# Lucas-Lehmer test

Lucas-Lehmer test
You are encouraged to solve this task according to the task description, using any language you may know.

Lucas-Lehmer Test: for ${\displaystyle p}$ an odd prime, the Mersenne number ${\displaystyle 2^{p}-1}$ is prime if and only if ${\displaystyle 2^{p}-1}$ divides ${\displaystyle S(p-1)}$ where ${\displaystyle S(n+1)=(S(n))^{2}-2}$, and ${\displaystyle S(1)=4}$.

Calculate all Mersenne primes up to the implementation's maximum precision, or the 47th Mersenne prime   (whichever comes first).

## 360 Assembly

For maximum compatibility, this program uses only the basic instruction set.

*        Lucas-Lehmer testLUCASLEH CSECT           USING  LUCASLEH,R12SAVEARA  B      STM-SAVEARA(R15)         DC     17F'0'         DC     CL8'LUCASLEH'STM      STM    R14,R12,12(R13) save calling context         ST     R13,4(R15)               ST     R15,8(R13)         LR     R12,R15         set addessability*        ----   CODE         LA     R2,2            R2=2         LA     R11,0           R11:N'         BCTR   R11,0           N':=X'FFFFFFFF'         LA     R10,1           R10:N N=1         LA     R4,1            R4:IEXP         LA     R6,1            step          LH     R7,IEXPMAX      R7:IEXPMAX limitLOOPE    BXH    R4,R6,ENDLOOPE	do iexp=2 to iexpmax         SR     R3,R3           R3:S S=0          CR     R4,R2           if iexp=2 then S=0         BE     OKS         LA     R3,4            else S=4OKS      EQU    *         SLDA   R10,1           n=(n+1)*2-1         LA     R5,0            I         LA     R8,1            step          LR     R9,R4           IEXP         SR     R9,R2           IEXP-2 limitLOOPI    BXH    R5,R8,ENDLOOPI	do i=1 to iexp-2	 *        ----   compute s=(s*s-2) MOD n          SR     R14,R14         R14=0         LR     R15,R3          R15=S         MR     R14,R3          R{14-15}=S*S         SLR    R15,R2          R15=R15-2=S*S-2         BNM    *+6             skip next if no borrow         BCTR   R14,0           perform borrow         DR     R14,R10         R10=N          LR     R3,R14          R14=MOD         B      LOOPIENDLOOPI EQU    *         LTR    R3,R3         BNZ    NOPRT           if s<>0 then no print         CVD    R4,P            store to packed P         UNPK   Z,P             Z=P         MVC    C,Z             C=Z         OI     C+L'C-1,X'F0'   zap sign         MVC    WTOBUF(4),C+12         MVI    WTOBUF,C'M'         WTO    MF=(E,WTOMSG)		  NOPRT    EQU    *         B      LOOPEENDLOOPE EQU    **        ----   END CODERETURN   EQU    *         LM     R14,R12,12(R13)         XR     R15,R15         BR     R14*        ----   DATAIEXPMAX  DC     H'31'I        DS     HIEXP     DS     HS        DS     FN        DS     FP        DS     PL8             packedZ        DS     ZL16            zonedC        DS     CL16            character WTOMSG   DS     0F         DC     H'80',XL2'0000'WTOBUF   DC     80C' '         LTORG           YREGS           END    LUCASLEH
Output:
M002
M003
M005
M007
M013
M017
M019
M031

with Ada.Text_Io; use Ada.Text_Io;with Ada.Integer_Text_Io; use Ada.Integer_Text_Io; procedure Lucas_Lehmer_Test is   type Ull is mod 2**64;   function Mersenne(Item : Integer) return Boolean is      S : Ull := 4;      MP : Ull := 2**Item - 1;   begin      if Item = 2 then         return True;      else         for I in 3..Item loop            S := (S * S - 2) mod MP;         end loop;         return S = 0;      end if;   end Mersenne;   Upper_Bound : constant Integer := 64;begin   Put_Line(" Mersenne primes:");   for P in 2..Upper_Bound loop      if Mersenne(P) then         Put(" M");         Put(Item => P, Width => 1);      end if;   end loop;end Lucas_Lehmer_Test;
Output:
Mersenne primes:
M2 M3 M5 M7 M13 M17 M19 M31


## Agena

Because of the very large numbers computed, the mapm binding is used to calculate with arbitrary precision.

readlib 'mapm'; mapm.xdigits(100); mersenne := proc(p::number) is   local s, m;   s := 4;   m := mapm.xnumber(2^p) - 1;   if p = 2 then      return true   else      for i from 3 to p do         s := (mapm.xnumber(s)^2 - 2) % m      od;      return mapm.xtoNumber(s) = 0   fiend; for i from 3 to 64 do   if mersenne(i) then      write('M' & i & ' ')   fiod;

produces:

M3 M5 M7 M13 M17 M19 M31

## ALGOL 68

Works with: ALGOL 68 version Revision 1 - no extensions to language used
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny
PRAGMAT stack=1M precision=20000 PRAGMAT PROC is prime = ( INT p )BOOL:  IF p = 2 THEN TRUE  ELIF p <= 1 OR p MOD 2 = 0 THEN FALSE  ELSE    BOOL prime := TRUE;    FOR i FROM 3 BY 2 TO ENTIER sqrt(p)      WHILE prime := p MOD i /= 0 DO SKIP OD;    prime  FI; PROC is mersenne prime = ( INT p )BOOL:  IF p = 2 THEN TRUE  ELSE    LONG LONG INT m p :=  LONG LONG 2 ** p - 1, s := 4;    FROM 3 TO p DO      s := (s ** 2 - 2) MOD m p    OD;    s = 0  FI; test:(  INT upb prime = ( long long bits width - 1 ) OVER 2; # no unsigned #  INT upb count = 45; # find 45 mprimes if INT has enough bits #   printf(($" Finding Mersenne primes in M[2.."g(0)"]: "l$,upb prime));   INT count:=0;  FOR p FROM 2 TO upb prime WHILE    IF is prime(p) THEN      IF is mersenne prime(p) THEN        printf (($" M"g(0)$,p));        count +:= 1      FI    FI;    count <= upb count  DO SKIP OD)
Output:
Finding Mersenne primes in M[2..33252]:
M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9689 M9941 M11213 M19937 M21701 M23209


## ARM Assembly

Works with: as version Raspberry Pi
 /* ARM assembly Raspberry PI  *//*  program lucaslehmer.s   *//* use library gmp     *//* link with gcc option -lgmp */ /* Constantes    */.equ STDOUT, 1                           @ Linux output console.equ EXIT,   1                           @ Linux syscall.equ WRITE,  4                           @ Linux syscall .equ NBRECH,          30 /* Initialized data */.dataszMessResult:       .ascii "Prime : M"sMessValeur:        .fill 11, 1, ' '            @ size => 11                    .asciz "\n" szCarriageReturn:   .asciz "\n"szformat:           .asciz "nombre= %Zd\n" /* UnInitialized data */.bss .align 4spT:                .skip 100mpT:                .skip 100Deux:               .skip 100snT:                .skip 100/*  code section */.text.global main main:     ldr r0,iAdrDeux                       @ create big number = 2    mov r1,#2    bl __gmpz_init_set_ui    ldr r0,iAdrspT                        @ init big number    bl __gmpz_init    ldr r0,iAdrmpT                        @ init big number    bl __gmpz_init    mov r5,#3                             @ start number    mov r6,#0                             @ result counter1:    ldr r0,iAdrspT                        @ conversion integer in big number gmp    mov r1,r5     bl __gmpz_set_ui    ldr r0,iAdrspT                        @ control if exposant is prime !    ldr r0,iAdrspT    mov r1,#25    bl __gmpz_probab_prime_p    cmp r0,#0    beq 5f 2:    //ldr r1,iAdrspT                      @ example number display    //ldr r0,iAdrszformat    //bl __gmp_printf/******** Compute (2 pow p) - 1   ******/    ldr r0,iAdrmpT                        @ compute 2 pow p    ldr r1,iAdrDeux    mov r2,r5    bl __gmpz_pow_ui    ldr r0,iAdrmpT                          ldr r1,iAdrmpT    mov r2,#1    bl __gmpz_sub_ui                      @ then (2 pow p) - 1      ldr r0,iAdrsnT    mov r1,#4    bl __gmpz_init_set_ui                 @ init big number with 4 /**********  Test lucas_lehner  *******/    mov r4,#2                             @ loop counter3:                                        @ begin loop    ldr r0,iAdrsnT    ldr r1,iAdrsnT    mov r2,#2    bl __gmpz_pow_ui                      @ compute square big number     ldr r0,iAdrsnT    ldr r1,iAdrsnT    mov r2,#2    bl __gmpz_sub_ui                      @ = (sn *sn) - 2     ldr r0,iAdrsnT                        @ compute remainder -> sn    ldr r1,iAdrsnT                        @ sn    ldr r2,iAdrmpT                        @ p    bl __gmpz_tdiv_r     //ldr r1,iAdrsnT                      @ display number for control    //ldr r0,iAdrszformat    //bl __gmp_printf     add r4,#1                             @ increment counter    cmp r4,r5                             @ end ?    blt 3b                                @ no -> loop                                          @ compare result with zero    ldr r0,iAdrsnT    mov r1,#0    bl __gmpz_cmp_d    cmp r0,#0    bne 5f/********* is prime display result      *********/    mov r0,r5    ldr r1,iAdrsMessValeur                @ display value    bl conversion10                       @ call conversion decimal    ldr r0,iAdrszMessResult               @ display message    bl affichageMess    add r6,#1                             @ increment counter result    cmp r6,#NBRECH    bge 10f5:    add r5,#2                             @ increment number by two    b 1b                                  @ and loop 10:    ldr r0,iAdrDeux                       @ clear memory big number    bl __gmpz_clear    ldr r0,iAdrsnT    bl __gmpz_clear    ldr r0,iAdrmpT    bl __gmpz_clear    ldr r0,iAdrspT   bl __gmpz_clear100:                                      @ standard end of the program    mov r0, #0                            @ return code    mov r7, #EXIT                         @ request to exit program    svc 0                                 @ perform system calliAdrszMessResult:         .int szMessResultiAdrsMessValeur:          .int sMessValeuriAdrszCarriageReturn:     .int szCarriageReturniAdrszformat:             .int szformatiAdrspT:                  .int spTiAdrmpT:                  .int mpTiAdrDeux:                 .int DeuxiAdrsnT:                  .int snT/******************************************************************//*     display text with size calculation                         */ /******************************************************************//* r0 contains the address of the message */affichageMess:    push {r0,r1,r2,r7,lr}                       @ save  registers     mov r2,#0                                   @ counter length */1:                                              @ loop length calculation    ldrb r1,[r0,r2]                             @ read octet start position + index     cmp r1,#0                                   @ if 0 its over    addne r2,r2,#1                              @ else add 1 in the length    bne 1b                                      @ and loop                                                 @ so here r2 contains the length of the message     mov r1,r0                                   @ address message in r1     mov r0,#STDOUT                              @ code to write to the standard output Linux    mov r7, #WRITE                              @ code call system "write"     svc #0                                      @ call system    pop {r0,r1,r2,r7,lr}                        @ restaur registers    bx lr                                       @ return/******************************************************************//*     Converting a register to a decimal unsigned                */ /******************************************************************//* r0 contains value and r1 address area   *//* r0 return size of result (no zero final in area) *//* area size => 11 bytes          */.equ LGZONECAL,   10conversion10:    push {r1-r4,lr}                                 @ save registers     mov r3,r1    mov r2,#LGZONECAL 1:                                                  @ start loop    bl divisionpar10U                               @ unsigned  r0 <- dividende. quotient ->r0 reste -> r1    add r1,#48                                      @ digit    strb r1,[r3,r2]                                 @ store digit on area    cmp r0,#0                                       @ stop if quotient = 0     subne r2,#1                                     @ else previous position    bne 1b                                          @ and loop                                                    @ and move digit from left of area    mov r4,#02:    ldrb r1,[r3,r2]    strb r1,[r3,r4]    add r2,#1    add r4,#1    cmp r2,#LGZONECAL    ble 2b                                                      @ and move spaces in end on area    mov r0,r4                                         @ result length     mov r1,#' '                                       @ space3:    strb r1,[r3,r4]                                   @ store space in area    add r4,#1                                         @ next position    cmp r4,#LGZONECAL    ble 3b                                            @ loop if r4 <= area size 100:    pop {r1-r4,lr}                                    @ restaur registres     bx lr                                             @return /***************************************************//*   division par 10   unsigned                    *//***************************************************//* r0 dividende   *//* r0 quotient */	/* r1 remainder  */divisionpar10U:    push {r2,r3,r4, lr}    mov r4,r0                                          @ save value    //mov r3,#0xCCCD                                   @ r3 <- magic_number lower  raspberry 3    //movt r3,#0xCCCC                                  @ r3 <- magic_number higter raspberry 3    ldr r3,iMagicNumber                                @ r3 <- magic_number    raspberry 1 2    umull r1, r2, r3, r0                               @ r1<- Lower32Bits(r1*r0) r2<- Upper32Bits(r1*r0)     mov r0, r2, LSR #3                                 @ r2 <- r2 >> shift 3    add r2,r0,r0, lsl #2                               @ r2 <- r0 * 5     sub r1,r4,r2, lsl #1                               @ r1 <- r4 - (r2 * 2)  = r4 - (r0 * 10)    pop {r2,r3,r4,lr}    bx lr                                              @ leave function iMagicNumber:  	.int 0xCCCCCCCD
Output:
Prime : M3
Prime : M5
Prime : M7
Prime : M13
Prime : M17
Prime : M19
Prime : M31
Prime : M61
Prime : M89
Prime : M107
Prime : M127
Prime : M521
Prime : M607
Prime : M1279
Prime : M2203
Prime : M2281
Prime : M3217
Prime : M4253
Prime : M4423
Exception en point flottant


## AWK

 # syntax: GAWK -f LUCAS-LEHMER_TEST.AWK# converted from PascalBEGIN {    printf("Mersenne primes:")    n = 1    for (exponent=2; exponent<=32; exponent++) {      s = (exponent == 2) ? 0 : 4      n = (n+1)*2-1      for (i=1; i<=exponent-2; i++) {        s = (s*s-2)%n      }      if (s == 0) {        printf(" M%s",exponent)      }    }    printf("\n")    exit(0)}
Output:
Mersenne primes: M2 M3 M5 M7 M13 M17 M19


## BBC BASIC

Using its native arithmetic BBC BASIC can only test up to M23.

      *FLOAT 64      PRINT "Mersenne Primes:"      FOR p% = 2 TO 23        IF FNlucas_lehmer(p%) PRINT "M" ; p%      NEXT      END       DEF FNlucas_lehmer(p%)      LOCAL i%, mp, sn      IF p% = 2 THEN = TRUE      IF (p% AND 1) = 0 THEN = FALSE      mp = 2^p% - 1      sn = 4      FOR i% = 3 TO p%        sn = sn^2 - 2        sn -= (mp * INT(sn / mp))      NEXT      = (sn = 0)
Output:
Mersenne Primes:
M2
M3
M5
M7
M13
M17
M19


## Bracmat

Only exponents that are prime are tried. The primality test of these numbers uses a side effect of Bracmat's attempt at computing a root of a small enough number. ('small enough' meaning that the number must fit in a computer word, normally 32 or 64 bits.) To do that, Bracmat first creates a list of factors of the number and then takes the root of each factor. For example, to compute 54^2/3, Bracmat first creates the expression (2*3^3)^2/3 and then 2^2/3*3^(3*2/3), which becomes 2^2/3*9. If a number cannot be factorized, (either because it is prime or because it is to great to fit in a computer word) the root expression doesn't change much. For example, the expression 13^(13^-1) becomes 13^1/13, and this matches the pattern 13^%.

  ( clk$:?t0:?now & ( time = ( print = . put$ ( str              $( div$(!arg,1)                  ","                  (   div$(mod$(!arg*100,100),1):?arg                    & !arg:<10                    & 0                  |                   )                  !arg                  " "                )              )        )      & -1*!now+(clk$:?now):?SEC & print$!SEC      & print$(!now+-1*!t0) & put$"s: "    )  & 3:?exponent  &   whl    ' ( !exponent:~>12000      & (   !exponent^(!exponent^-1):!exponent^%          & 4:?s          & 2^!exponent+-1:?n          & 0:?i          &   whl            ' ( 1+!i:?i              & !exponent+-2:~<!i              & mod$(!s^2+-2.!n):?s ) & ( !s:0 & !time & out$(str$(M !exponent " is PRIME!")) | ) | ) & 1+!exponent:?exponent ) & done ); Output: (after 4.5 hours): 0,00 0,00 s: M3 is PRIME! 0,00 0,00 s: M5 is PRIME! 0,00 0,00 s: M7 is PRIME! 0,00 0,00 s: M13 is PRIME! 0,00 0,00 s: M17 is PRIME! 0,00 0,01 s: M19 is PRIME! 0,00 0,01 s: M31 is PRIME! 0,00 0,01 s: M61 is PRIME! 0,01 0,02 s: M89 is PRIME! 0,01 0,03 s: M107 is PRIME! 0,00 0,04 s: M127 is PRIME! 0,50 0,54 s: M521 is PRIME! 0,29 0,84 s: M607 is PRIME! 6,81 7,65 s: M1279 is PRIME! 38,35 46,01 s: M2203 is PRIME! 6,32 52,33 s: M2281 is PRIME! 116,01 168,34 s: M3217 is PRIME! 293,09 461,44 s: M4253 is PRIME! 64,61 526,05 s: M4423 is PRIME! 8863,90 9389,95 s: M9689 is PRIME! 1101,12 10491,08 s: M9941 is PRIME! 5618,45 16109,53 s: M11213 is PRIME! ## C ### GMP This uses some pre-tests to show how we can skip some numbers with relatively inexpensive methods. This also does a simple optimization of the modulus. It takes about 30 seconds to get to M11213. This is substantially faster than many of the other solutions, though certainly not comparable to dedicated programs such as Prime95. Takes an optional argument to test up to the given value. Library: GMP #include <stdio.h>#include <stdlib.h>#include <limits.h>#include <gmp.h> int lucas_lehmer(unsigned long p){ mpz_t V, mp, t; unsigned long k, tlim; int res; if (p == 2) return 1; if (!(p&1)) return 0; mpz_init_set_ui(t, p); if (!mpz_probab_prime_p(t, 25)) /* if p is composite, 2^p-1 is not prime */ { mpz_clear(t); return 0; } if (p < 23) /* trust the PRP test for these values */ { mpz_clear(t); return (p != 11); } mpz_init(mp); mpz_setbit(mp, p); mpz_sub_ui(mp, mp, 1); /* If p=3 mod 4 and p,2p+1 both prime, then 2p+1 | 2^p-1. Cheap test. */ if (p > 3 && p % 4 == 3) { mpz_mul_ui(t, t, 2); mpz_add_ui(t, t, 1); if (mpz_probab_prime_p(t,25) && mpz_divisible_p(mp, t)) { mpz_clear(mp); mpz_clear(t); return 0; } } /* Do a little trial division first. Saves quite a bit of time. */ tlim = p/2; if (tlim > (ULONG_MAX/(2*p))) tlim = ULONG_MAX/(2*p); for (k = 1; k < tlim; k++) { unsigned long q = 2*p*k+1; /* factor must be 1 or 7 mod 8 and a prime */ if ( (q%8==1 || q%8==7) && q % 3 && q % 5 && q % 7 && mpz_divisible_ui_p(mp, q) ) { mpz_clear(mp); mpz_clear(t); return 0; } } mpz_init_set_ui(V, 4); for (k = 3; k <= p; k++) { mpz_mul(V, V, V); mpz_sub_ui(V, V, 2); /* mpz_mod(V, V, mp) but more efficiently done given mod 2^p-1 */ if (mpz_sgn(V) < 0) mpz_add(V, V, mp); /* while (n > mp) { n = (n >> p) + (n & mp) } if (n==mp) n=0 */ /* but in this case we can have at most one loop plus a carry */ mpz_tdiv_r_2exp(t, V, p); mpz_tdiv_q_2exp(V, V, p); mpz_add(V, V, t); while (mpz_cmp(V, mp) >= 0) mpz_sub(V, V, mp); } res = !mpz_sgn(V); mpz_clear(t); mpz_clear(mp); mpz_clear(V); return res;} int main(int argc, char* argv[]) { unsigned long i, n = 43112609; if (argc >= 2) n = strtoul(argv[1], 0, 10); for (i = 1; i <= n; i++) { if (lucas_lehmer(i)) { printf("M%lu ", i); fflush(stdout); } } printf("\n"); return 0;} Output: (partial output after 50 minutes) M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9689 M9941 M11213 M11213 M19937 M21701 M23209 M44497  ### Small inputs with native types Works with: gcc version 4.1.2 20070925 (Red Hat 4.1.2-27) Works with: C99 Compiler options: gcc -std=c99 -lm Lucas-Lehmer_test.c -o Lucas-Lehmer_test #include <math.h>#include <stdio.h>#include <limits.h>#pragma precision=log10l(ULLONG_MAX)/2 typedef enum { FALSE=0, TRUE=1 } BOOL; BOOL is_prime( int p ){ if( p == 2 ) return TRUE; else if( p <= 1 || p % 2 == 0 ) return FALSE; else { BOOL prime = TRUE; const int to = sqrt(p); int i; for(i = 3; i <= to; i+=2) if (!(prime = p % i))break; return prime; }} BOOL is_mersenne_prime( int p ){ if( p == 2 ) return TRUE; else { const long long unsigned m_p = ( 1LLU << p ) - 1; long long unsigned s = 4; int i; for (i = 3; i <= p; i++){ s = (s * s - 2) % m_p; } return s == 0; }} int main(int argc, char **argv){ const int upb = log2l(ULLONG_MAX)/2; int p; printf(" Mersenne primes:\n"); for( p = 2; p <= upb; p += 1 ){ if( is_prime(p) && is_mersenne_prime(p) ){ printf (" M%u",p); } } printf("\n");} Output:  Mersenne primes: M2 M3 M5 M7 M13 M17 M19 M31  ## C++ Straightforward method. Library: GMP #include <iostream>#include <gmpxx.h> static bool is_mersenne_prime(mpz_class p){ if( 2 == p ) return true; else { mpz_class s(4); mpz_class div( (mpz_class(1) << p.get_ui()) - 1 ); for( mpz_class i(3); i <= p; ++i ) { s = (s * s - mpz_class(2)) % div ; } return ( s == mpz_class(0) ); }} int main(){ mpz_class maxcount(45); mpz_class found(0); mpz_class check(0); for( mpz_nextprime(check.get_mpz_t(), check.get_mpz_t()); found < maxcount; mpz_nextprime(check.get_mpz_t(), check.get_mpz_t())) { //std::cout << "P" << check << " " << std::flush; if( is_mersenne_prime(check) ) { ++found; std::cout << "M" << check << " " << std::flush; } }} Output: (Incomplete; It takes a long time.)  M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9689 M9941 M11213 M19937 M21701 M23209 M44497  ## C# Works with: Visual Studio version 2010 Works with: .NET Framework version 4.0 using System;using System.Collections.Generic;using System.Numerics;using System.Threading.Tasks; namespace LucasLehmerTestForRosettaCode{ public class LucasLehmerTest { static BigInteger ZERO = new BigInteger(0); static BigInteger ONE = new BigInteger(1); static BigInteger TWO = new BigInteger(2); static BigInteger FOUR = new BigInteger(4); private static bool isMersennePrime(int p) { if (p % 2 == 0) return (p == 2); else { for (int i = 3; i <= (int)Math.Sqrt(p); i += 2) if (p % i == 0) return false; //not prime BigInteger m_p = BigInteger.Pow(TWO, p) - ONE; BigInteger s = FOUR; for (int i = 3; i <= p; i++) s = (s * s - TWO) % m_p; return s == ZERO; } } public static int[] GetMersennePrimeNumbers(int upTo) { List<int> response = new List<int>(); Parallel.For(2, upTo + 1, i => { if (isMersennePrime(i)) response.Add(i); }); response.Sort(); return response.ToArray(); } static void Main(string[] args) { int[] mersennePrimes = LucasLehmerTest.GetMersennePrimeNumbers(11213); foreach (int mp in mersennePrimes) Console.Write("M" + mp+" "); Console.ReadLine(); } }} Output: (Run only to 11213)  M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9689 M9941 M11213  ### Quick Remainder The mod function, (%) has a computation cost equivalent to the divide operation. In this case, a combination of ands, shifts and adds can replace the mod function. Another change is creating the list of candidate Mersenne numbers in descending order, the point being to start the more time consuming calculations first. This avoids a long calculation occurring by itself at the end of the Parallel.For queue. Also added trial division step, translated from the Rust and C versions. using System;using System.Collections.Generic;using System.Numerics;using System.Threading.Tasks; public class Program { static int[] oddPrimes = new int[] { 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 }; static void Main() { int iExpMax = 11213; List<int> mn = new List<int>(), res = new List<int>(); DateTime st = DateTime.Now; for (bool skip = false; iExpMax >= 2; iExpMax--, skip = false) { for (int i = 2; i * i <= iExpMax; i += i == 2 ? 1 : 2) if (iExpMax % i == 0) { skip = true; continue; } if (!skip) mn.Add(iExpMax); } Parallel.ForEach(mn, e => { if (e == 2) { res.Add(2); return; } // trial division BigInteger m = BigInteger.Pow(2, e) - 1; for (long k = 1, ee = e << 1, q = ee + 1; k <= 100000 && q < m; k++, q += ee) { bool cont = false; foreach (int j in oddPrimes) if (q % j == 0) { cont = true; break; } if (cont || ((q & 7) != 1 && (q & 7) != 7)) continue; if (m % q == 0) return; } // main event BigInteger s = 4, mask = BigInteger.Pow(2, e) - 1, msk2 = mask + 2; for (int j = e; j > 2; j--) { s = ((s *= s) & mask) + (s >> e); s -= s >= mask ? msk2 : 2; } if (s == 0) res.Add(e); }); res.Sort(); foreach (int item in res) Console.Write("M{0} ", item); Console.WriteLine("\n{0}", DateTime.Now - st); if (System.Diagnostics.Debugger.IsAttached) Console.ReadLine(); }} Output: M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9689 M9941 M11213 00:00:26.8747764 Execution time of this quicker version is less than one-quarter, the former program taking well over 2 minutes to reach M11213, and this version completing in under half a minute. Heh, still 4 times slower than the Rust version... ## Clojure (defn prime? [i] (cond (< i 4) (>= i 2) (zero? (rem i 2)) false :else (not-any? #(zero? (rem i %)) (range 3 (inc (Math/sqrt i)))))))) (defn mersenne? [p] (or (= p 2) (let [mp (dec (bit-shift-left 1 p))] (loop [n 3 s 4] (if (> n p) (zero? s) (recur (inc n) (rem (- (* s s) 2) mp))))))) (filter mersenne? (filter prime? (iterate inc 1))) Output:  Infinite list of Mersenne primes: (2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281 3217 4253...  ## Common Lisp Translation of: Clojure  (defun or-f (&optional a b) (or a b));necessary for reduce, as 'or' is implemented as a macro (defun prime-p (n) (cond ((< n 4) (>= n 2)) ((zerop (rem n 2)) nil) (t (not (reduce #'or-f (mapcar (lambda (x) (zerop (rem n x))) (loop for i from 3 to (sqrt n) collect i))))))) (defun mersenne-p (p) (or (= p 2) (let ((mp (- 1 (expt 2 p)))) (do ((n 3) (s 4)) ((> n p) (zerop s)) (incf n) (setf s (rem (- (* s s) 2) mp)))))) (princ (remove-if-not #'mersenne-p (remove-if-not #'prime-p (loop for i to 5000 collect i))))  Output: (2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281 3217 4253 4423)  ## D Translation of: Python import std.stdio, std.math, std.bigint; bool isPrime(in uint p) pure nothrow @safe @nogc { if (p < 2 || p % 2 == 0) return p == 2; foreach (immutable i; 3 .. cast(uint)real(p).sqrt + 1) if (p % i == 0) return false; return true;} bool isMersennePrime(in uint p) pure nothrow /*@safe*/ { if (!p.isPrime) return false; if (p == 2) return true; immutable mp = (1.BigInt << p) - 1; auto s = 4.BigInt; foreach (immutable _; 3 .. p + 1) s = (s ^^ 2 - 2) % mp; return s == 0;} void main() { foreach (immutable p; 2 .. 2_300) if (p.isMersennePrime) { write('M', p, ' '); stdout.flush; }} Output: M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281  With p up to 10_000 it prints: M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9941  ## DWScript Using Integer type, which is 64bit, limits the search to M31. function IsMersennePrime(p : Integer) : Boolean;var i, s, m_p : Integer;begin if p=2 then Result:=True else begin m_p := (1 shl p)-1; s := 4; for i:=3 to p do s:=(s*s-2) mod m_p; Result:=(s=0); end;end; const upperBound = Round(Log2(High(Integer))/2); PrintLn('Finding Mersenne primes in M[2..' + IntToStr(upperBound) + ']: ');Print('M2'); var p : Integer;for p:=3 to upperBound step 2 do begin if IsMersennePrime(p) then Print(' M'+IntToStr(p));end;PrintLn(''); Output:  M2 M3 M5 M7 M13 M17 M19 M31 ## EchoLisp  (require 'bigint)(define (mersenne-prime? odd-prime: p) (define mp (1- (expt 2 p))) (define s #4) (for [(i (in-range 3 (1+ p)))] (set! s (% (- (* s s) 2) mp))) (when (zero? s) (printf "M%d" p))) ;; run it in the background(require 'tasks)(define LP (primes 10000)) ; list of candidate primes (define (mp-task LP) (mersenne-prime? (first LP)) (rest LP)) ;; return next state (task-run (make-task mp-task LP)) → M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281  ## Elixir Translation of: Erlang defmodule LucasLehmer do use Bitwise def test do for p <- 2..1300, p==2 or s(bsl(1,p)-1, p-1)==0, do: IO.write "M#{p} " end defp s(mp, 1), do: rem(4, mp) defp s(mp, n) do x = s(mp, n-1) rem(x*x-2, mp) endend LucasLehmer.test Output: M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279  ## Erlang -module(mp).-export([main/0]). main() -> [ io:format("M~p ", [P]) || P <- lists:seq(2,700), (P == 2) orelse (s((1 bsl P) - 1, P-1) == 0) ]. s(MP,1) -> 4 rem MP;s(MP,N) -> X=s(MP,N-1), (X*X - 2) rem MP. In 3 seconds will print M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607  Testing larger numbers (i.e. 5000) is possible but will take few minutes. ## ERRE With native arithmetic up to 23: for bigger numbers you must use MULPREC program. PROGRAM LL_TEST !$DOUBLE PROCEDURE LUCAS_LEHMER(P%->RES)     LOCAL I%,MP,SN     IF P%=2 THEN RES%=TRUE EXIT PROCEDURE END IF     IF (P% AND 1)=0 THEN RES%=FALSE EXIT PROCEDURE END IF     MP=2^P%-1     SN=4     FOR I%=3 TO P% DO        SN=SN^2-2        SN-=(MP*INT(SN/MP))     END FOR     RES%=(SN=0)END PROCEDURE BEGIN     PRINT("Mersenne Primes:")     FOR P%=2 TO 23 DO        LUCAS_LEHMER(P%->RES%)        IF RES% THEN PRINT("M";P%) END IF     END FOREND PROGRAM
Output:
Mersenne Primes:
M 2
M 3
M 5
M 7
M 13
M 17
M 19


## F#

Simple arbitrary-precision version:

let rec s mp n =  if n = 1 then 4I % mp else ((s mp (n - 1)) ** 2 - 2I) % mp [ for p in 2..47 do    if p = 2 || s ((1I <<< p) - 1I) (p - 1) = 0I then      yield p ]

Tail-recursive version:

let IsMersennePrime exponent =    if exponent <= 1 then failwith "Exponent must be >= 2"    let prime = 2I ** exponent - 1I;     let rec LucasLehmer i acc =        match i with        | x when x = exponent - 2 -> acc        | x -> LucasLehmer (x + 1) ((acc*acc - 2I) % prime)     LucasLehmer 0 4I = 0I

Version using library folding function (way shorter and faster than the above):

let IsMersennePrime exponent =    if exponent <= 1 then failwith "Exponent must be >= 2"    let prime = 2I ** exponent - 1I;     let LucasLehmer =        [| 1 .. exponent-2 |] |> Array.fold (fun acc _ -> (acc*acc - 2I) % prime) 4I     LucasLehmer = 0I

## Factor

USING: io math.primes.lucas-lehmer math.ranges prettyprintsequences ; 47 [1,b] [ lucas-lehmer ] filter"Mersenne primes:" print[ "M" write pprint bl ] each nl
Output:
Mersenne primes:
M2 M3 M5 M7 M13 M17 M19 M31


## Forth

: lucas-lehmer  1+ 2 do    4 i 2 <> * abs swap 1+ dup + 1- swap    i 1- 1 ?do dup * 2 - over mod loop 0= if ." M" i . then  loop cr; 1 15 lucas-lehmer

## Frink

Frink's isPrime function automatically detects numbers of the form 2n-1 and performs a Lucas-Lehmer test on them, including testing if n is prime, which is sufficient to prove primality for this form.

 for n = primes[]   if isPrime[2^n-1]      println[n]

## Fortran

Works with: Fortran version 90 and later

Only Mersenne number with prime exponent can be themselves prime but for the small numbers used in this example it was not worth the effort to include this check. As the size of the exponent increases this becomes more important.

PROGRAM LUCAS_LEHMER  IMPLICIT NONE   INTEGER, PARAMETER :: i64 = SELECTED_INT_KIND(18)  INTEGER(i64) :: s, n  INTEGER :: i, exponent   DO exponent = 2, 31     IF (exponent == 2) THEN        s = 0     ELSE        s = 4     END IF     n = 2_i64**exponent - 1     DO i = 1, exponent-2        s = MOD(s*s - 2, n)     END DO     IF (s==0) WRITE(*,"(A,I0,A)") "M", exponent, " is PRIME"  END DO END PROGRAM LUCAS_LEHMER

## FreeBASIC

### Native types for Mersenne primes <= M63

' version 18-09-2015' compile with: fbc -s console #Ifndef TRUE        ' define true and false for older freebasic versions    #Define FALSE 0    #Define TRUE Not FALSE#EndIf Function mul_mod(a As ULongInt, b As ULongInt, modulus As ULongInt) As ULongInt    ' returns a * b mod modulus     Dim As ULongInt x , y = a ' a mod modulus, but a is already smaller then modulus     While b > 0        If (b And 1) = 1 Then            x = (x + y) Mod modulus        End If        y = (y Shl 1) Mod modulus        b = b Shr 1    Wend    Return x End Function Function LLT(p As UInteger) As Integer     Dim As ULongInt s = 4, m = 1    m = m Shl p : m = m - 1       ' m = 2 ^ p - 1     For i As Integer = 2 To p - 1        s = mul_mod(s, s, m) - 2    Next     If s = 0 Then Return TRUE Else Return FALSE End Function ' ------=< MAIN >=------ Dim As UInteger p Print' M2 can not be tested, we start with 3for p = 3 To 63    If LLT(p) = TRUE Then Print " M";Str(p);Next Print' empty keyboard bufferWhile Inkey <> "" : WendPrint : Print "hit any key to end program"SleepEnd
Output:
 M3 M5 M7 M13 M17 M19 M31 M61

### Library: GMP

Uses the trick from the C entry to avoid the slow Mod

' version 18-09-2015' compile with: fbc -s console #Include Once "gmp.bi" #Macro init_big_int (a)    Dim As Mpz_ptr a = Allocate( Len(__mpz_struct))    Mpz_init(a)#EndMacro ' ------=< MAIN >=------ Const As UInteger max = 12000  ' 230 sec., 10000 about 125 sec. Dim As UInteger p, xDim As Byte sieve(max) Dim As String buffer = Space(Len(Str(max))+1) init_big_int(m)init_big_int(s)init_big_int(r) ' sieve to find the primes' remove even numbers except 2For p = 4 To Sqr(max) Step 2    sieve(p) = 1Next For p = 3 To Sqr(max) Step 2    For x = p * p To max Step p * 2        sieve(x) = 1    NextNext ' exception: the test will not work for p = 2 For p = 3 To max Step 2            ' odd numbers only     If sieve(p) = 1 Then Continue For     Mpz_set_ui(s, 4)                 ' s(0) = 4    Mpz_set_ui(m, 1)                 ' set m to 1    Mpz_mul_2exp(m, m, p)            ' m = m shl p =  2 ^ p    Mpz_sub_ui(m, m, 1)              ' m = m - 1 =  2 ^ p - 1     For x = 2 To p - 1        Mpz_mul(s, s, s)               ' s = s * s        Mpz_sub_ui(s, s, 2)            ' s = s - 2        ' Mpz_fdiv_r(s, s, m)          ' s = s mod m        If Mpz_sgn(s) < 0 Then            Mpz_add(s, s ,m)        Else            Mpz_tdiv_r_2exp(r, s, p)            Mpz_tdiv_q_2exp(s, s, p)            Mpz_add(s, s, r)        End If        If (Mpz_cmp(s, m) >= 0) Then Mpz_sub(s, s, m)    Next     'If Mpz_cmp_ui(s, 0) = 0 Then    '   LSet buffer = Str(p)    '   Print "M"; buffer; " is prime"    'End If    If Mpz_cmp_ui(s, 0) = 0 Then        Print "M";Str(p),    End IfNextPrint Mpz_clear (m)  ' cleanupDeAllocate(m)Mpz_clear (s)DeAllocate(s)Mpz_clear (r)DeAllocate(r) ' empty keyboard bufferWhile InKey <> "" : WendPrint : Print "hit any key to end program"SleepEnd
Output:
M3            M5            M7            M13           M17
M19           M31           M61           M89           M107
M127          M521          M607          M1279         M2203
M2281         M3217         M4253         M4423         M9689
M9941         M11213

## FunL

def mersenne( p ) =  if p == 2 then return true   var s = 4  var M = 2^p - 1   repeat p - 2    s = (s*s - 2) mod M   s == 0 import integers.primes for p <- primes().filter( mersenne ).take( 20 )  println( 'M' + p )
Output:
M2
M3
M5
M7
M13
M17
M19
M31
M61
M89
M107
M127
M521
M607
M1279
M2203
M2281
M3217
M4253
M4423


## GAP

LucasLehmer := function(n)    local i, m, s;    if n = 2 then        return true;    elif not IsPrime(n) then        return false;    else        m := 2^n - 1;        s := 4;        for i in [3 .. n] do            s := RemInt(s*s, m) - 2;        od;        return s = 0;    fi;end; Filtered([1 .. 2000], LucasLehmer);[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279]

## Go

Processing the first list indicates that the test works. Processing the second shows it working on some larger numbers.

package main import (    "fmt"    "math/big") var primes = []uint{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} var mersennes = []uint{521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689,    9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091,    756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917,    20996011, 24036583} func main() {    llTest(primes)    fmt.Println()    llTest(mersennes)} func llTest(ps []uint) {    var s, m big.Int    one := big.NewInt(1)    two := big.NewInt(2)    for _, p := range ps {        m.Sub(m.Lsh(one, p), one)        s.SetInt64(4)        for i := uint(2); i < p; i++ {            s.Mod(s.Sub(s.Mul(&s, &s), two), &m)        }        if s.BitLen() == 0 {            fmt.Printf("M%d ", p)        }    }}
Output:
M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127
M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9689 M9941 M11213 M19937...


Works with: GHCi version 6.8.2
Works with: GHC version 6.8.2
module Main  where main = printMersennes $take 45$ filter lucasLehmer $sieve [2..] s mp 1 = 4 mod mps mp n = ((s mp$ n-1)^2-2) mod mp lucasLehmer 2 = TruelucasLehmer p = s (2^p-1) (p-1) == 0 printMersennes = mapM_ (\x -> putStrLn $"M" ++ show x) It is pointed out on the Sieve of Eratosthenes page that the following "sieve" is inefficient. Nonetheless it takes very little time compared to the Lucas-Lehmer test itself. sieve (p:xs) = p : sieve [x | x <- xs, x mod p > 0] It takes about 30 minutes to get up to: M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9689 M9941 M11213  ## HicEst s = 0DO exponent = 2, 31 IF(exponent > 2) s = 4 n = 2^exponent - 1 DO i = 1, exponent-2 s = MOD(s*s - 2, n) ENDDO IF(s == 0) WRITE(Messagebox) 'M', exponent, ' is prime;', nENDDO END ## J ## Java We use arbitrary-precision integers in order to be able to test any arbitrary prime. import java.math.BigInteger;public class Mersenne{ public static boolean isPrime(int p) { if (p == 2) return true; else if (p <= 1 || p % 2 == 0) return false; else { int to = (int)Math.sqrt(p); for (int i = 3; i <= to; i += 2) if (p % i == 0) return false; return true; } } public static boolean isMersennePrime(int p) { if (p == 2) return true; else { BigInteger m_p = BigInteger.ONE.shiftLeft(p).subtract(BigInteger.ONE); BigInteger s = BigInteger.valueOf(4); for (int i = 3; i <= p; i++) s = s.multiply(s).subtract(BigInteger.valueOf(2)).mod(m_p); return s.equals(BigInteger.ZERO); } } // an arbitrary upper bound can be given as an argument public static void main(String[] args) { int upb; if (args.length == 0) upb = 500; else upb = Integer.parseInt(args[0]); System.out.print(" Finding Mersenne primes in M[2.." + upb + "]:\nM2 "); for (int p = 3; p <= upb; p += 2) if (isPrime(p) && isMersennePrime(p)) System.out.print(" M" + p); System.out.println(); }} Output: (after about eight hours):  Finding Mersenne primes in M[2..2147483647]: M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9689 M9941 M11213  ## JavaScript In JavaScript we using BigInt ( numbers with 'n' suffix ) - so we can use really big numbers  ////////// In JavaScript we don't have sqrt for BigInt - so here is implementation function newtonIteration(n, x0) { const x1 = ((n / x0) + x0) >> 1n; if (x0 === x1 || x0 === (x1 - 1n)) { return x0; } return newtonIteration(n, x1); } function sqrt(value) { if (value < 0n) { throw 'square root of negative numbers is not supported' } if (value < 2n) { return value; } return newtonIteration(value, 1n); }////////// End of sqrt implementation function isPrime(p) { if (p == 2n) { return true; } else if (p <= 1n || p % 2n === 0n) { return false; } else { var to = sqrt(p); for (var i = 3n; i <= to; i += 2n) if (p % i == 0n) { return false; } return true; } } function isMersennePrime(p) { if (p == 2n) { return true; } else { var m_p = (1n << p) - 1n; var s = 4n; for (var i = 3n; i <= p; i++) { s = (s * s - 2n) % m_p; } return s === 0n; } } var upb = 5000; var tm = Date.now(); console.log(Finding Mersenne primes in M[2..${upb}]:);    console.log('M2');    for (var p = 3n; p <= upb; p += 2n){        if (isPrime(p) && isMersennePrime(p)) {            console.log("M" + p);        }    }    console.log(... Took: ${Date.now()-tm} ms);  Output: Finding Mersenne primes in M[2..5000]: M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 ... Took: 107748 ms  ## Julia  using Primes function getmersenneprimes(n) t1 = time() count = 0 i = 2 while(n > count) if(isprime(i) && ismersenneprime(2^BigInt(i) - 1)) println("M$i, cumulative time elapsed: $(time() - t1) seconds") count += 1 end i += 1 endend getmersenneprimes(50)  Output:  M2, cumulative time elapsed: 0.019999980926513672 seconds M3, cumulative time elapsed: 0.02200007438659668 seconds M5, cumulative time elapsed: 0.02200007438659668 seconds M7, cumulative time elapsed: 0.02200007438659668 seconds M13, cumulative time elapsed: 0.02200007438659668 seconds M17, cumulative time elapsed: 0.02200007438659668 seconds M19, cumulative time elapsed: 0.02200007438659668 seconds M31, cumulative time elapsed: 0.02200007438659668 seconds M61, cumulative time elapsed: 0.023000001907348633 seconds M89, cumulative time elapsed: 0.024000167846679688 seconds M107, cumulative time elapsed: 0.02500009536743164 seconds M127, cumulative time elapsed: 0.026000022888183594 seconds M521, cumulative time elapsed: 0.12400007247924805 seconds M607, cumulative time elapsed: 0.14300012588500977 seconds M1279, cumulative time elapsed: 0.6940000057220459 seconds M2203, cumulative time elapsed: 2.5870001316070557 seconds M2281, cumulative time elapsed: 2.88700008392334 seconds M3217, cumulative time elapsed: 8.276000022888184 seconds M4253, cumulative time elapsed: 20.874000072479248 seconds M4423, cumulative time elapsed: 23.56000018119812 seconds M9689, cumulative time elapsed: 338.970999956131 seconds M9941, cumulative time elapsed: 373.2020001411438 seconds M11213, cumulative time elapsed: 557.3210000991821 seconds M19937, cumulative time elapsed: 3963.986000061035 seconds M21701, cumulative time elapsed: 5330.933000087738 seconds M23209, cumulative time elapsed: 6783.236999988556 seconds M44497, cumulative time elapsed: 57961.360000133514 seconds  ## Kotlin In view of the Java result, I've set the program to stop at M4423 so it will run in a reasonable time (about 85 seconds) on a typical laptop: // version 1.0.6 import java.math.BigInteger const val MAX = 19 val bigTwo = BigInteger.valueOf(2L)val bigFour = bigTwo * bigTwo 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 = 3 // first odd prime var s: BigInteger var m: BigInteger while (true) { m = bigTwo.shiftLeft(p - 1) - BigInteger.ONE s = bigFour for (i in 1 .. p - 2) s = (s * s - bigTwo) % m if (s == BigInteger.ZERO) { count +=1 print("M$p ")            if (count == MAX) {                println()                break             }        }            // obtain next odd prime        while(true) {            p += 2                     if (isPrime(p)) break        }     } }
Output:
M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423


## Mathematica

This version is very speedy and is bounded.

Select[Table[M = 2^p - 1;   For[i = 1; s = 4, i <= p - 2, i++, s = Mod[s^2 - 2, M]];   If[s == 0, "M" <> [email protected], p], {p,    Prime /@ Range[300]}], StringQ] => {M3, M5, M7, M13, M17, M19, M31, M61, M89, M107, M127, M521, M607, M1279}

This version is unbounded (and timed):

t = SessionTime[];For[p = 2, True, p = NextPrime[p], M = 2^p - 1;  For[i = 1; s = 4, i <= p - 2, i++, s = Mod[s^2 - 2, M]];  If[s == 0, Print["M" <> [email protected]]]](SessionTime[] - t) {Seconds, Minutes/60, Hours/3600, Days/86400}

I'll see what this gets.

## MATLAB

MATLAB suffers from a lack of an arbitrary precision math (bignums) library. It also doesn't have great support for 64-bit integer arithmetic...or at least MATLAB 2007 doesn't. So, the best precision we have is doubles; therefore, this script can only find up to M19 and no greater.

function [mNumber,mersennesPrime] = mersennePrimes()     function isPrime = lucasLehmerTest(thePrime)         llResidue = 4;        mersennesPrime = (2^thePrime)-1;         for i = ( 1:thePrime-2 )            llResidue = mod( ((llResidue^2) - 2),mersennesPrime );                end                  isPrime = (llResidue == 0);     end     %Because IEEE764 Double is the highest precision number we can    %represent in MATLAB, the highest Mersenne Number we can test is 2^52.    %In addition, because we have this cap, we can only test up to the    %number 30 for Mersenne Primeness. When we input 31 into the    %Lucas-Lehmer test, during the computation of the residue, the    %algorithm multiplies two numbers together the result of which is    %greater than 2^53. Because we require every digit to be significant,    %this leads to an error. The Lucas-Lehmer test should say that M31 is a    %Mersenne Prime, but because of the rounding error in calculating the    %residues caused by floating-point arithmetic, it does not. So M30 is    %the largest number we test.     mNumber = (3:30);     [isPrime] = arrayfun(@lucasLehmerTest,mNumber);     mNumber = [2 mNumber(isPrime)];    mersennesPrime = (2.^mNumber) - 1; end
Output:
[mNumber,mersennesPrime] = mersennePrimes mNumber =      2     3     5     7    13    17    19  mersennesPrime =            3           7          31         127        8191      131071      524287

lucas_lehmer(p) := block([s, n, i],   if not primep(p) then false elseif p = 2 then true else   (s: 4,   n: 2^p - 1,   for i: 2 thru p - 1 do s: mod(s*s - 2, n),   is(s = 0)))$sublist(makelist(i, i, 1, 200), lucas_lehmer);/* [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127] */ ## Modula-3 Modula-3 uses L as the literal for LONGINT. MODULE LucasLehmer EXPORTS Main; IMPORT IO, Fmt, Long; PROCEDURE Mersenne(p: CARDINAL): BOOLEAN = VAR s := 4L; m := Long.Shift(1L, p) - 1L; (* 2^p - 1 *) BEGIN IF p = 2 THEN RETURN TRUE; ELSE FOR i := 3 TO p DO s := (s * s - 2L) MOD m; END; RETURN s = 0L; END; END Mersenne; BEGIN FOR i := 2 TO 63 DO IF Mersenne(i) THEN IO.Put("M" & Fmt.Int(i) & " "); END; END; IO.Put("\n");END LucasLehmer. Output: M2 M3 M5 M7 M13 M17 M19 M31  ## Nim import math proc isPrime(a: int): bool = if a == 2: return true if a < 2 or a mod 2 == 0: return false for i in countup(3, int sqrt(float a), 2): if a mod i == 0: return false return true proc isMersennePrime(p: int): bool = if p == 2: return true let mp = (1'i64 shl p) - 1 var s = 4'i64 for i in 3 .. p: s = (s * s - 2) mod mp result = s == 0 let upb = int((log2 float int64.high) / 2)echo " Mersenne primes:"for p in 2 .. upb: if isPrime(p) and isMersennePrime(p): stdout.write " M",pecho "" Output:  Mersenne primes: M2 M3 M5 M7 M13 M17 M19 M31 ## Oz Oz's multiple precision number system use GMP core. %% compile : ozc -x <file.oz>functorimport Application Systemdefine fun {Arg Idx Default} Cmd = {Application.getArgs plain} Len = {Length Cmd} in if Len < Idx then Default else {StringToInt {Nth Cmd Idx}} end end fun {LLtest N} Mp = {Pow 2 N} - 1 fun {S K} X T in if K == 1 then 4 else T = {S K-1} X = T * T - 2 X mod Mp end end in if N == 2 then true else {S N-1} == 0 end end proc {FindLL X} fun {Sieve Ls} case Ls of nil then nil [] X|Xs then fun {DIV M} M mod X \= 0 end in X|{Sieve {Filter Xs DIV}} end end in if {IsList X} then case X of nil then skip [] M|Ms then {System.printInfo "M"#M#" "} {FindLL Ms} end else {FindLL {Filter {Sieve 2|{List.number 3 X 2}} LLtest}} end end Num = {Arg 1 607} {FindLL Num} {Application.exit 0}end ## PARI/GP LL(p)={ my(m=Mod(4,1<<p-1)); for(i=3,p,m=m^2-2); m==0}; search()={ print("2^2-1"); forprime(p=3,43112609, if(LL(p), print("2^"p"-1")) )}; ## Pascal int64 is good enough up to M31: Program LucasLehmer(output);var s, n: int64; i, exponent: integer;begin n := 1; for exponent := 2 to 31 do begin if exponent = 2 then s := 0 else s := 4; n := (n + 1)*2 - 1; // This saves from needing the math unit for exponentiation for i := 1 to exponent-2 do s := (s*s - 2) mod n; if s = 0 then writeln('M', exponent, ' is PRIME!'); end;end. Output: :> ./LucasLehmer M2 is PRIME! M3 is PRIME! M5 is PRIME! M7 is PRIME! M13 is PRIME! M17 is PRIME! M19 is PRIME! M31 is PRIME!  ## Perl Using Math::GMP: use Math::GMP qw/:constant/; sub is_prime { Math::GMP->new(shift)->probab_prime(12); } sub is_mersenne_prime { my$p = shift;  return 1 if $p == 2; my$mp = 2 ** $p - 1; my$s = 4;  $s = ($s * $s - 2) %$mp  for 3..$p;$s == 0;} foreach my $p (2 .. 43_112_609) { print "M$p\n" if is_prime($p) && is_mersenne_prime($p);}

The ntheory module offers a couple options. This is direct:

Library: ntheory
use ntheory qw/:all/;$|=1; # flush output on every printmy$n = 0;for (1..47) {  1 while !is_mersenne_prime(++$n); print "M$n ";}print "\n";

However it uses knowledge from the thousands of CPU years spent by GIMPS to accelerate results for known values, so doesn't actually run the L-L test until after the 44th value, although code is included for C, Perl, and C+GMP. If we substitute Math::Prime::Util::GMP::is_mersenne_prime we can force the test to run.

A less opaque method uses the modular Lucas sequence, though it has no pretesting other than primality and calculates both ${\displaystyle U_{k}}$ and ${\displaystyle V_{k}}$ so won't be as fast:

use ntheory qw/:all/;use bigint try=>"GMP,Pari";forprimes {  my $p =$_;  my $mp1 = 2**$p;  print "M$p\n" if$p == 2 || 0 == (lucas_sequence($mp1-1, 4, 1,$mp1))[0];} 43_112_609;

We can also use the core module Math::BigInt:

Translation of: Python
sub is_prime {    my $p = shift; if ($p == 2) {        return 1;    } elsif ($p <= 1 ||$p % 2 == 0) {        return 0;    } else {        my $limit = sqrt($p);        for (my $i = 3;$i <= $limit;$i += 2) {            return 0 if $p %$i == 0;        }        return 1;    }} sub is_mersenne_prime {    use bigint;    my $p = shift; if ($p == 2) {        return 1;    } else {        my $m_p = 2 **$p - 1;        my $s = 4; foreach my$i (3 .. $p) {$s = ($s ** 2 - 2) %$m_p;        }        return $s == 0; }} my$precision = 20000;   # maximum requested number of decimal places of 2 ** MP-1 #my $long_bits_width =$precision / log(2) * log(10);my $upb_prime = int(($long_bits_width - 1)/2);    # no unsigned #my $upb_count = 45; # find 45 mprimes if int was given enough bits # print " Finding Mersenne primes in M[2..$upb_prime]:\n"; my $count = 0;foreach my$p (2 .. $upb_prime) { if (is_prime($p) && is_mersenne_prime($p)) { print "M$p\n";        $count++; } last if$count >= $upb_count;} ## Perl 6 multi is_mersenne_prime(2) { True }multi is_mersenne_prime(Int$p) {    my $m_p = 2 **$p - 1;    my $s = 4;$s = $s.expmod(2,$m_p) - 2 for 3 .. $p; !$s} .say for (2,3,5,7 … *).hyper(:8degree).grep( *.is-prime ).map: { next unless .&is_mersenne_prime; "M$_" }; On my system: Letting it run for about a minute... M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9689 M9941 M11213 ^C real 0m55.527s user 6m47.106s sys 0m0.404s ## Phix Library: mpfr Native types work up to M31, after which inaccuracies mean that we need to wheel out gmp. Uses the mod replacement trick from C/FreeBASIC(gmp) bool full = true -- (see extended output below)constant limit = iff(full?20:23) include mpfr.e function mersenne(integer p) if p = 2 then return true end if if not is_prime(p) then return false end if mpz s := mpz_init(4), m := mpz_init(), r = mpz_init() mpz_ui_pow_ui(m, 2, p) mpz_sub_si(m,m,1) for i=3 to p do mpz_mul(s,s,s) mpz_sub_si(s,s,2)-- mpz_mod(s,s,m) if mpz_sign(s) < 0 then mpz_add(s, s ,m) else mpz_tdiv_r_2exp(r, s, p) mpz_tdiv_q_2exp(s, s, p) mpz_add(s, s, r) end if if (mpz_cmp(s, m) >= 0) then mpz_sub(s, s, m) end if end for bool res = mpz_cmp_si(s,0)=0 {s,m,r} = mpz_free({s,m,r}) return resend function atom t0 = time(), t1 = t0integer i=2, j = 1, count = 0constant mersennes = {1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609} while count<limit do if mersenne(i) then count += 1 string e = iff(time()-t1<0.1?"",", "&elapsed(time()-t1)) printf(1,"M%d (%d%s)\n",{i,count,e}) t1 = time() end if if full or i<1000 then i += 1 else i = mersennes[j] j += 1 end ifend whileprintf(1,"completed in %s\n",{elapsed(time()-t0)}) Output: M2 (1) M3 (2) M5 (3) M7 (4) M13 (5) M17 (6) M19 (7) M31 (8) M61 (9) M89 (10) M107 (11) M127 (12) M521 (13, 0.1s) M607 (14) M1279 (15, 0.7s) M2203 (16, 2.0s) M2281 (17, 0.3s) M3217 (18, 4.0s) M4253 (19, 8.0s) M4423 (20, 1.7s) completed in 16.9s  Using the idea from Go of using a mersennes table above 1000 to speed things up, ie by setting full to false we get: (ditto) M1279 (15, 0.3s) M2203 (16) M2281 (17) M3217 (18) M4253 (19) M4423 (20) M9689 (21, 0.5s) M9941 (22, 0.5s) M11213 (23, 0.6s) completed in 2.5s  Three more entries in one sixth of the time. Increasing the limit to 31 (with full still false) we can also get (ditto) M19937 (24, 2.1s) M21701 (25, 2.5s) M23209 (26, 3.0s) M44497 (27, 15.3s) M86243 (28, 1 minute and 12s) M110503 (29, 1 minute and 53s) M132049 (30, 2 minutes and 46s) M216091 (31, 7 minutes and 45s) completed in 14 minutes and 01s  but beyond that I gave up. ## PicoLisp (de prime? (N) (or (= N 2) (and (> N 1) (bit? 1 N) (let S (sqrt N) (for (D 3 T (+ D 2)) (T (> D S) T) (T (=0 (% N D)) NIL) ) ) ) ) ) (de mersenne? (P) (or (= P 2) (let (MP (dec (>> (- P) 1)) S 4) (do (- P 2) (setq S (% (- (* S S) 2) MP)) ) (=0 S) ) ) ) Output: : (for N 10000 (and (prime? N) (mersenne? N) (println N)) ) 2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281 3217 4253 4423 9689 9941 ## Pop11 Checking large numbers takes a lot of time so we limit p to be smaller than 1000. define Lucas_Lehmer_Test(p); lvars mp = 2**p - 1, sn = 4, i; for i from 2 to p - 1 do (sn*sn - 2) rem mp -> sn; endfor; sn = 0;enddefine; lvars p = 3;printf('M2', '%p\n');while p < 1000 do if Lucas_Lehmer_Test(p) then printf('M', '%p'); printf(p, '%p\n'); endif; p + 2 -> p;endwhile; Output: (obtained in few seconds) M2M3M5M7M13M17M19M31M61M89M107M127M521M607 ## PowerShell This is just a translation of VBScript using [bigint], it could be optimized. Flirt with the girl in the cubicle next door while it runs:  function Get-MersennePrime ([bigint]$Maximum = 4800){    [bigint]$n = [bigint]::One for ($exp = 2; $exp -lt$Maximum; $exp++) { if ($exp -eq 2)        {            $s = 0 } else {$s = 4        }         $n = ($n + 1) * 2 - 1         for ($i = 1;$i -le $exp - 2;$i++)        {             $s = ($s * $s - 2) %$n        }         if ($s -eq 0) {$exp        }    }}
 Get-MersennePrime | Format-Wide {"{0,4}" -f $_} -Column 4 -Force  Output:  2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281 3217 4253 4423  ## Prolog  show(Count) :- findall(N, limit(Count, (between(2, infinite, N), mersenne_prime(N))), S), forall(member(P, S), (write(P), write(" "))), nl. lucas_lehmer_seq(M, L) :- lazy_list(ll_iter, 4-M, L). ll_iter(S-M, T-M, T) :- T is ((S*S) - 2) mod M. drop(N, Lz1, Lz2) :- append(Pfx, Lz2, Lz1), length(Pfx, N), !. mersenne_prime(2).mersenne_prime(P) :- P > 2, prime(P), M is (1 << P) - 1, lucas_lehmer_seq(M, Residues), Skip is P - 3, drop(Skip, Residues, [R|_]), R =:= 0. % check if a number is prime%wheel235(L) :- W = [4, 2, 4, 2, 4, 6, 2, 6 | W], L = [1, 2, 2 | W]. prime(N) :- N >= 2, wheel235(W), prime(N, 2, W). prime(N, D, _) :- D*D > N, !.prime(N, D, [A|As]) :- N mod D =\= 0, D2 is D + A, prime(N, D2, As).  Output: ?- show(20). 2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281 3217 4253 4423 true.  ## PureBasic PureBasic has no large integer support. Calculations are limited to the range of a signed quad integer type. Procedure Lucas_Lehmer_Test(p) Protected mp.q = (1 << p) - 1, sn.q = 4, i For i = 3 To p sn = (sn * sn - 2) % mp Next If sn = 0 ProcedureReturn #True EndIf ProcedureReturn #FalseEndProcedure #upperBound = SizeOf(Quad) * 8 - 1 ;equivalent to significant bits in a signed quad integerIf OpenConsole() Define p = 3 PrintN("M2") While p <= #upperBound If Lucas_Lehmer_Test(p) PrintN("M" + Str(p)) EndIf p + 2 Wend Print(#CRLF$ + #CRLF+ "Press ENTER to exit"): Input() CloseConsole()EndIf Output: M2 M3 M5 M7 M13 M17 M19 M31 ## Python  from sys import stdoutfrom math import sqrt, log def is_prime ( p ): if p == 2: return True # Lucas-Lehmer test only works on odd primes elif p <= 1 or p % 2 == 0: return False else: for i in range(3, int(sqrt(p))+1, 2 ): if p % i == 0: return False return True def is_mersenne_prime ( p ): if p == 2: return True else: m_p = ( 1 << p ) - 1 s = 4 for i in range(3, p+1): s = (s ** 2 - 2) % m_p return s == 0 precision = 20000 # maximum requested number of decimal places of 2 ** MP-1 #long_bits_width = precision * log(10, 2)upb_prime = int( long_bits_width - 1 ) / 2 # no unsigned #upb_count = 45 # find 45 mprimes if int was given enough bits # print (" Finding Mersenne primes in M[2..%d]:"%upb_prime) count=0for p in range(2, int(upb_prime+1)): if is_prime(p) and is_mersenne_prime(p): print("M%d"%p), stdout.flush() count += 1 if count >= upb_count: breakprint  Output:  Finding Mersenne primes in M[2..33218]: M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9689 M9941 M11213 M19937 M21701 M23209 ### Faster loop without division  def isqrt(n): if n < 0: raise ValueError elif n < 2: return n else: a = 1 << ((1 + n.bit_length()) >> 1) while True: b = (a + n // a) >> 1 if b >= a: return a a = b def isprime(n): if n < 5: return n == 2 or n == 3 elif n%2 == 0: return False else: r = isqrt(n) k = 3 while k <= r: if n%k == 0: return False k += 2 return True def lucas_lehmer_fast(n): if n == 2: return True elif not isprime(n): return False else: m = 2**n - 1 s = 4 for i in range(2, n): sqr = s*s s = (sqr & m) + (sqr >> n) if s >= m: s -= m s -= 2 return s == 0 # test taken from the previous rosetta implementation from math import logfrom sys import stdout precision = 20000 # maximum requested number of decimal places of 2 ** MP-1 #long_bits_width = precision * log(10, 2)upb_prime = int( long_bits_width - 1 ) / 2 # no unsigned ## upb_count = 45 # find 45 mprimes if int was given enough bits #upb_count = 15 # find 45 mprimes if int was given enough bits # print (" Finding Mersenne primes in M[2..%d]:"%upb_prime) count=0# for p in range(2, upb_prime+1): for p in range(2, int(upb_prime+1)): if lucas_lehmer_fast(p): print("M%d"%p), stdout.flush() count += 1 if count >= upb_count: breakprint  The main loop may be run much faster using gmpy2 : import gmpy2 as mp def lucas_lehmer(n): if n == 2: return True if not mp.is_prime(n): return False two = mp.mpz(2) m = two**n - 1 s = two*two for i in range(2, n): sqr = s*s s = (sqr & m) + (sqr >> n) if s >= m: s -= m s -= two return mp.is_zero(s) With this, one can test all primes below 10^5 in around 24 hours on a Core i5 processor, with only one running thread. The primes found are 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243 Of course, they agree with OEIS A000043. ## R  # vectorized approach based on scalar code from primeSieve and mersenne in CRAN package numbersrequire(gmp)n <- 4423 # note that the sieve below assumes n > 9 # sieve the set of primes up to np <- seq(1, n, by = 2)q <- length(p)p[1] <- 2for (k in seq(3, sqrt(n), by = 2)) if (p[(k + 1)/2] != 0) p[seq((k * k + 1)/2, q, by = k)] <- 0p <- p[p > 0]cat(p[1]," special case M2 == 3\n")p <- p[-1] z2 <- gmp::as.bigz(2)z4 <- z2 * z2zp <- gmp::as.bigz(p)zmp <- z2^zp - 1S <- rep(z4, length(p)) for (i in 1:(p[length(p)] - 2)){ S <- gmp::mod.bigz(S * S - z2, zmp) if( i+2 == p[1] ){ if( S[1] == 0 ){ cat( p[1], "\n") flush.console() } p <- p[-1] zmp <- zmp[-1] S <- S[-1] }}  ## Racket  #lang racket(require math) (define (mersenne-prime? p) (divides? (- (expt 2 p) 1) (S (- p 1)))) (define (S n) (if (= n 1) 4 (- (sqr (S (- n 1))) 2))) (define (loop p) (when (mersenne-prime? p) (displayln p)) (loop (next-prime p))) (loop 3)  ## REXX REXX won't have a problem with the large number of digits involved, but since it's an interpreted language, such massive number crunching isn't conducive in searching for large primes. /*REXX pgm uses the Lucas─Lehmer primality test for prime powers of 2 (Mersenne primes)*/@.=0; @.2=1; @.3=1; @.5=1; @.7=1; @.11=1; @.13=1 /*a partial list of some low primes. */!.[email protected].; !.0=1; !.2=1; !.4=1; !.5=1; !.6=1; !.8=1 /*#'s with these last digs aren't prime*/parse arg limit . /*obtain optional arguments from the CL*/if limit=='' then limit= 200 /*Not specified? Then use the default.*/say center('Mersenne prime index list',70-3,"═") /*show a fancy─dancy header (or title).*/say right('M'2, 25) " [1 decimal digit]" /*left─justify them to align&look nice.*/ /* [►] note that J==1 is a special case*/ do j=1 by 2 to limit /*there're only so many hours in a day.*/ power= j + (j==1) /*POWER ≡ J except for when J=1. */ if \isPrime(power) then iterate /*if POWER isn't prime, then ignore it.*/= LL2(power)                           /*perform the Lucas─Lehmer 2 (LL2) test*/         if $=='' then iterate /*Did it flunk LL2? Then skip this #.*/ say right($, 25)   MPsize              /*left─justify them to align&look nice.*/         end   /*j*/exit                                             /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/isPrime: procedure expose !. @.                  /*allow 2 stemmed arrays to be accessed*/         parse arg  x    ''  -1  z               /*obtain variable   X   and last digit.*/         if @.x      then return 1               /*is  X  already found to be a prime?  */         if !.z      then return 0               /*is last decimal digit even or a five?*/         if x//3==0  then return 0               /*divisible by three?  Then not a prime*/         if x//7==0  then return 0               /*divisible by seven?    "   "  "   "  */                do j=11  by 6   until j*j > x    /*ensures that J isn't divisible by 3. */                if x //  j   ==0  then return 0  /*Is X divisible by  J   ?             */                if x // (j+2)==0  then return 0  /* " "     "      "  J+2 ?         ___ */                end   /*j*/                      /* [↑]  perform  DO  loop through √ x  */         @.x=1;                         return 1 /*indicate number  X  is a prime.      *//*──────────────────────────────────────────────────────────────────────────────────────*/LL2: procedure expose MPsize;    parse arg ?     /*Lucas─Lehmer test on    2**?  -  1   */     if ?==2  then s=0                           /*handle special case for an even prime*/              else s=4                           /* [↓]  same as NUMERIC FORM SCIENTIFIC*/     numeric form;               q= 2**?         /*ensure correct form for REXX numbers.*/           /*╔═══════════════════════════════════════════════════════════════════════════╗           ╔═╝ Compute a power of 2 using only 9 decimal digits.  One million digits     ║           ║ could be used, but that really slows up computations.  So, we start with the║           ║ default of 9 digits, and then find the ten's exponent in the product (2**?),║           ║ double it,  and then add 6.    {2  is all that's needed,  but  6  is a lot  ║           ║ safer.}   The doubling is for the squaring of   S    (below, for  s*s).   ╔═╝           ╚═══════════════════════════════════════════════════════════════════════════╝*/     if pos('E', q)\==0  then do                 /*is number in exponential notation ?  */                                  parse var q 'E' tenPow            /*get the exponent. */                                  numeric digits  tenPow * 2 + 6    /*expand precision. */                              end                                   /*REXX used dec FP. */                         else numeric digits    digits() * 2 + 6    /*use 9*2 + 6 digits*/     q=2**? - 1                                  /*compute a power of two,  minus one.  */        r= q // 8                                /*obtain   Q   modulus  eight.         */     if r==1 | r==7  then nop                    /*before crunching, do a simple test.  */                     else return ''              /*modulus   Q   isn't one  or  seven.  */                 do ?-2;       s= (s*s -2) // q  /*lather,  rinse,  repeat   ···        */                 end                             /* [↑]   compute and test for a  MP.   */     if s\==0  then return ''                    /*Not a Mersenne prime?  Return a null.*/     sz= length(q)                               /*obtain number of decimal digs in MP. */     MPsize=' ['sz      "decimal digit"s(sz)']'  /*define a literal to display after MP.*/                    return 'M'?                  /*return "modified" # (Mersenne index).*//*──────────────────────────────────────────────────────────────────────────────────────*/s:   if arg(1)==1  then return arg(3);  return word(arg(2) 's', 1)   /*simple pluralizer*/
output   when the following is used for input:     10000
═════════════════════Mersenne prime index list═════════════════════
M2  [1 decimal digit]
M3  [1 decimal digit]
M5  [2 decimal digits]
M7  [3 decimal digits]
M13  [4 decimal digits]
M17  [6 decimal digits]
M19  [6 decimal digits]
M31  [10 decimal digits]
M61  [19 decimal digits]
M89  [27 decimal digits]
M107  [33 decimal digits]
M127  [39 decimal digits]
M521  [157 decimal digits]
M607  [183 decimal digits]
M1279  [386 decimal digits]
M2203  [664 decimal digits]
M2281  [687 decimal digits]
M3217  [969 decimal digits]
M4253  [1281 decimal digits]
M4423  [1332 decimal digits]
M9689  [2917 decimal digits]
M9941  [2993 decimal digits]


## Ring

 see "Mersenne Primes :" + nlfor p = 2 to 18    if lucasLehmer(p) see "M"  + p + nl oknext func lucasLehmer p     i = 0 mp = 0 sn = 0     if p = 2 return true ok     if (p and 1) = 0 return false ok     mp = pow(2,p) - 1     sn = 4     for i = 3 to p         sn = pow(sn,2) - 2         sn -= (mp * floor(sn / mp))     next     return (sn=0)

## RPL

 %%HP: T(3)A(R)F(.);                                                          ; ASCII transfer header \<< DUP LN DUP \pi * 4 SWAP / 1 + UNROT / * IP 2 { 2 } ROT 2 SWAP            ; input n; n := Int(n/ln(n)*(1 + 4/(pi*ln(n)))), p:=2; (n ~ number of primes less then n, pi used here only as a convenience),  2 is assumed to be the 1st elemente in the list  START SWAP NEXTPRIME DUP UNROT DUP 2 SWAP ^ 1 - 4 PICK3 2 - 1 SWAP         ; for i := 2 to n,  p := nextprime;  s := 4; m := 2^p - 1;    START SQ 2 - OVER MOD                                                    ;   for j := 1 to p - 2;  s := s^2 mod m;      NEXT NIP NOT { + } { DROP } IFTE                                         ;   next j;  if s = 0 then add p to the list else discard p;   NEXT NIP                                                                   ; next i;     \>>
Output:
Outputs for arguments 130, 607 and 2281, respectively:

{ 2 3 5 7 13 17 19 31 61 89 107 127 }
{ 2 3 5 7 13 17 19 31 61 89 107 127 521 607 }
{ 2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281 }

These take respectively 1m 22s on the real HP 50g, 4m 29s and 10h 29m 23s on the emulator (Debug4 running on PC under WinXP, Intel(R) Core(TM) Duo CPU T2350 @ 1.86GHz).


## Ruby

def is_prime ( p )  return true  if p == 2  return false if p <= 1 || p.even?  (3 .. Math.sqrt(p)).step(2) do |i|    return false  if p % i == 0  end  trueend def is_mersenne_prime ( p )  return true  if p == 2  m_p = ( 1 << p ) - 1  s = 4  (p-2).times { s = (s ** 2 - 2) % m_p }  s == 0end precision = 20000   # maximum requested number of decimal places of 2 ** MP-1 #long_bits_width = precision / Math.log(2) * Math.log(10)upb_prime = (long_bits_width - 1).to_i / 2    # no unsigned #upb_count = 45      # find 45 mprimes if int was given enough bits # puts " Finding Mersenne primes in M[2..%d]:" % upb_prime count = 0for p in 2..upb_prime  if is_prime(p) && is_mersenne_prime(p)    print "M%d " % p    count += 1  end  break  if count >= upb_countendputs
Output:
 Finding Mersenne primes in M[2..33218]:
M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9689 M9941 M11213 M19937 M21701 M23209

## Rust

  extern crate rug;extern crate primal; use rug::Integer;use rug::ops::Pow;use std::thread::spawn; fn is_mersenne (p : usize) {    let p = p as u32;    let mut m = Integer::from(1);    m = m << p;      m = Integer::from(&m - 1);    let mut flag1 = false;    for k in 1..10_000 {        let mut flag2 = false;        let mut div : u32 = 2*k*p + 1;        if &div >= &m {break; }        for j in [3,5,7,11,13,17,19,23,29,31,37].iter() {            if div % j == 0 {                flag2 = true;                break;            }           }        if flag2 == true {continue;}        if div % 8 != 1 && div % 8 != 7 { continue; }        if m.is_divisible_u(div) {             flag1 = true;            break;        }    }    if flag1 == true {return ()}    let mut s = Integer::from(4);    let two = Integer::from(2);    for _i in 2..p {		let mut sqr = s.pow(2);		s = Integer::from(&Integer::from(&sqr & &m) + &Integer::from(&sqr >> p));		if &s >= &m {s = s - &m}		s = Integer::from(&s - &two);    }	if s == 0 {println!("Mersenne : {}",p);} } fn main () {    println!("Mersenne : 2");    let limit = 11_214;    let mut thread_handles = vec![];    for p in primal::Primes::all().take_while(|p| *p < limit) {        thread_handles.push(spawn(move || is_mersenne(p)));     }    for handle in thread_handles {        handle.join().unwrap();    }}

with Intel(R) Core(TM) i7-5500U CPU @ 2.40GHz : Less than 8,6 seconds to get the Mersenne primes up to 11213

Output:
Mersenne : 2
Mersenne : 5
Mersenne : 3
Mersenne : 7
Mersenne : 13
Mersenne : 17
Mersenne : 19
Mersenne : 31
Mersenne : 61
Mersenne : 89
Mersenne : 127
Mersenne : 107
Mersenne : 521
Mersenne : 607
Mersenne : 1279
Mersenne : 2281
Mersenne : 2203
Mersenne : 3217
Mersenne : 4423
Mersenne : 4253
Mersenne : 9689
Mersenne : 9941
Mersenne : 11213

real	0m8.581s
user	0m33.894s
sys	0m0.107s


## Scala

Library: Scala

In accordance with definition of Mersenne primes it could only be a Mersenne number with prime exponent.

object LLT extends App {  import Stream._   def primeSieve(s: Stream[Int]): Stream[Int] =    s.head #:: primeSieve(s.tail filter { _ % s.head != 0 })  val primes = primeSieve(from(2))   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 }   val upbPrime = 9941  println(s"Finding Mersenne primes in M[2..$upbPrime]") ((primes takeWhile (_ <= upbPrime)).par map { p => (p, mersenne(p)) } map { p => if (p._1 == 2) (p, 0) else (p, s(p._2, p._1 - 1)) } filter { _._2 == 0 }) .foreach { p => println(s"prime M${(p._1)._1}: " +        { if ((p._1)._1 < 200) (p._1)._2 else s"(${(p._1)._2.toString.size} digits)" }) } println("That's All Folks!")} Output: After approx 20 minutes (2.10 GHz dual core) Finding Mersenne primes in M[2..9999] prime M2: 3 prime M3: 7 prime M5: 31 prime M7: 127 prime M13: 8191 prime M17: 131071 prime M19: 524287 prime M31: 2147483647 prime M61: 2305843009213693951 prime M89: 618970019642690137449562111 prime M107: 162259276829213363391578010288127 prime M127: 170141183460469231731687303715884105727 prime M521: (157 digits) prime M607: (183 digits) prime M1279: (386 digits) prime M2203: (664 digits) prime M2281: (687 digits) prime M3217: (969 digits) prime M4253: (1281 digits) prime M4423: (1332 digits) prime M9689: (2917 digits) prime M9941: (2993 digits) That's All Folks! ## Scheme ;;;The heart of the algorithm(define (S n) (let ((m (- (expt 2 n) 1))) (let loop ((c (- n 2)) (a 4)) (if (zero? c) a (loop (- c 1) (remainder (- (* a a) 2) m)))))) (define (mersenne-prime? n) (if (= n 2) #t (zero? (S n)))) ;;;Trivial unoptimized implementation for the base primes(define (next-prime x) (if (prime? (+ x 1)) (+ x 1) (next-prime (+ x 1)))) (define (prime? x) (let loop ((c 2)) (cond ((>= c x) #t) ((zero? (remainder x c)) #f) (else (loop (+ c 1)))))) ;;Main loop(let loop ((i 45) (p 2)) (if (not (zero? i)) (if (mersenne-prime? p) (begin (display "M") (display p) (display " ") (loop (- i 1) (next-prime p))) (loop i (next-prime p))))) M2 M3 M5 M7 M13...  ## Scilab  iexpmax=30 n=1 for iexp=2:iexpmax if iexp==2 then s=0; else s=4; end n=(n+1)*2-1 for i=1:iexp-2 s=modulo((s*s-2),n) end if s==0 then printf("M%d ",iexp); end end Output: M2 M3 M5 M7 M13 M17 M19 ## Seed7 To get maximum speed the program should be compiled with -O2. $ include "seed7_05.s7i";  include "bigint.s7i"; const func boolean: isPrime (in integer: number) is func  result    var boolean: prime is FALSE;  local    var integer: upTo is 0;    var integer: testNum is 3;  begin    if number = 2 then      prime := TRUE;    elsif number rem 2 = 0 or number <= 1 then      prime := FALSE;    else      upTo := sqrt(number);      while number rem testNum <> 0 and testNum <= upTo do        testNum +:= 2;      end while;      prime := testNum > upTo;    end if;  end func; const func boolean: lucasLehmerTest (in integer: p) is func  result    var boolean: prime is TRUE;  local    var bigInteger: m_p is 0_;    var bigInteger: s is 4_;    var integer: i is 0;  begin    if p <> 2 then      m_p := 2_ ** p - 1_;      for i range 2 to pred(p) do        s := (s ** 2 - 2_) rem m_p;      end for;      prime := s = 0_;    end if;  end func; const proc: main is func  local    var integer: p is 2;  begin    writeln(" Mersenne primes:");    while p <= 3217 do      if isPrime(p) and lucasLehmerTest(p) then        write(" M" <& p);        flush(OUT);      end if;      incr(p);    end while;    writeln;  end func;

Original source: lucasLehmerTest, isPrime

Output:
 Mersenne primes:
M2 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217


## Sidef

Translation of: Perl 6
func is_mersenne_prime(p) {    return true if (p == 2)    var s = 4    var M = (2**p - 1)    { s = powmod(s, 2, M)-2 } * (p-2)    s == 0} Inf.times {|n|    if (n.is_prime && is_mersenne_prime(n)) {        say "M#{n}"    }}
Output:
M2
M3
M5
M7
M13
M17
M19
M31
M61
M89
M107
M127
M521
M607
M1279
M2203
M2281
^C


## Swift

Uses a sieve of Eratosthenes.

func lucasLehmer(_ p: Int) -> Bool {  let m = BigInt(2).power(p) - 1  var s = BigInt(4)   for _ in 0..<p-2 {    s = ((s * s) - 2) % m  }   return s == 0} for prime in Eratosthenes(upTo: 70) where lucasLehmer(prime) {  let m = Int(pow(2, Double(prime))) - 1   print("2^\(prime) - 1 = \(m) is prime")}
Output:
2^3 - 1 = 7 is prime
2^5 - 1 = 31 is prime
2^7 - 1 = 127 is prime
2^13 - 1 = 8191 is prime
2^17 - 1 = 131071 is prime
2^19 - 1 = 524287 is prime
2^31 - 1 = 2147483647 is prime
2^61 - 1 = 2305843009213693951 is prime

## Tcl

Translation of: Pop11
proc main argv {    set n 0    set t [clock seconds]    show_mersenne 2 [incr n] t     for {set p 3} {$p <= [lindex$argv 0]} {incr p 2} {        if {![prime $p]} continue if {[LucasLehmer$p]} {            show_mersenne $p [incr n] t } }}proc show_mersenne {p n timevar} { upvar 1$timevar time    set now [clock seconds]    puts [format "%2d: %5ds  M%s" $n [expr {$now - $time}]$p]    set time $now}proc prime i { if {$i in {2 3}} {return 1}   prime0 $i [expr {int(sqrt($i))}]}proc prime0 {i div} {    expr {!($i %$div)? 0: $div <= 2? 1: [prime0$i [incr div -1]]}}proc LucasLehmer p {    set mp [expr {2**$p-1}] set s 4 for {set i 2} {$i < $p} {incr i} { set s [expr {($s**2 - 2) % $mp}] } expr {$s == 0}} main 33218
Output:

The program was still running, but as the next Mersenne prime is 19937 there will be a long wait until the program finds it.

 1:     0s  M2
2:     0s  M3
3:     0s  M5
4:     0s  M7
5:     0s  M13
6:     0s  M17
7:     0s  M19
8:     0s  M31
9:     0s  M61
10:     0s  M89
11:     0s  M107
12:     0s  M127
13:     1s  M521
14:     0s  M607
15:     4s  M1279
16:    21s  M2203
17:     4s  M2281
18:    69s  M3217
19:   180s  M4253
20:    39s  M4423
21:  5543s  M9689
22:   655s  M9941
23:  3546s  M11213

## TI-83 BASIC

19→M1→NFor(E,2,M)If E=2 Then:0→SElse:4→SEnd(N+1)*2-1→NFor(I,1,E-2)Reminder(S*S-2,N)→SEndIf S=0 Then:Disp EEndEnd
Output:
2
3
5
7
13
17
19

## uBasic/4tH

Translation of: VBScript
m = 15n = 1For j = 2 To m    If j = 2 Then        s = 0    Else        s = 4    EndIf    n = (n + 1) * 2 - 1    For i = 1 To j - 2        s = (s * s - 2) % n    Next i    If s = 0 Then Print "M";jNext

## VBScript

iexpmax = 15n=1out=""For iexp = 2 To iexpmax	If iexp = 2 Then		s = 0	Else		s = 4	End If	n = (n + 1) * 2 - 1	For i = 1 To iexp - 2		s = (s * s - 2) Mod n	Next 	If s = 0 Then		out=out & "M" & iexp & " "	End IfNext Wscript.echo out
Output:
M2 M3 M5 M7 M13

## Visual Basic .NET

Works with: Visual Basic .NET version 2011
Public Class LucasLehmer    Private Sub btnGo_Click(sender As Object, e As EventArgs) Handles btnGo.Click        Const iexpmax = 31        Dim s, n As Long        Dim i, iexp As Integer        n = 1        txtOut.Text = ""        For iexp = 2 To iexpmax            If iexp = 2 Then                s = 0            Else                s = 4            End If            n = (n + 1) * 2 - 1            For i = 1 To iexp - 2                s = (s * s - 2) Mod n            Next i            If s = 0 Then                txtOut.Text = txtOut.Text & "M" & iexp & " "            End If        Next iexp    End SubEnd Class
Output:
M2 M3 M5 M7 M13 M17 M19 M31

## zkl

Using Extensible prime generator#zkl and the GMP library.

var [const] BN=Import.lib("zklBigNum");	// lib GMPprimes:=Utils.Generator(Import("sieve").postponed_sieve);fcn isMersennePrime(p){   if(p==2) return(True);   mp:=BN(1).shiftLeft(p) - 1; // 2^p - 1, a BIG number, like 1000s of digits   s:=BN(4); do(p-2){ s.mul(s).sub(2).mod(mp) } // the % REALLY cuts down on mem usage   return(s==0);}

Calculating S(n) is done in place (overwriting the value in the BigInt with the result); this really cuts down on memory usage.

mersennePrimes:=primes.tweak(fcn(p){ isMersennePrime(p) and p or Void.Skip });println("Mersenne primes:");foreach mp in (mersennePrimes) { print(" M",mp); }

This will "continuously" spew out Mersenne Primes.

Tweaking a Walker (aka iterator, Generators are a class of Walker) basically puts a filter on the underlying iterator, in this case, ignoring prime numbers that are not Mersenne primes and passing those that are.

Output:
Mersenne primes:
M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203
M2281 M3217 M4253 M4423 M9689 M9941 M11213 M19937 M21701 M23209 M44497 ^C


Additionally, this problem is readily threaded and has a linear speedup. Since there are lots of calculations between results, the [bigger] results are basically time sorted. However, N times faster doesn't mean much given the huge calculations used by the LL test (math with thousands of digits ain't quick).

ps,mpOut := Thread.Pipe(),Thread.Pipe(); // how the threads will communicatefcn(ps){   // a thread to generate primes, sleeps most of the time   Utils.Generator(Import("sieve").postponed_sieve).pump(ps)}.launch(ps); do(4){ // four threads to perform the Lucas-Lehmer test   fcn(ps,out){ ps.pump(out,isMersennePrime,Void.Filter) }.launch(ps,mpOut)}println("Mersenne primes:");foreach mp in (mpOut) { print(" M",mp); }