if (x < 2) return x;
BigInteger y = x / 2;
while (y > x / y) {
y = ((x / y) + y) / 2;
}
return y;
}
 
static bool IsPrime(BigInteger bi) {
if (bi < 2) return false;
if (bi % 2 == 0) return bi == 2;
if (bi % 3 == 0) return bi == 3;
if (bi % 5 == 0) return bi == 5;
if (bi % 7 == 0) return bi == 7;
if (bi % 11 == 0) return bi == 11;
if (bi % 13 == 0) return bi == 13;
if (bi % 17 == 0) return bi == 17;
if (bi % 19 == 0) return bi == 19;
 
BigInteger limit = Sqrt(bi);
BigInteger test = 23;
while (test < limit) {
if (bi % test == 0) return false;
test += 2;
if (bi % test == 0) return false;
test += 4;
}
 
return true;
}
 
static void Main(string[] args) {
const int MAX = 9;
 
int pow = 2;
int count = 0;
 
while (true) {
if (IsPrime(pow)) {
BigInteger p = BigInteger.Pow(2, pow) - 1;
if (IsPrime(p)) {
Console.WriteLine("2 ^ {0} - 1", pow);
if (++count >= MAX) {
break;
}
}
}
pow++;
}
}
}
}
Output:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1
2 ^ 61 - 1

D[edit]

Simplest thing that could possibly work. Using better primality tests will allow for more results to be calculated in a reasonable amount of time.

import std.bigint;
import std.stdio;
 
bool isPrime(BigInt bi) {
if (bi < 2) return false;
if (bi % 2 == 0) return bi == 2;
if (bi % 3 == 0) return bi == 3;
 
auto test = BigInt(5);
while (test * test < bi) {
if (bi % test == 0) return false;
test += 2;
if (bi % test == 0) return false;
test += 4;
}
 
return true;
}
 
void main() {
auto base = BigInt(2);
 
for (int pow=1; pow<32; pow++) {
if (isPrime(base-1)) {
writeln("2 ^ ", pow, " - 1");
}
base *= 2;
}
}
Output:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1

F#[edit]

Translation of: C#
open System
open System.Numerics
 
let Sqrt (n:BigInteger) =
if n < (BigInteger 0) then raise (ArgumentException "Negative argument.")
if n < (BigInteger 2) then n
else
let rec H v r s =
if v < s then
r
else
H (v - s) (r + (BigInteger 1)) (s + (BigInteger 2))
H n (BigInteger 0) (BigInteger 1)
 
let IsPrime (n:BigInteger) =
if n < (BigInteger 2) then false
elif n % (BigInteger 2) = (BigInteger 0) then n = (BigInteger 2)
elif n % (BigInteger 3) = (BigInteger 0) then n = (BigInteger 3)
elif n % (BigInteger 5) = (BigInteger 0) then n = (BigInteger 5)
elif n % (BigInteger 7) = (BigInteger 0) then n = (BigInteger 7)
elif n % (BigInteger 11) = (BigInteger 0) then n = (BigInteger 11)
elif n % (BigInteger 13) = (BigInteger 0) then n = (BigInteger 13)
elif n % (BigInteger 17) = (BigInteger 0) then n = (BigInteger 17)
elif n % (BigInteger 19) = (BigInteger 0) then n = (BigInteger 19)
else
let limit = (Sqrt n)
let rec H t =
if t <= limit then
if n % t = (BigInteger 0) then false
else
let t2 = t + (BigInteger 2)
if n % t2 = (BigInteger 0) then false
else H (t2 + (BigInteger 4))
else
true
H (BigInteger 23)
 
[<EntryPoint>]
let main _ =
let MAX = BigInteger 9
 
let rec loop (pow:int) (count:int) =
if IsPrime (BigInteger pow) then
let p = BigInteger.Pow((BigInteger 2), pow) - (BigInteger 1)
if IsPrime p then
printfn "2 ^ %A - 1" pow
if (BigInteger (count + 1)) >= MAX then count
else loop (pow + 1) (count + 1)
else loop (pow + 1) count
else loop (pow + 1) count
 
loop 2 0 |> ignore
 
0 // return an integer exit code
Output:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1
2 ^ 61 - 1

Factor[edit]

Factor comes with a Lucas-Lehmer primality test.

USING: formatting math.primes.lucas-lehmer math.ranges sequences ;
 
: mersennes-upto ( n -- seq ) [1,b] [ lucas-lehmer ] filter ;
 
3500 mersennes-upto [ "2 ^ %d - 1\n" printf ] each
Output:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1
2 ^ 61 - 1
2 ^ 89 - 1
2 ^ 107 - 1
2 ^ 127 - 1
2 ^ 521 - 1
2 ^ 607 - 1
2 ^ 1279 - 1
2 ^ 2203 - 1
2 ^ 2281 - 1
2 ^ 3217 - 1

Fortran[edit]

Translation of: C
 
program mersenne
use iso_fortran_env, only: output_unit, INT64
implicit none
 
integer, parameter :: l=INT64
integer(kind=l) :: base
integer :: pow
 
base = 2
 
do pow = 1, 32
if (is_prime(base-1)) then
write(output_unit,'(A2,x,I0,x,A3)') "2^", pow, "- 1"
end if
base = base * 2
end do
contains
pure function is_prime(n)
integer(kind=l), intent(in) :: n
logical :: is_prime
integer(kind=l) :: test
 
is_prime = .false.
if (n < 2) return
if (modulo(n, 2) == 0) then
is_prime = n==2
return
end if
if (modulo(n, 3) == 0) then
is_prime = n==3
return
end if
 
test = 5
do
if (test**2 >= n) then
is_prime = .true.
return
end if
 
if (modulo(n, test) == 0) return
test = test + 2
if (modulo(n, test) == 0) return
test = test + 4
end do
end function is_prime
end program mersenne
 
Output:
2^ 2 - 1
2^ 3 - 1
2^ 5 - 1
2^ 7 - 1
2^ 13 - 1
2^ 17 - 1
2^ 19 - 1
2^ 31 - 1

Frink[edit]

Frink has built-in routines for iterating through prime numbers. Frink's isPrime[n] function automatically detects numbers of the form 2n-1 and performs a more efficient Lucas-Lehmer primality test on the number. This works with arbitrarily large numbers.

Did you know that all Java-based JVM languages are many many orders of magnitude faster because Frink's developer contributed vastly faster BigInteger algorithms to Java? It took the Java developers 11 years to integrate them but they became part of 1.8 and later! Operations that used to take days now can be done in seconds thanks to Frink's contributions to Java.

for n = primes[]
if isPrime[2^n - 1]
println["2^$n - 1"]
Output:
2^2 - 1
2^3 - 1
2^5 - 1
2^7 - 1
2^13 - 1
2^17 - 1
2^19 - 1
2^31 - 1
2^61 - 1
2^89 - 1
2^107 - 1
2^127 - 1
2^521 - 1
2^607 - 1
2^1279 - 1
2^2203 - 1
2^2281 - 1
2^3217 - 1
2^4253 - 1
2^4423 - 1
2^9689 - 1
2^9941 - 1
2^11213 - 1
2^19937 - 1
2^21701 - 1
...

Go[edit]

The github.com/ncw/gmp package is a drop-in replacement for Go's math/big package. It's a CGo wrapper around the C GMP library and under these circumstances is two to four times as fast as the native Go package. Editing just the import line you can use whichever is more convenient for you (CGo has drawbacks, including limited portability). Normally build tags would be used to control this instead of editing imports in the source, but this keeps the example simpler.

Note that the use of ProbablyPrime(0) requires Go 1.8 or later. When using the math/big package, passing a parameter of zero to this method forces it to apply only the Baillie-PSW test to check for primality. This is 100% accurate for numbers up to 2^64 and at the time of writing (June 2018) no known composite number above that bound passes the test.

package main
 
import (
"fmt"
"time"
 
// Use one or the other of these:
"math/big"
//big "github.com/ncw/gmp"
)
 
func main() {
start := time.Now()
one := big.NewInt(1)
mp := big.NewInt(0)
bp := big.NewInt(0)
const max = 22
for count, p := 0, uint(2); count < max; {
mp.Lsh(one, p)
mp.Sub(mp, one)
if mp.ProbablyPrime(0) {
elapsed := time.Since(start).Seconds()
if elapsed >= 0.01 {
fmt.Printf("2 ^ %-4d - 1 took %6.2f secs\n", p, elapsed)
} else {
fmt.Printf("2 ^ %-4d - 1\n", p)
}
count++
}
for {
if p > 2 {
p += 2
} else {
p = 3
}
bp.SetUint64(uint64(p))
if bp.ProbablyPrime(0) {
break
}
}
}
}
Output using the GMP package on a 3.4 GHz Xeon E3-1245:
2 ^ 2    - 1
2 ^ 3    - 1
2 ^ 5    - 1
2 ^ 7    - 1
2 ^ 13   - 1
2 ^ 17   - 1
2 ^ 19   - 1
2 ^ 31   - 1
2 ^ 61   - 1
2 ^ 89   - 1
2 ^ 107  - 1
2 ^ 127  - 1
2 ^ 521  - 1
2 ^ 607  - 1
2 ^ 1279 - 1 took   0.05 secs
2 ^ 2203 - 1 took   0.38 secs
2 ^ 2281 - 1 took   0.44 secs
2 ^ 3217 - 1 took   1.53 secs
2 ^ 4253 - 1 took   4.39 secs
2 ^ 4423 - 1 took   5.02 secs
2 ^ 9689 - 1 took  73.78 secs
2 ^ 9941 - 1 took  81.24 secs

(A previous run on more modest hardware - Celeron N3050 @ 1.60GHz × 2 - was ~365 seconds for M9941.)

This can be sped up quite a bit for modern multi-core CPUs by some simple changes to use goroutines.

package main
 
import (
"fmt"
"runtime"
"time"
 
// Use one or the other of these:
"math/big"
//big "github.com/ncw/gmp"
)
 
func main() {
start := time.Now()
 
nworkers := runtime.GOMAXPROCS(0)
fmt.Println("Using", nworkers, "workers.")
workC := make(chan uint, 1)
resultC := make(chan uint, nworkers)
 
// Generate possible Mersenne exponents and send them to workC.
go func() {
workC <- 2
bp := big.NewInt(0)
for p := uint(3); ; p += 2 {
// Possible exponents must be prime.
bp.SetUint64(uint64(p))
if bp.ProbablyPrime(0) {
workC <- p
}
}
}()
 
// Start up worker go routines, each takes
// possible Mersenne exponents from workC as `p`
// and if 2^p-1 is prime sends `p` to resultC.
one := big.NewInt(1)
for i := 0; i < nworkers; i++ {
go func() {
mp := big.NewInt(0)
for p := range workC {
mp.Lsh(one, p)
mp.Sub(mp, one)
if mp.ProbablyPrime(0) {
resultC <- p
}
}
}()
}
 
// Receive some maximum number of Mersenne prime exponents
// from resultC and show the Mersenne primes.
const max = 24
for count := 0; count < max; count++ {
// Note: these could come back out of order, although usually
// only the first few. If that is an issue, correcting it is
// left as an excercise to the reader :).
p := <-resultC
elapsed := time.Since(start).Seconds()
if elapsed >= 0.01 {
fmt.Printf("2 ^ %-5d - 1 took %6.2f secs\n", p, elapsed)
} else {
fmt.Printf("2 ^ %-5d - 1\n", p)
}
}
}
Output using the GMP package on the same 3.4 GHz Xeon E3-1245 (4 core × 2 SMT threads) as above:
Using 8 workers.
2 ^ 2     - 1
2 ^ 5     - 1
2 ^ 3     - 1
2 ^ 7     - 1
2 ^ 13    - 1
2 ^ 19    - 1
2 ^ 61    - 1
2 ^ 31    - 1
2 ^ 107   - 1
2 ^ 17    - 1
2 ^ 127   - 1
2 ^ 89    - 1
2 ^ 521   - 1
2 ^ 607   - 1
2 ^ 1279  - 1 took   0.01 secs
2 ^ 2203  - 1 took   0.09 secs
2 ^ 2281  - 1 took   0.12 secs
2 ^ 3217  - 1 took   0.36 secs
2 ^ 4253  - 1 took   0.94 secs
2 ^ 4423  - 1 took   1.06 secs
2 ^ 9689  - 1 took  16.28 secs
2 ^ 9941  - 1 took  18.02 secs
2 ^ 11213 - 1 took  26.76 secs
2 ^ 19937 - 1 took 194.16 secs

Using this approach, the Celeron machine (dual core) takes ~180 seconds to reach M9941 and ~270 seconds to reach M11213.

Haskell[edit]

import Data.Numbers.Primes (primes)
import Text.Printf (printf)
 
lucasLehmer :: Int -> Bool
lucasLehmer p = iterate f 4 !! p-2 == 0
where
f b = (b^2 - 2) `mod` m
m = 2^p - 1
 
main = mapM_ (printf "M %d\n") $ take 20 mersenne
where
mersenne = filter lucasLehmer primes
Output:
M 3
M 5
M 7
M 13
M 17
M 19
M 31
M 61
M 89
M 107
M 127
M 521
M 607
M 1279
M 2203
M 2281
M 3217
M 4253
M 4423
M 9689

Java[edit]

Translation of: Kotlin
import java.math.BigInteger;
 
public class MersennePrimes {
private static final int MAX = 20;
 
private static final BigInteger ONE = BigInteger.ONE;
private static final BigInteger TWO = BigInteger.valueOf(2);
 
private static boolean isPrime(int n) {
if (n < 2) return false;
if (n % 2 == 0) return n == 2;
if (n % 3 == 0) return n == 3;
int d = 5;
while (d * d <= n) {
if (n % d == 0) return false;
d += 2;
if (n % d == 0) return false;
d += 4;
}
return true;
}
 
public static void main(String[] args) {
int count = 0;
int p = 2;
while (true) {
BigInteger m = TWO.shiftLeft(p - 1).subtract(ONE);
if (m.isProbablePrime(10)) {
System.out.printf("2 ^ %d - 1\n", p);
if (++count == MAX) break;
}
// obtain next prime, p
do {
p = (p > 2) ? p + 2 : 3;
} while (!isPrime(p));
}
}
}
Output:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1
2 ^ 61 - 1
2 ^ 89 - 1
2 ^ 107 - 1
2 ^ 127 - 1
2 ^ 521 - 1
2 ^ 607 - 1
2 ^ 1279 - 1
2 ^ 2203 - 1
2 ^ 2281 - 1
2 ^ 3217 - 1
2 ^ 4253 - 1
2 ^ 4423 - 1

Julia[edit]

Works with: Julia version 0.6

Julia module Primes uses Miller-Rabin primality test.

using Primes
 
mersenne(n::Integer) = convert(typeof(n), 2) ^ n - one(n)
function main(nmax::Integer)
n = ith = zero(nmax)
while ith ≤ nmax
if isprime(mersenne(n))
println("M$n")
ith += 1
end
n += 1
end
end
 
main(big(20))
Output:
M2
M3
M5
M7
M13
M17
M19
M31
M61
M89
M107
M127
M521
M607
M1279
M2203
M2281
M3217
M4253
M4423
M9689

Kotlin[edit]

This task is similar to the Lucas-Lehmer test task except that you can use whatever method you like to test the primality of the Mersenne numbers. Here, I've chosen to use the JDK's BigInteger.isProbablePrime(certainty) method. The exact algorithm is implementation dependent --- GNU classpath uses only Miller-Rabin, while Oracle JDK uses Miller-Rabin and sometimes adds a Lucas test (this is not the Lucas-Lehmer test).

A 'certainty' parameter of 10 is enough to find the first 20 Mersenne primes but as even this takes about 90 seconds on my modest machine I've not bothered going beyond that.

// version 1.2.10
 
import java.math.BigInteger
 
const val MAX = 20
 
val bigOne = BigInteger.ONE
val bigTwo = 2.toBigInteger()
 
/* for checking 'small' primes */
fun isPrime(n: Int): Boolean {
if (n < 2) return false
if (n % 2 == 0) return n == 2
if (n % 3 == 0) return n == 3
var d : Int = 5
while (d * d <= n) {
if (n % d == 0) return false
d += 2
if (n % d == 0) return false
d += 4
}
return true
}
 
fun main(args: Array<String>) {
var count = 0
var p = 2
while (true) {
val m = (bigTwo shl (p - 1)) - bigOne
if (m.isProbablePrime(10)) {
println("2 ^ $p - 1")
if (++count == MAX) break
}
// obtain next prime, p
while(true) {
p = if (p > 2) p + 2 else 3
if (isPrime(p)) break
}
}
}
Output:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1
2 ^ 61 - 1
2 ^ 89 - 1
2 ^ 107 - 1
2 ^ 127 - 1
2 ^ 521 - 1
2 ^ 607 - 1
2 ^ 1279 - 1
2 ^ 2203 - 1
2 ^ 2281 - 1
2 ^ 3217 - 1
2 ^ 4253 - 1
2 ^ 4423 - 1

Lua[edit]

This checks for primality using a trial division function. The limitation is 'until p == p + 1', meaning that the program will halt when Lua's number type (a 64-bit floating point value) no longer has enough precision to distiguish between one integer and the next.

-- Returns a boolean to show whether x is prime
function isPrime (x)
if x < 2 then return false end
if x < 4 then return true end
if x % 2 == 0 then return false end
for d = 3, math.sqrt(x), 2 do
if x % d == 0 then return false end
end
return true
end
 
-- Main procedure
local i, p = 0
repeat
i = i + 1
p = 2 ^ i - 1
if isPrime(p) then
print("2 ^ " .. i .. " - 1 = " .. p)
end
until p == p + 1
Output:
2 ^ 2 - 1 = 3
2 ^ 3 - 1 = 7
2 ^ 5 - 1 = 31
2 ^ 7 - 1 = 127
2 ^ 13 - 1 = 8191
2 ^ 17 - 1 = 131071
2 ^ 19 - 1 = 524287
2 ^ 31 - 1 = 2147483647


Nim[edit]

Using only standard library[edit]

If we want to use only the standard library, we are limited to 64 bits. So we used a simple primality test.

func isOddPrime(n: uint64): bool =
if n == 1: return false
if n mod 3 == 0: return n == 3
var d = 5u
while d * d <= n:
if n mod d == 0: return false
inc d, 2
if n mod d == 0: return false
inc d, 4
result = true
 
var p = 2u64
for e in 1..63:
if isOddPrime(p - 1):
echo "2^", e, " - 1"
p *= 2
Output:
2^2 - 1
2^3 - 1
2^5 - 1
2^7 - 1
2^13 - 1
2^17 - 1
2^19 - 1
2^31 - 1
2^61 - 1

Using big integers[edit]

Library: bignum

The module bignum provides big integers and a probabilistic primality test. We searched the Mersenne numbers for exponents between 1 and 10_000.

import bignum
 
var p = newInt(2)
for e in 1..10_000:
if probablyPrime(p - 1, 25) != 0:
echo "2^", e, " - 1"
p *= 2
Output:
2^2 - 1
2^3 - 1
2^5 - 1
2^7 - 1
2^13 - 1
2^17 - 1
2^19 - 1
2^31 - 1
2^61 - 1
2^89 - 1
2^107 - 1
2^127 - 1
2^521 - 1
2^607 - 1
2^1279 - 1
2^2203 - 1
2^2281 - 1
2^3217 - 1
2^4253 - 1
2^4423 - 1
2^9689 - 1
2^9941 - 1

PARI/GP[edit]

LL(p)={
my(m=Mod(4,1<<p-1));
for(i=3,p,m=m^2-2);
m==0
};
forprime(p=2,, if(LL(p), print("2^"p"-1")))

Perl[edit]

Since GIMPS went to the trouble of dedicating thousands of CPU years to finding Mersenne primes, we should be kind enough to use the results. The ntheory module front end does this, so the results up to 43 million is extremely fast (4 seconds), and we can reduce this another 10x by only checking primes. After the GIMPS double-checked mark, a Lucas-Lehmer test is done using code similar to Rosetta Code Lucas-Lehmer in C+GMP.

If this is too contrived, we can use Math::Prime::Util::GMP::is_mersenne_prime instead, which will run the Lucas-Lehmer test on each input. The first 23 Mersenne primes are found in under 15 seconds.

Library: ntheory
use ntheory qw/forprimes is_mersenne_prime/;
forprimes { is_mersenne_prime($_) && say } 1e9;
Output:
2
3
5
7
13
17
19
31
61
...

Phix[edit]

Library: Phix/mpfr
with javascript_semantics
include mpfr.e
atom t0 = time()
mpz mp = mpz_init(),
    bp = mpz_init()
integer p = 0, count = 0
constant lim = iff(platform()=JS?14:17)
while true do
    mpz_ui_pow_ui(mp,2,p)
    mpz_sub_ui(mp,mp,1)
    if mpz_prime(mp) then
        string s = iff(time()-t0<1?"":", "&elapsed(time()-t0))
        printf(1, "2^%d-1%s\n",{p,s})
        count += 1
        if count>=lim then exit end if
    end if
    while true do
        p = iff(p>2?p+2:3)
        mpz_set_si(bp,p)
        if mpz_prime(bp) then exit end if
    end while   
end while
{mp,bp} = mpz_free({mp,bp})
Output:
2^3-1
2^5-1
2^7-1
2^13-1
2^17-1
2^19-1
2^31-1
2^61-1
2^89-1
2^107-1
2^127-1
2^521-1
2^607-1
2^1279-1
2^2203-1, 2.5s
2^2281-1, 2.9s
2^3217-1, 9.5s

PHP[edit]

<?php
 
function is_prime($n) {
if ($n <= 3) {
return $n > 1;
} elseif (($n % 2 == 0) or ($n % 3 == 0)) {
return false;
}
$i = 5;
while ($i * $i <= $n) {
if ($n % $i == 0) {
return false;
}
$i += 2;
if ($n % $i == 0) {
return false;
}
$i += 4;
}
return true;
}
 
for ($i = 0 ; $i <= 63 ; $i++) {
$pow = pow(2, $i) - 1;
$mersenne = is_prime($pow);
if ($mersenne) {
echo '2 ^ ', $i, ' - 1', PHP_EOL;
}
}
Output:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1
2 ^ 61 - 1

PicoLisp[edit]

(de **Mod (X Y N)
(let M 1
(loop
(when (bit? 1 Y)
(setq M (% (* M X) N)) )
(T (=0 (setq Y (>> 1 Y)))
M )
(setq X (% (* X X) N)) ) ) )
(de isprime (N)
(cache '(NIL) N
(if (== N 2)
T
(and
(> N 1)
(bit? 1 N)
(let (Q (dec N) N1 (dec N) K 0 X)
(until (bit? 1 Q)
(setq
Q (>> 1 Q)
K (inc K) ) )
(catch 'composite
(do 16
(loop
(setq X
(**Mod
(rand 2 (min (dec N) 1000000000000))
Q
N ) )
(T (or (=1 X) (= X N1)))
(T
(do K
(setq X (**Mod X 2 N))
(when (=1 X) (throw 'composite))
(T (= X N1) T) ) )
(throw 'composite) ) )
(throw 'composite T) ) ) ) ) ) )
(for N 1000
(and
(isprime (dec (** 2 N)))
(prinl "2 \^ " N " - 1") ) )
Output:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1
2 ^ 61 - 1
2 ^ 89 - 1
2 ^ 107 - 1
2 ^ 127 - 1
2 ^ 521 - 1
2 ^ 607 - 1

Pike[edit]

int power = 1;
while(power++) {
int candidate = 2->pow(power)-1;
if( candidate->probably_prime_p() )
write("2 ^ %d - 1\n", power);
}

Output:

2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1
2 ^ 61 - 1

Prolog[edit]

Lucas-Lehmer test, works with SWI Prolog

 
lucas_lehmer_seq(M, L) :-
lazy_list(ll_iter, 4-M, L).
 
ll_iter(S-M, T-M, T) :-
T is ((S*S) - 2) mod M.
 
drop(N, Lz1, Lz2) :-
append(Pfx, Lz2, Lz1), length(Pfx, N), !.
 
mersenne_prime(2).
mersenne_prime(P) :-
P > 2,
M is (1 << P) - 1,
lucas_lehmer_seq(M, Residues),
Skip is P - 3, drop(Skip, Residues, [R|_]),
R =:= 0.
 
Output:
?- findall(X, (between(1, 1000, X), mersenne_prime(X)), L), write(L).
[2,3,5,7,13,17,19,31,61,89,107,127,521,607]
L = [2, 3, 5, 7, 13, 17, 19, 31, 61|...].

Python[edit]

Translation of: Java
import random
 
#Take from https://www.codeproject.com/Articles/691200/%2FArticles%2F691200%2FPrimality-test-algorithms-Prime-test-The-fastest-w
def MillerRabinPrimalityTest(number):
'''
because the algorithm input is ODD number than if we get
even and it is the number 2 we return TRUE ( spcial case )
if we get the number 1 we return false and any other even
number we will return false.
'''

if number == 2:
return True
elif number == 1 or number % 2 == 0:
return False
 
''' first we want to express n as : 2^s * r ( were r is odd ) '''
 
''' the odd part of the number '''
oddPartOfNumber = number - 1
 
''' The number of time that the number is divided by two '''
timesTwoDividNumber = 0
 
''' while r is even divid by 2 to find the odd part '''
while oddPartOfNumber % 2 == 0:
oddPartOfNumber = oddPartOfNumber / 2
timesTwoDividNumber = timesTwoDividNumber + 1
 
'''
since there are number that are cases of "strong liar" we
need to check more then one number
'''

for time in range(3):
 
''' choose "Good" random number '''
while True:
''' Draw a RANDOM number in range of number ( Z_number ) '''
randomNumber = random.randint(2, number)-1
if randomNumber != 0 and randomNumber != 1:
break
 
''' randomNumberWithPower = randomNumber^oddPartOfNumber mod number '''
randomNumberWithPower = pow(randomNumber, oddPartOfNumber, number)
 
''' if random number is not 1 and not -1 ( in mod n ) '''
if (randomNumberWithPower != 1) and (randomNumberWithPower != number - 1):
# number of iteration
iterationNumber = 1
 
''' while we can squre the number and the squered number is not -1 mod number'''
while (iterationNumber <= timesTwoDividNumber - 1) and (randomNumberWithPower != number - 1):
''' squre the number '''
randomNumberWithPower = pow(randomNumberWithPower, 2, number)
 
# inc the number of iteration
iterationNumber = iterationNumber + 1
'''
if x != -1 mod number then it because we did not found strong witnesses
hence 1 have more then two roots in mod n ==>
n is composite ==> return false for primality
'''

if (randomNumberWithPower != (number - 1)):
return False
 
''' well the number pass the tests ==> it is probably prime ==> return true for primality '''
return True
 
# Main
MAX = 20
p = 2
count = 0
while True:
m = (2 << (p - 1)) - 1
if MillerRabinPrimalityTest(m):
print "2 ^ {} - 1".format(p)
count = count + 1
if count == MAX:
break
# obtain next prime, p
while True:
p = p + 2 if (p > 2) else 3
if MillerRabinPrimalityTest(p):
break
print "done"
Output:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1
2 ^ 61 - 1
2 ^ 89 - 1
2 ^ 107 - 1
2 ^ 127 - 1
2 ^ 521 - 1
2 ^ 607 - 1
2 ^ 1279 - 1
2 ^ 2203 - 1
2 ^ 2281 - 1
2 ^ 3217 - 1
2 ^ 4253 - 1
2 ^ 4423 - 1
done

Raku[edit]

(formerly Perl 6)

Works with: Rakudo version 2018.01

We already have a multitude of tasks that demonstrate how to find Mersenne primes; Prime decomposition, Primality by trial division, Trial factoring of a Mersenne number, Lucas-Lehmer test, Miller–Rabin primality_test, etc. that all have Raku entries. I'm not sure what I could add here that would be useful.

Hmmm.

Create code that will list all of the Mersenne primes until some limitation is reached.

It doesn't specify to calculate them, only to list them; why throw away all of the computer millenia of processing power that the GIMPS has invested?

use HTTP::UserAgent;
use Gumbo;
 
my $table = parse-html(HTTP::UserAgent.new.get('https://www.mersenne.org/primes/').content, :TAG<table>);
 
say 'All known Mersenne primes as of ', Date(now);
 
say 'M', ++$, ": 2$_ - 1"
for $table[1]».[*][0][*].comb(/'exp_lo='\d+/)».subst(/\D/, '',:g)
.trans([<0123456789>.comb] => [<⁰¹²³⁴⁵⁶⁷⁸⁹>.comb]).words;
 
Output:
All known Mersenne primes as of 2018-12-21
M1: 2² - 1
M2: 2³ - 1
M3: 2⁵ - 1
M4: 2⁷ - 1
M5: 2¹³ - 1
M6: 2¹⁷ - 1
M7: 2¹⁹ - 1
M8: 2³¹ - 1
M9: 2⁶¹ - 1
M10: 2⁸⁹ - 1
M11: 2¹⁰⁷ - 1
M12: 2¹²⁷ - 1
M13: 2⁵²¹ - 1
M14: 2⁶⁰⁷ - 1
M15: 2¹²⁷⁹ - 1
M16: 2²²⁰³ - 1
M17: 2²²⁸¹ - 1
M18: 2³²¹⁷ - 1
M19: 2⁴²⁵³ - 1
M20: 2⁴⁴²³ - 1
M21: 2⁹⁶⁸⁹ - 1
M22: 2⁹⁹⁴¹ - 1
M23: 2¹¹²¹³ - 1
M24: 2¹⁹⁹³⁷ - 1
M25: 2²¹⁷⁰¹ - 1
M26: 2²³²⁰⁹ - 1
M27: 2⁴⁴⁴⁹⁷ - 1
M28: 2⁸⁶²⁴³ - 1
M29: 2¹¹⁰⁵⁰³ - 1
M30: 2¹³²⁰⁴⁹ - 1
M31: 2²¹⁶⁰⁹¹ - 1
M32: 2⁷⁵⁶⁸³⁹ - 1
M33: 2⁸⁵⁹⁴³³ - 1
M34: 2¹²⁵⁷⁷⁸⁷ - 1
M35: 2¹³⁹⁸²⁶⁹ - 1
M36: 2²⁹⁷⁶²²¹ - 1
M37: 2³⁰²¹³⁷⁷ - 1
M38: 2⁶⁹⁷²⁵⁹³ - 1
M39: 2¹³⁴⁶⁶⁹¹⁷ - 1
M40: 2²⁰⁹⁹⁶⁰¹¹ - 1
M41: 2²⁴⁰³⁶⁵⁸³ - 1
M42: 2²⁵⁹⁶⁴⁹⁵¹ - 1
M43: 2³⁰⁴⁰²⁴⁵⁷ - 1
M44: 2³²⁵⁸²⁶⁵⁷ - 1
M45: 2³⁷¹⁵⁶⁶⁶⁷ - 1
M46: 2⁴²⁶⁴³⁸⁰¹ - 1
M47: 2⁴³¹¹²⁶⁰⁹ - 1
M48: 2⁵⁷⁸⁸⁵¹⁶¹ - 1
M49: 2⁷⁴²⁰⁷²⁸¹ - 1
M50: 2⁷⁷²³²⁹¹⁷ - 1
M51: 2⁸²⁵⁸⁹⁹³³ - 1

REXX[edit]

This REXX version   (using a 32-bit Regina REXX interpreter)   will find those Mersenne primes which are less than
8 million decimal digits   (which would be M43).

/*REXX program uses  exponent─and─mod  operator to test possible Mersenne numbers.      */
do j=1; /*process a range, or run out of time.*/
if \isPrime(j) then iterate /*if J isn't a prime, keep plugging.*/
r= testMer(j) /*If J is prime, give J the 3rd degree.*/
if r==0 then say right('M'j, 10) "──────── is a Mersenne prime."
else say right('M'j, 50) "is composite, a factor:" r
end /*j*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
isPrime: procedure; parse arg x; if wordpos(x, '2 3 5 7') \== 0 then return 1
if x<11 then return 0; if x//2 == 0 | x//3 == 0 then return 0
do j=5 by 6; if x//j == 0 | x//(j+2) == 0 then return 0
if j*j>x then return 1 /*◄─┐ ___ */
end /*j*/ /* └─◄ Is j>√ x ? Then return 1*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
iSqrt: procedure; parse arg x; #= 1; r= 0; do while #<=x; #=#*4; end
do while #>1; #=#%4; _= x-r-#; r= r%2; if _>=0 then do; x=_; r=r+#; end
end /*while*/ /*iSqrt ≡ integer square root.*/
return r /*───── ─ ── ─ ─ */
/*──────────────────────────────────────────────────────────────────────────────────────*/
testMer: procedure; parse arg x; p =2**x /* [↓] do we have enough digits?*/
$$=x2b( d2x(x) ) + 0
if pos('E',p)\==0 then do; parse var p "E" _; numeric digits _+2; p=2**x; end
 !.=1;  !.1=0;  !.7=0 /*array used for a quicker test. */
R=iSqrt(p) /*obtain integer square root of P*/
do k=2 by 2; q=k*x + 1 /*(shortcut) compute value of Q. */
m=q // 8 /*obtain the remainder when ÷ 8. */
if !.m then iterate /*M must be either one or seven.*/
parse var q '' -1 _; if _==5 then iterate /*last digit a five?*/
if q// 3==0 then iterate /* ÷ by three? */
if q// 7==0 then iterate /* " " seven? */
if q//11==0 then iterate /* " " eleven?*/
/* ____ */
if q>R then return 0 /*Is q>√2**x ? A Mersenne prime*/
sq=1; $=$$ /*obtain binary version from $. */
do until $==''; sq=sq*sq
parse var $ _ 2 $ /*obtain 1st digit and the rest. */
if _ then sq=(sq+sq) // q
end /*until*/
if sq==1 then return q /*Not a prime? Return a factor.*/
end /*k*/



Ring[edit]

 
# Project : Mersenne primes
 
n = 0
while true
n = n +1
if isprime(pow(2,n)-1) = 1
see n + nl
ok
end
 
func isprime num
if (num <= 1) return 0 ok
if (num % 2 = 0) and num != 2 return 0 ok
for i = 3 to floor(num / 2) -1 step 2
if (num % i = 0) return 0 ok
next
return 1
 

Output:

2
3
5
7
13
17
19

Scala[edit]

 
object MersennePrimes extends App {
 
val primes = primeSieve(LazyList.from(2))
val upbPrime = 9941
 
def primeSieve(s: LazyList[Int]): LazyList[Int] =
s.head #:: primeSieve(s.tail filter {
_ % s.head != 0
})
 
def mersenne(p: Int): BigInt = (BigInt(2) pow p) - 1
 
def s(mp: BigInt, p: Int): BigInt = {
if (p == 1) 4 else ((s(mp, p - 1) pow 2) - 2) % mp
}
println(s"Finding Mersenne primes in M[2..$upbPrime]")
((primes takeWhile (_ <= upbPrime)).map { p => (p, mersenne(p)) }
map { p => if (p._1 == 2) (p, 0) else (p, s(p._2, p._1 - 1)) } filter {
_._2 == 0
})
.foreach { p =>
println(s"prime M${(p._1)._1}: " + {
if ((p._1)._1 < 200) (p._1)._2 else s"(${(p._1)._2.toString.length} digits)"
})
}
println("That's All Folks!")
}
 

Sidef[edit]

Uses the is_mersenne_prime() function from Math::Prime::Util::GMP.

for p in (^Inf -> lazy.grep { .is_mersenne_prime }) {
say "2^#{p} - 1"
}
Output:
2^2 - 1
2^3 - 1
2^5 - 1
2^7 - 1
2^13 - 1
2^17 - 1
2^19 - 1
2^31 - 1
2^61 - 1
2^89 - 1
2^107 - 1
2^127 - 1
2^521 - 1
2^607 - 1
2^1279 - 1
2^2203 - 1
2^2281 - 1
2^3217 - 1
2^4253 - 1
2^4423 - 1
2^9689 - 1
2^9941 - 1
^C
sidef mersenne.sf  12.47s user 0.02s system 99% cpu 12.495 total

Visual Basic .NET[edit]

Translation of: C#
Imports System.Numerics
 
Module Module1
 
Function Sqrt(x As BigInteger) As BigInteger
If x < 0 Then
Throw New ArgumentException("Negative argument.")
End If
If x < 2 Then
Return x
End If
Dim y = x / 2
While y > (x / y)
y = ((x / y) + y) / 2
End While
Return y
End Function
 
Function IsPrime(bi As BigInteger) As Boolean
If bi < 2 Then
Return False
End If
If bi Mod 2 = 0 Then
Return bi = 2
End If
If bi Mod 3 = 0 Then
Return bi = 3
End If
If bi Mod 5 = 0 Then
Return bi = 5
End If
If bi Mod 7 = 0 Then
Return bi = 7
End If
If bi Mod 11 = 0 Then
Return bi = 11
End If
If bi Mod 13 = 0 Then
Return bi = 13
End If
If bi Mod 17 = 0 Then
Return bi = 17
End If
If bi Mod 19 = 0 Then
Return bi = 19
End If
 
Dim limit = Sqrt(bi)
Dim test As BigInteger = 23
While test < limit
If bi Mod test = 0 Then
Return False
End If
test += 2
If bi Mod test = 0 Then
Return False
End If
test += 4
End While
 
Return True
End Function
 
Sub Main()
Const MAX = 9
 
Dim pow = 2
Dim count = 0
 
While True
If IsPrime(pow) Then
Dim p = BigInteger.Pow(2, pow) - 1
If IsPrime(p) Then
Console.WriteLine("2 ^ {0} - 1", pow)
count += 1
If count >= MAX Then
Exit While
End If
End If
End If
pow += 1
End While
End Sub
 
End Module
Output:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1
2 ^ 61 - 1

Wren[edit]

Library: Wren-math
Library: Wren-big

A bit slow so limited to first 14 Mersenne primes.

import "/math" for Int
import "/big" for BigInt
 
var MAX = 14
System.print("The first %(MAX) Mersenne primes are:")
var count = 0
var p = 2
while (true) {
var m = (BigInt.one << p) - 1
if (m.isProbablePrime(10)) {
System.print("2 ^ %(p) - 1")
count = count + 1
if (count == MAX) break
}
while (true) {
p = (p > 2) ? p + 2 : 3
if (Int.isPrime(p)) break
}
}
Output:
The first 14 Mersenne primes are:
2 ^ 2 - 1
2 ^ 3 - 1
2 ^ 5 - 1
2 ^ 7 - 1
2 ^ 13 - 1
2 ^ 17 - 1
2 ^ 19 - 1
2 ^ 31 - 1
2 ^ 61 - 1
2 ^ 89 - 1
2 ^ 107 - 1
2 ^ 127 - 1
2 ^ 521 - 1
2 ^ 607 - 1

zkl[edit]

Library: GMP

Uses libGMP (GNU MP Bignum Library) and its Miller-Rabin probabilistic primality testing algorithm.

var [const] BN=Import.lib("zklBigNum");  // libGMP
fcn mprimes{
n,m := BN(2),0;
foreach e in ([2..]){
n,m = n.shiftLeft(1), n-1;
if(m.probablyPrime()) println("2^%d - 1".fmt(e));
}
}()
// gets rather slow after M(4423)
Output:
2^2 - 1
2^3 - 1
2^5 - 1
2^7 - 1
2^13 - 1
2^17 - 1
2^19 - 1
2^31 - 1
2^61 - 1
2^89 - 1
2^107 - 1
2^127 - 1
2^521 - 1
2^607 - 1
2^1279 - 1
2^2203 - 1
2^2281 - 1
2^3217 - 1
2^4253 - 1
2^4423 - 1
2^9689 - 1
2^9941 - 1
2^11213 - 1
...