Mersenne primes
Create code that will list (preferably calculate) all of the Mersenne primes until some limitation is reached.
- Task
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
- the Wikipedia entry: Mersenne prime.
- the MathWorld entry; Mersenne prime.
- For a list of all the know Mersenne primes: [https://primes.utm.edu/mersenne/index.html#known list of Mersenne
- For a general website about primes: prime pages.
- the OEIS entry: Mersenne primes.
- the OEIS entry: A000043 Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime.
11l
<lang 11l>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</lang>
- 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
<lang> 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 </lang>
AWK
<lang AWK>
- 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)
} </lang>
- 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
BASIC
BASIC256
<lang BASIC256>m = 0 while True m += 1 if isPrime((2^m)-1) = True then print m end while end
function isPrime(v) if v <= 1 then return False for i = 2 To int(sqr(v)) if v % i = 0 then return False next i return True end function</lang>
- Output:
Igual que la entrada de FreeBASIC.
FreeBASIC
<lang freebasic>Function isPrime(Byval ValorEval As Integer) As Boolean
If ValorEval <= 1 Then Return False
For i As Integer = 2 To Int(Sqr(ValorEval))
If ValorEval Mod i = 0 Then Return False
Next i
Return True
End Function
Dim As Integer m = 0 While True
m += 1 If isPrime((2^m)-1) Then Print m
Wend Sleep</lang>
- Output:
2 3 5 7 13 17 19 31
Yabasic
<lang yabasic>m = 0 while True
m = m + 1 if isPrime((2^m)-1) = True then print m : fi
wend end
sub isPrime(v)
if v < 2 then return False : fi
if mod(v, 2) = 0 then return v = 2 : fi
if mod(v, 3) = 0 then return v = 3 : fi
d = 5
while d * d <= v
if mod(v, d) = 0 then return False else d = d + 2 : fi
end while
return True
end sub</lang>
- Output:
Igual que la entrada de FreeBASIC.
C
<lang c>#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;
}</lang>
- 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++
<lang cpp>#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;
}</lang>
- 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#
Needs a better primality checking algorithm to do really large prime numbers. <lang csharp>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++;
}
}
}
}</lang>
- 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
Simplest thing that could possibly work. Using better primality tests will allow for more results to be calculated in a reasonable amount of time. <lang D>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;
}
}</lang>
- 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#
<lang fsharp>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</lang>
- 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
Factor comes with a Lucas-Lehmer primality test. <lang factor>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</lang>
- 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
<lang fortran> 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 </lang>
- 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
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. <lang frink>for n = primes[]
if isPrime[2^n - 1]
println["2^$n - 1"]</lang>
- 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
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.
<lang go>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 } } } }</lang>
- 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. <lang Go>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) } } }</lang>
- 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
<lang haskell>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</lang>
- 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
<lang Java>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));
}
}
}</lang>
- 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
Julia module Primes uses Miller-Rabin primality test.
<lang julia>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))</lang>
- Output:
M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9689
Kotlin
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. <lang scala>// 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
}
}
}</lang>
- 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
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. <lang lua>-- 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</lang>
- 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
Using only standard library
If we want to use only the standard library, we are limited to 64 bits. So we used a simple primality test. <lang Nim>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</lang>
- 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
The module bignum provides big integers and a probabilistic primality test. We searched the Mersenne numbers for exponents between 1 and 10_000.
<lang Nim>import bignum
var p = newInt(2) for e in 1..10_000:
if probablyPrime(p - 1, 25) != 0: echo "2^", e, " - 1" p *= 2</lang>
- 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
<lang parigp>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")))</lang>
Perl
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.
<lang perl>use ntheory qw/forprimes is_mersenne_prime/; forprimes { is_mersenne_prime($_) && say } 1e9;</lang>
- Output:
2 3 5 7 13 17 19 31 61 ...
Phix
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
<lang PHP><?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;
}
}</lang>
- 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
<lang PicoLisp>(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") ) )</lang>
- 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
<lang Pike>int power = 1; while(power++) {
int candidate = 2->pow(power)-1;
if( candidate->probably_prime_p() )
write("2 ^ %d - 1\n", power);
}</lang> 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
Lucas-Lehmer test, works with SWI Prolog <lang 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.
</lang>
- 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
<lang python>import random
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"</lang>
- 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
(formerly Perl 6)
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?
<lang perl6>use HTTP::UserAgent; use Gumbo;
my $table = parse-html(HTTP::UserAgent.new.get('https://www.mersenne.org/primes/').content, :TAG
); say 'All known Mersenne primes as of ', Date(now); say 'M', ++$, ": 2$_ - 1" for $table[1]».[*][0][*].comb(/'exp_lo='\d+/)».subst(/\D/, ,:g) .trans([<0123456789>.comb] => [<⁰¹²³⁴⁵⁶⁷⁸⁹>.comb]).words; </lang>- 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
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).
<lang rexx>/*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*/</lang>
Ring
<lang ring>
- 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
</lang> Output:
2 3 5 7 13 17 19
Scala
<lang Scala> 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!")
} </lang>
Sidef
Uses the is_mersenne_prime() function from Math::Prime::Util::GMP. <lang ruby>for p in (^Inf -> lazy.grep { .is_mersenne_prime }) {
say "2^#{p} - 1"
}</lang>
- 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
<lang vbnet>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</lang>
- 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
A bit slow so limited to first 14 Mersenne primes. <lang ecmascript>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
}
}</lang>
- 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
XPL0
<lang XPL0>func IsPrime(N); \Return 'true' if N is prime int N, I; [if N <= 2 then return N = 2; if (N&1) = 0 then \even >2\ return false; for I:= 3 to sqrt(N) do
[if rem(N/I) = 0 then return false; I:= I+1; ];
return true; ];
int N; [for N:= 1 to 31 do
if IsPrime(1<<N-1) then
[Text(0, "2^^"); IntOut(0, N); Text(0, " - 1");
CrLf(0);
];
]</lang>
- 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
zkl
Uses libGMP (GNU MP Bignum Library) and its Miller-Rabin probabilistic primality testing algorithm. <lang zkl>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)</lang>
- 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 ...