Prime decomposition: Difference between revisions
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=={{header|S-BASIC}}== |
=={{header|S-BASIC}}== |
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<lang S-BASIC> |
<lang S-BASIC> |
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rem - |
rem - return p mod q |
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function mod(p, q = integer) = integer |
function mod(p, q = integer) = integer |
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end = p - q * (p/q) |
end = p - q * (p/q) |
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Find the prime factors of n and store in global array factors |
Find the prime factors of n and store in global array factors |
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(arrays cannot be passed as parameters) and return the number |
(arrays cannot be passed as parameters) and return the number |
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found. If n is prime, it will be stored as the |
found. If n is prime, it will be stored as the only factor. |
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factor. |
|||
end |
end |
||
function primefactors(n = integer) = integer |
function primefactors(n = integer) = integer |
Revision as of 17:15, 8 August 2022
You are encouraged to solve this task according to the task description, using any language you may know.
The prime decomposition of a number is defined as a list of prime numbers which when all multiplied together, are equal to that number.
- Example
12 = 2 × 2 × 3, so its prime decomposition is {2, 2, 3}
- Task
Write a function which returns an array or collection which contains the prime decomposition of a given number greater than 1.
If your language does not have an isPrime-like function available, you may assume that you have a function which determines whether a number is prime (note its name before your code).
If you would like to test code from this task, you may use code from trial division or the Sieve of Eratosthenes.
Note: The program must not be limited by the word size of your computer or some other artificial limit; it should work for any number regardless of size (ignoring the physical limits of RAM etc).
- Related tasks
- count in factors
- factors of an integer
- Sieve of Eratosthenes
- primality by trial division
- factors of a Mersenne number
- trial factoring of a Mersenne number
- partition an integer X into N primes
- sequence of primes by Trial Division
11l
<lang 11l>F decompose(BigInt number)
[BigInt] result V n = number BigInt i = 2 L n % i == 0 result.append(i) n I/= i i = 3 L n >= i * i L n % i == 0 result.append(i) n I/= i i += 2 I n != 1 result.append(n) R result
L(i) 2..9
print(decompose(i))
print(decompose(1023 * 1024)) print(decompose(2 * 3 * 5 * 7 * 11 * 11 * 13 * 17)) print(decompose(BigInt(16860167264933) * 179951))</lang>
- Output:
[2] [3] [2, 2] [5] [2, 3] [7] [2, 2, 2] [3, 3] [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 11, 31] [2, 3, 5, 7, 11, 11, 13, 17] [179951, 16860167264933]
360 Assembly
For maximum compatibility, this program uses only the basic instruction set. <lang 360asm>PRIMEDE CSECT
USING PRIMEDE,R13 B 80(R15) skip savearea DC 17F'0' savearea DC CL8'PRIMEDE' STM R14,R12,12(R13) ST R13,4(R15) ST R15,8(R13) LR R13,R15 end prolog LA R2,0 LA R3,1023 LA R4,1024 MR R2,R4 ST R3,N n=1023*1024 LA R5,WBUFFER LA R6,0 L R1,N n XDECO R1,0(R5) LA R5,12(R5) MVC 0(3,R5),=C' : ' LA R5,3(R5) LA R0,2 ST R0,I i=2
WHILE1 EQU * do while(i<=n/2)
L R2,N SRA R2,1 L R4,I CR R4,R2 i<=n/2 BH EWHILE1
WHILE2 EQU * do while(n//i=0)
L R3,N LA R2,0 D R2,I LTR R2,R2 n//i=0 BNZ EWHILE2 ST R3,N n=n/i ST R3,M m=n L R1,I i XDECO R1,WDECO MVC 0(5,R5),WDECO+7 LA R5,5(R5) MVI OK,X'01' ok B WHILE2
EWHILE2 EQU *
L R4,I CH R4,=H'2' if i=2 then BNE NE2 LA R0,3 ST R0,I i=3 B EIFNE2
NE2 L R2,I else
LA R2,2(R2) ST R2,I i=i+2
EIFNE2 B WHILE1 EWHILE1 EQU *
CLI OK,X'01' if ^ok then BE NOTPRIME MVC 0(7,R5),=C'[prime]' LA R5,7(R5) B EPRIME
NOTPRIME L R1,M m
XDECO R1,WDECO MVC 0(5,R5),WDECO+7
EPRIME XPRNT WBUFFER,80 put
L R13,4(0,R13) epilog LM R14,R12,12(R13) XR R15,R15 BR R14
N DS F I DS F M DS F OK DC X'00' WBUFFER DC CL80' ' WDECO DS CL16
YREGS END PRIMEDE</lang>
- Output:
1047552 : 2 2 2 2 2 2 2 2 2 2 3 11 31
AArch64 Assembly
<lang AArch64 Assembly> /* ARM assembly AARCH64 Raspberry PI 3B */ /* program primeDecomp64.s */
/*******************************************/ /* Constantes file */ /*******************************************/ /* for this file see task include a file in language AArch64 assembly*/ .include "../includeConstantesARM64.inc" .equ NBFACT, 100
/*******************************************/ /* Structures */ /********************************************/ /* structurea area factors */
.struct 0
fac_value: // factor
.struct fac_value + 8
fac_number: // number of identical factors
.struct fac_number + 8
fac_end: /*******************************************/ /* Initialized data */ /*******************************************/ .data szMessStartPgm: .asciz "Program start \n" szMessEndPgm: .asciz "Program normal end.\n" szMessNotPrime: .asciz "Not prime.\n" szMessPrime: .asciz "Prime\n" szCarriageReturn: .asciz "\n" szSpaces: .asciz " " szMessNumber: .asciz " The factors of @ are :\n" /*******************************************/ /* UnInitialized data */ /*******************************************/ .bss sZoneConv: .skip 32 .align 4 tbZoneDecom: .skip fac_end * NBFACT /*******************************************/ /* code section */ /*******************************************/ .text .global main main: // program start
ldr x0,qAdrszMessStartPgm // display start message bl affichageMess ldr x20,qVal //mov x20,17 mov x0,x20 ldr x1,qAdrtbZoneDecom bl decompFact // decomposition cmp x0,#0 beq 1f mov x2,x0 mov x0,x20 ldr x1,qAdrtbZoneDecom bl displayFactors // display factors b 2f
1:
ldr x0,qAdrszMessPrime // prime bl affichageMess
2:
ldr x0,qAdrszMessEndPgm // display end message bl affichageMess
100: // standard end of the program
mov x0,0 // return code mov x8,EXIT // request to exit program svc 0 // perform system call
qAdrszMessStartPgm: .quad szMessStartPgm qAdrszMessEndPgm: .quad szMessEndPgm qAdrszCarriageReturn: .quad szCarriageReturn qAdrszMessNotPrime: .quad szMessNotPrime qAdrszMessPrime: .quad szMessPrime qAdrtbZoneDecom: .quad tbZoneDecom //qVal: .quad 2 <<31 qVal: .quad 1047552 // test not prime //qVal: .quad 1429671721 // test not prime (37811 * 37811)
/******************************************************************/ /* prime decomposition */ /******************************************************************/ /* x0 contains the number */ /* x1 contains address factors array */ /* REMARK no save register x9-x19 */ decompFact:
stp x1,lr,[sp,-16]! // save registers mov x12,x0 // save number bl isPrime // prime ? cbnz x0,12f // yes -> no decomposition mov x19,fac_end // element area size mov x18,0 // raz indice mov x16,0 // prev divisor mov x17,0 // number of identical divisors mov x13,2 // first divisor
2:
cmp x12,1 beq 10f udiv x14,x12,x13 // division msub x15,x14,x13,x12 // remainder = x12 -(x13*x14) cbnz x15,5f // if remainder <> zero x13 not divisor mov x12,x14 // quotient -> new dividende cmp x13,x16 // same divisor ? beq 4f // yes cbz x16,3f // yes it is first divisor ? madd x11,x18,x19,x1 // no -> store prev divisor in the area str x16,[x11,fac_value] str x17,[x11,fac_number] // and store number of same factor add x18,x18,1 // increment indice mov x17,0 // raz number of same factor
3:
mov x16,x13 // save new divisor
4:
add x17,x17,1 // increment number of same factor mov x0,x12 // the new dividende is prime ? bl isPrime cbnz x0,10f // yes b 2b // else loop
5: // divisor is not a factor
cmp x13,2 // begin ? cinc x13,x13,ne // if divisor <> 2 add 1 add x13,x13,1 b 2b // and loop
10: // new dividende is prime
cmp x16,x12 // divisor = dividende ? cinc x17,x17,eq //add 1 if last dividende = diviseur madd x11,x18,x19,x1 str x16,[x11,fac_value] // store divisor in area str x17,[x11,fac_number] // and store number add x18,x18,1 // increment indice cmp x16,x12 //store last dividende if <> diviseur beq 11f madd x11,x18,x19,x1 str x12,[x11,fac_value] // sinon stockage dans la table mov x17,1 str x17,[x11,fac_number] // store 1 in number add x18,x18,1
11:
mov x0,x18 // return nb factors b 100f
12:
mov x0,#0 // number is prime b 100f
100:
ldp x1,lr,[sp],16 // restaur des 2 registres ret // retour adresse lr x30
/******************************************************************/ /* prime decomposition */ /******************************************************************/ /* x0 contains the number */ /* x1 contains address factors array */ /* x2 number of factors */ displayFactors:
stp x1,lr,[sp,-16]! // save registres mov x19,fac_end // element area size mov x13,x1 // save area address ldr x1,qAdrsZoneConv // load zone conversion address bl conversion10 ldr x0,qAdrszMessNumber bl strInsertAtCharInc // insert result at Second @ character bl affichageMess mov x9,0 // indice
1:
madd x10,x9,x19,x13 // compute address area element ldr x0,[x10,fac_value] ldr x12,[x10,fac_number] bl conversion10 // decimal conversion
2:
mov x0,x1 bl affichageMess ldr x0,qAdrszSpaces bl affichageMess subs x12,x12,#1 bgt 2b add x9,x9,1 cmp x9,x2 blt 1b ldr x0,qAdrszCarriageReturn bl affichageMess
100:
ldp x1,lr,[sp],16 // restaur des 2 registres ret // retour adresse lr x30
qAdrsZoneConv: .quad sZoneConv qAdrszSpaces: .quad szSpaces qAdrszMessNumber: .quad szMessNumber /******************************************************************/ /* test if number is prime */ /******************************************************************/ /* x0 contains the number */ /* x0 return 1 if prime else return 0 */ isPrime:
stp x1,lr,[sp,-16]! // save registers cmp x0,1 // <= 1 ? ble 98f cmp x0,3 // 2 and 3 prime ble 97f tst x0,1 // even ? beq 98f mov x9,3 // first divisor
1:
udiv x11,x0,x9 msub x10,x11,x9,x0 // compute remainder cbz x10,98f // end if zero add x9,x9,#2 // increment divisor cmp x9,x11 // divisors<=quotient ? ble 1b // loop
97:
mov x0,1 // return prime b 100f
98:
mov x0,0 // not prime b 100f
100:
ldp x1,lr,[sp],16 // restaur 2 registers ret // return to address lr x30
/********************************************************/ /* File Include fonctions */ /********************************************************/ /* for this file see task include a file in language AArch64 assembly */ .include "../includeARM64.inc"
</lang>
- Output:
Program start The factors of 1047552 are : 2 2 2 2 2 2 2 2 2 2 3 11 31 Program normal end.
ABAP
<lang ABAP>class ZMLA_ROSETTA definition
public create public .
public section.
types: enumber TYPE N LENGTH 60, listof_enumber TYPE TABLE OF enumber .
class-methods FACTORS importing value(N) type ENUMBER exporting value(ORET) type LISTOF_ENUMBER . protected section. private section.
ENDCLASS.
CLASS ZMLA_ROSETTA IMPLEMENTATION.
- <SIGNATURE>---------------------------------------------------------------------------------------+
- | Static Public Method ZMLA_ROSETTA=>FACTORS
- +-------------------------------------------------------------------------------------------------+
- | [--->] N TYPE ENUMBER
- | [<---] ORET TYPE LISTOF_ENUMBER
- +--------------------------------------------------------------------------------------</SIGNATURE>
method FACTORS. CLEAR oret. WHILE n mod 2 = 0. n = n / 2. APPEND 2 to oret. ENDWHILE. DATA: lim type enumber, i type enumber. lim = sqrt( n ). i = 3. WHILE i <= lim. WHILE n mod i = 0. APPEND i to oret. n = n / i. lim = sqrt( n ). ENDWHILE. i = i + 2. ENDWHILE. IF n > 1. APPEND n to oret. ENDIF. endmethod.
ENDCLASS.</lang>
ACL2
<lang Lisp>(include-book "arithmetic-3/top" :dir :system)
(defun prime-factors-r (n i)
(declare (xargs :mode :program)) (cond ((or (zp n) (zp (- n i)) (zp i) (< i 2) (< n 2)) (list n)) ((= (mod n i) 0) (cons i (prime-factors-r (floor n i) 2))) (t (prime-factors-r n (1+ i)))))
(defun prime-factors (n)
(declare (xargs :mode :program)) (prime-factors-r n 2))</lang>
Ada
The solution is generic.
The package Prime_Numbers is instantiated by a type that supports necessary operations +, *, /, mod, >. The constants 0, 1, 2 are parameters too, because the type might have no literals. The same package is used for Almost prime#Ada, Semiprime#Ada, Count in factors#Ada, Primality by Trial Division#Ada, Sequence of primes by Trial Division#Ada, and Ulam_spiral_(for_primes)#Ada.
This is the specification of the generic package Prime_Numbers:
<lang ada>generic
type Number is private; Zero : Number; One : Number; Two : Number; with function "+" (X, Y : Number) return Number is <>; with function "*" (X, Y : Number) return Number is <>; with function "/" (X, Y : Number) return Number is <>; with function "mod" (X, Y : Number) return Number is <>; with function ">" (X, Y : Number) return Boolean is <>;
package Prime_Numbers is
type Number_List is array (Positive range <>) of Number; function Decompose (N : Number) return Number_List; function Is_Prime (N : Number) return Boolean;
end Prime_Numbers;</lang>
The function Decompose first estimates the maximal result length as log2 of the argument. Then it allocates the result and starts to enumerate divisors. It does not care to check if the divisors are prime, because non-prime divisors will be automatically excluded.
This is the implementation of the generic package Prime_Numbers:
<lang ada>package body Prime_Numbers is
-- auxiliary (internal) functions function First_Factor (N : Number; Start : Number) return Number is K : Number := Start; begin while ((N mod K) /= Zero) and then (N > (K*K)) loop K := K + One; end loop; if (N mod K) = Zero then return K; else return N; end if; end First_Factor; function Decompose (N : Number; Start : Number) return Number_List is F: Number := First_Factor(N, Start); M: Number := N / F; begin if M = One then -- F is the last factor return (1 => F); else return F & Decompose(M, Start); end if; end Decompose; -- functions visible from the outside function Decompose (N : Number) return Number_List is (Decompose(N, Two)); function Is_Prime (N : Number) return Boolean is (N > One and then First_Factor(N, Two)=N);
end Prime_Numbers;</lang>
In the example provided, the package Prime_Numbers is instantiated with plain integer type:
<lang ada>with Prime_Numbers, Ada.Text_IO;
procedure Test_Prime is
package Integer_Numbers is new Prime_Numbers (Natural, 0, 1, 2); use Integer_Numbers; procedure Put (List : Number_List) is begin for Index in List'Range loop Ada.Text_IO.Put (Positive'Image (List (Index))); end loop; end Put;
begin
Put (Decompose (12));
end Test_Prime;</lang>
- Output:
(decomposition of 12)
2 2 3
ALGOL 68
- note: This specimen retains the original Python coding style.
<lang algol68>#IF long int possible THEN #
MODE LINT = LONG INT; LINT lmax int = long max int; OP LLENG = (INT i)LINT: LENG i,
LSHORTEN = (LINT i)INT: SHORTEN i;
- ELSE
MODE LINT = INT; LINT lmax int = max int; OP LLENG = (INT i)LINT: i,
LSHORTEN = (LINT i)INT: i;
FI#
OP LLONG = (INT i)LINT: LLENG i;
MODE YIELDLINT = PROC(LINT)VOID;
PROC (LINT, YIELDLINT)VOID gen decompose;
INT upb cache = bits width;
BITS cache := 2r0; BITS cached := 2r0;
PROC is prime = (LINT n)BOOL: (
BOOL has factor := FALSE, out := TRUE; # FOR LINT factor IN # gen decompose(n, # ) DO ( # ## (LINT factor)VOID:( IF has factor THEN out := FALSE; GO TO done FI; has factor := TRUE # OD # )); done: out
);
PROC is prime cached := (LINT n)BOOL: (
LINT l half n = n OVER LLONG 2 - LLONG 1; IF l half n <= LLENG upb cache THEN INT half n = LSHORTEN l half n; IF half n ELEM cached THEN BOOL(half n ELEM cache) ELSE BOOL out = is prime(n); BITS mask = 2r1 SHL (upb cache - half n); cached := cached OR mask; IF out THEN cache := cache OR mask FI; out FI ELSE is prime(n) # above useful cache limit # FI
);
PROC gen primes := (YIELDLINT yield)VOID:(
yield(LLONG 2); LINT n := LLONG 3; WHILE n < l maxint - LLONG 2 DO yield(n); n +:= LLONG 2; WHILE n < l maxint - LLONG 2 AND NOT is prime cached(n) DO n +:= LLONG 2 OD OD
);
- PROC # gen decompose := (LINT in n, YIELDLINT yield)VOID: (
LINT n := in n; # FOR LINT p IN # gen primes( # ) DO ( # ## (LINT p)VOID: IF p*p > n THEN GO TO done ELSE WHILE n MOD p = LLONG 0 DO yield(p); n := n OVER p OD FI # OD # ); done: IF n > LLONG 1 THEN yield(n) FI
);
main:(
- FOR LINT m IN # gen primes( # ) DO ( #
- (LINT m)VOID:(
LINT p = LLONG 2 ** LSHORTEN m - LLONG 1; print(("2**",whole(m,0),"-1 = ",whole(p,0),", with factors:")); # FOR LINT factor IN # gen decompose(p, # ) DO ( # ## (LINT factor)VOID: print((" ",whole(factor,0))) # OD # ); print(new line); IF m >= LLONG 59 THEN GO TO done FI
- OD # ));
done: EMPTY
)</lang>
- Output:
2**2-1 = 3, with factors: 3 2**3-1 = 7, with factors: 7 2**5-1 = 31, with factors: 31 2**7-1 = 127, with factors: 127 2**11-1 = 2047, with factors: 23 89 2**13-1 = 8191, with factors: 8191 2**17-1 = 131071, with factors: 131071 2**19-1 = 524287, with factors: 524287 2**23-1 = 8388607, with factors: 47 178481 2**29-1 = 536870911, with factors: 233 1103 2089 2**31-1 = 2147483647, with factors: 2147483647 2**37-1 = 137438953471, with factors: 223 616318177 2**41-1 = 2199023255551, with factors: 13367 164511353 2**43-1 = 8796093022207, with factors: 431 9719 2099863 2**47-1 = 140737488355327, with factors: 2351 4513 13264529 2**53-1 = 9007199254740991, with factors: 6361 69431 20394401 2**59-1 = 576460752303423487, with factors: 179951 3203431780337
Note: ALGOL 68G took 49,109,599 BogoMI and ELLA ALGOL 68RS took 1,127,634 BogoMI to complete the example.
ALGOL-M
Sadly, ALGOL-M does not allow arrays to be passed as parameters to procedures or functions, so the routine must store its results in (and know the name of) the external array used for that purpose. <lang ALGOL> BEGIN
INTEGER I, K, NFOUND; INTEGER ARRAY FACTORS[1:16];
COMMENT - RETURN P MOD Q; INTEGER FUNCTION MOD (P, Q); INTEGER P, Q; BEGIN
MOD := P - Q * (P / Q);
END;
COMMENT
FIND THE PRIME FACTORS OF N AND STORE IN THE EXTERNAL ARRAY "FACTORS", RETURNING THE NUMBER FOUND. IF N IS PRIME, IT WILL BE STORED AS THE FIRST AND ONLY FACTOR;
INTEGER FUNCTION PRIMEFACTORS(N); INTEGER N; BEGIN
INTEGER P, COUNT; P := 2; COUNT := 1; WHILE N >= P * P DO BEGIN IF MOD(N, P) = 0 THEN BEGIN FACTORS[COUNT] := P; COUNT := COUNT + 1; N := N / P; END ELSE P := P + 1; END; FACTORS[COUNT] := N; PRIMEFACTORS := COUNT;
END;
COMMENT -- EXERCISE THE ROUTINE;
FOR I := 77 STEP 2 UNTIL 99 DO
BEGIN WRITE(I,":"); NFOUND := PRIMEFACTORS(I); COMMENT - PRINT OUT THE FACTORS THAT WERE FOUND; FOR K := 1 STEP 1 UNTIL NFOUND DO BEGIN WRITEON(FACTORS[K]); END; END;
END </lang>
- Output:
77: 7 11 79: 79 81: 3 3 3 3 83: 83 85: 5 17 87: 3 29 89: 89 91: 7 13 93: 3 31 95: 5 19 97: 97 99: 3 3 11
Applesoft BASIC
<lang ApplesoftBasic>9040 PF(0) = 0 : SC = 0 9050 FOR CA = 2 TO INT( SQR(I)) 9060 IF I = 1 THEN RETURN 9070 IF INT(I / CA) * CA = I THEN GOSUB 9200 : GOTO 9060 9080 CA = CA + SC : SC = 1 9090 NEXT CA 9100 IF I = 1 THEN RETURN 9110 CA = I
9200 PF(0) = PF(0) + 1 9210 PF(PF(0)) = CA 9220 I = I / CA 9230 RETURN</lang>
Arturo
<lang rebol>decompose: function [num][
facts: to [:string] factors.prime num print [ pad.right (to :string num) ++ " = " ++ join.with:" x " facts 30 "{"++ (join.with:", " unique facts) ++ "}" ]
]
loop 2..40 => decompose</lang>
- Output:
2 = 2 {2} 3 = 3 {3} 4 = 2 x 2 {2} 5 = 5 {5} 6 = 2 x 3 {2, 3} 7 = 7 {7} 8 = 2 x 2 x 2 {2} 9 = 3 x 3 {3} 10 = 2 x 5 {2, 5} 11 = 11 {11} 12 = 2 x 2 x 3 {2, 3} 13 = 13 {13} 14 = 2 x 7 {2, 7} 15 = 3 x 5 {3, 5} 16 = 2 x 2 x 2 x 2 {2} 17 = 17 {17} 18 = 2 x 3 x 3 {2, 3} 19 = 19 {19} 20 = 2 x 2 x 5 {2, 5} 21 = 3 x 7 {3, 7} 22 = 2 x 11 {2, 11} 23 = 23 {23} 24 = 2 x 2 x 2 x 3 {2, 3} 25 = 5 x 5 {5} 26 = 2 x 13 {2, 13} 27 = 3 x 3 x 3 {3} 28 = 2 x 2 x 7 {2, 7} 29 = 29 {29} 30 = 2 x 3 x 5 {2, 3, 5} 31 = 31 {31} 32 = 2 x 2 x 2 x 2 x 2 {2} 33 = 3 x 11 {3, 11} 34 = 2 x 17 {2, 17} 35 = 5 x 7 {5, 7} 36 = 2 x 2 x 3 x 3 {2, 3} 37 = 37 {37} 38 = 2 x 19 {2, 19} 39 = 3 x 13 {3, 13} 40 = 2 x 2 x 2 x 5 {2, 5}
AutoHotkey
<lang AutoHotkey>MsgBox % factor(8388607) ; 47 * 178481
factor(n) {
if (n = 1) return f = 2 while (f <= n) { if (Mod(n, f) = 0) { next := factor(n / f) return, % f "`n" next } f++ }
}</lang>
Optimized Version
<lang AutoHotkey>prime_numbers(n) {
if (n <= 3) return [n] ans := [] done := false while !done { if !Mod(n,2){ ans.push(2) n /= 2 continue } if !Mod(n,3) { ans.push(3) n /= 3 continue } if (n = 1) return ans sr := sqrt(n) done := true ; try to divide the checked number by all numbers till its square root. i := 6 while (i <= sr+6){ if !Mod(n, i-1) { ; is n divisible by i-1? ans.push(i-1) n /= i-1 done := false break } if !Mod(n, i+1) { ; is n divisible by i+1? ans.push(i+1) n /= i+1 done := false break } i += 6 } } ans.push(n) return ans
}</lang> Examples:<lang AutoHotkey>num := 8388607, output := "" for i, p in prime_numbers(num)
output .= p " * "
MsgBox % num " = " Trim(output, " * ") return</lang>
- Output:
8388607 = 47 * 178481
AWK
As the examples show, pretty large numbers can be factored in tolerable time:
<lang awk># Usage: awk -f primefac.awk function pfac(n, r, f){
r = ""; f = 2 while (f <= n) { while(!(n % f)) { n = n / f r = r " " f } f = f + 2 - (f == 2) } return r
}
- For each line of input, print the prime factors.
{ print pfac($1) } </lang>
- Output:
entering input on stdin
$ 36 2 2 3 3 77 7 11 536870911 233 1103 2089 8796093022207 431 9719 2099863
Batch file
Unfortunately Batch does'nt have a BigNum library so the maximum number that can be decomposed is 2^31-1 <lang Batch file> @echo off
- usage: cmd /k primefactor.cmd number
setlocal enabledelayedexpansion
set /a compo=%1 if "%compo%"=="" goto:eof set list=%compo%= (
set /a div=2 & call :loopdiv set /a div=3 & call :loopdiv set /a div=5,inc=2
- looptest
call :loopdiv set /a div+=inc,inc=6-inc,div2=div*div if %div2% lss %compo% goto looptest if %compo% neq 1 set list= %list% %compo% echo %list%) & goto:eof
- loopdiv
set /a "res=compo%%div if %res% neq 0 goto:eof set list=%list% %div%, set/a compo/=div goto:loopdiv </lang>
Befunge
Handles safely integers only up to 250 (or ones which don't have prime divisors greater than 250). <lang Befunge>& 211p > : 1 - #v_ 25*, @ > 11g:. / v
> : 11g %!| > 11g 1+ 11p v ^ <</lang>
BQN
An efficient Factor
function using trial division and Pollard's rho algorithm is given in bqn-libs primes.bqn. The following standalone version is based on the trial division there, and builds in the sieve from Extensible prime generator.
<lang bqn>Factor ← { 𝕊n:
# Prime sieve primes ← ↕0 Sieve ← { p 𝕊 a‿b: p(⍋↑⊣)↩√b ⋄ l←b-a E ← {↕∘⌈⌾(((𝕩|-a)+𝕩×⊢)⁼)l} # Indices of multiples of 𝕩 a + / (1⥊˜l) E⊸{0¨⌾(𝕨⊸⊏)𝕩}´ p # Primes in segment [a,b) } # Factor by trial division r ← ↕0 # Result list Try ← { m ← (1+⌊√n) ⌊ 2×𝕩 # Upper bound for factors this round 𝕩<m ? # Stop if no factors primes ∾↩ np ← primes Sieve 𝕩‿m # New primes {0=𝕩|n? r∾↩𝕩 ⋄ n÷↩𝕩 ⋄ 𝕊𝕩 ;@}¨ np # Try each one 𝕊 m # Next segment ;@} Try 2 r ∾ 1⊸<⊸⥊n
}</lang>
- Output:
<lang bqn> > ⋈⟜Factor¨ 1232123+↕4 # Some factored numbers ┌─ ╵ 1232123 ⟨ 29 42487 ⟩
1232124 ⟨ 2 2 3 102677 ⟩ 1232125 ⟨ 5 5 5 9857 ⟩ 1232126 ⟨ 2 7 17 31 167 ⟩ ┘</lang>
Burlesque
<lang burlesque> blsq ) 12fC {2 2 3} </lang>
C
Version 1
Relatively sophiscated sieve method based on size 30 prime wheel. The code does not pretend to handle prime factors larger than 64 bits. All 32-bit primes are cached with 137MB data. Cache data takes about a minute to compute the first time the program is run, which is also saved to the current directory, and will be loaded in a second if needed again. <lang c>#include <inttypes.h>
- include <stdio.h>
- include <stdlib.h>
- include <string.h>
- include <assert.h>
typedef uint32_t pint; typedef uint64_t xint; typedef unsigned int uint;
- define PRIuPINT PRIu32 /* printf macro for pint */
- define PRIuXINT PRIu64 /* printf macro for xint */
- define MAX_FACTORS 63 /* because 2^64 is too large for xint */
uint8_t *pbits;
- define MAX_PRIME (~(pint)0)
- define MAX_PRIME_SQ 65535U
- define PBITS (MAX_PRIME / 30 + 1)
pint next_prime(pint); int is_prime(xint); void sieve(pint);
uint8_t bit_pos[30] = {
0, 1<<0, 0, 0, 0, 0, 0, 1<<1, 0, 0, 0, 1<<2, 0, 1<<3, 0, 0, 0, 1<<4, 0, 1<<5, 0, 0, 0, 1<<6, 0, 0, 0, 0, 0, 1<<7,
};
uint8_t rem_num[] = { 1, 7, 11, 13, 17, 19, 23, 29 };
void init_primes() {
FILE *fp; pint s, tgt = 4;
if (!(pbits = malloc(PBITS))) { perror("malloc"); exit(1); }
if ((fp = fopen("primebits", "r"))) { fread(pbits, 1, PBITS, fp); fclose(fp); return; }
memset(pbits, 255, PBITS); for (s = 7; s <= MAX_PRIME_SQ; s = next_prime(s)) { if (s > tgt) { tgt *= 2; fprintf(stderr, "sieve %"PRIuPINT"\n", s); } sieve(s); } fp = fopen("primebits", "w"); fwrite(pbits, 1, PBITS, fp); fclose(fp);
}
int is_prime(xint x) {
pint p; if (x > 5) { if (x < MAX_PRIME) return pbits[x/30] & bit_pos[x % 30];
for (p = 2; p && (xint)p * p <= x; p = next_prime(p)) if (x % p == 0) return 0;
return 1; } return x == 2 || x == 3 || x == 5;
}
void sieve(pint p) {
unsigned char b[8]; off_t ofs[8]; int i, q;
for (i = 0; i < 8; i++) { q = rem_num[i] * p; b[i] = ~bit_pos[q % 30]; ofs[i] = q / 30; }
for (q = ofs[1], i = 7; i; i--) ofs[i] -= ofs[i-1];
for (ofs[0] = p, i = 1; i < 8; i++) ofs[0] -= ofs[i];
for (i = 1; q < PBITS; q += ofs[i = (i + 1) & 7]) pbits[q] &= b[i];
}
pint next_prime(pint p) {
off_t addr; uint8_t bits, rem;
if (p > 5) { addr = p / 30; bits = bit_pos[ p % 30 ] << 1; for (rem = 0; (1 << rem) < bits; rem++); while (pbits[addr] < bits || !bits) { if (++addr >= PBITS) return 0; bits = 1; rem = 0; } if (addr >= PBITS) return 0; while (!(pbits[addr] & bits)) { rem++; bits <<= 1; } return p = addr * 30 + rem_num[rem]; }
switch(p) { case 2: return 3; case 3: return 5; case 5: return 7; } return 2;
}
int decompose(xint n, xint *f) {
pint p = 0; int i = 0;
/* check small primes: not strictly necessary */ if (n <= MAX_PRIME && is_prime(n)) { f[0] = n; return 1; }
while (n >= (xint)p * p) { if (!(p = next_prime(p))) break; while (n % p == 0) { n /= p; f[i++] = p; } } if (n > 1) f[i++] = n; return i;
}
int main() {
int i, len; pint p = 0; xint f[MAX_FACTORS], po;
init_primes();
for (p = 1; p < 64; p++) { po = (1LLU << p) - 1; printf("2^%"PRIuPINT" - 1 = %"PRIuXINT, p, po); fflush(stdout); if ((len = decompose(po, f)) > 1) for (i = 0; i < len; i++) printf(" %c %"PRIuXINT, i?'x':'=', f[i]); putchar('\n'); }
return 0;
}</lang>
Using GNU Compiler Collection gcc extensions
Note: The following code sample is experimental as it implements python style iterators for (potentially) infinite sequences. C is not normally written this way, and in the case of this sample it requires the GCC "nested procedure" extension to the C language. <lang c>#include <limits.h>
- include <stdio.h>
- include <math.h>
typedef enum{false=0, true=1}bool; const int max_lint = LONG_MAX;
typedef long long int lint;
- assert sizeof_long_long_int (LONG_MAX>=8) /* XXX */
/* the following line is the only time I have ever required "auto" */
- define FOR(i,iterator) auto bool lambda(i); yield_init = (void *)λ iterator; bool lambda(i)
- define DO {
- define YIELD(x) if(!yield(x))return
- define BREAK return false
- define CONTINUE return true
- define OD CONTINUE; }
/* Warning: _Most_ FOR(,){ } loops _must_ have a CONTINUE as the last statement.
* Otherwise the lambda will return random value from stack, and may terminate early */
typedef void iterator, lint_iterator; /* hint at procedure purpose */ static volatile void *yield_init; /* not thread safe */
- define YIELDS(type) bool (*yield)(type) = yield_init
typedef unsigned int bits;
- define ELEM(shift, bits) ( (bits >> shift) & 0b1 )
bits cache = 0b0, cached = 0b0; const lint upb_cache = 8 * sizeof(cache);
lint_iterator decompose(lint); /* forward declaration */
bool is_prime(lint n){
bool has_factor = false, out = true;
/* for factor in decompose(n) do */
FOR(lint factor, decompose(n)){ if( has_factor ){ out = false; BREAK; } has_factor = true; CONTINUE; } return out;
}
bool is_prime_cached (lint n){
lint half_n = n / 2 - 2; if( half_n <= upb_cache){ /* dont cache the initial four, nor the even numbers */ if (ELEM(half_n,cached)){ return ELEM(half_n,cache); } else { bool out = is_prime(n); cache = cache | out << half_n; cached = cached | 0b1 << half_n; return out; } } else { return is_prime(n); }
}
lint_iterator primes (){
YIELDS(lint); YIELD(2); lint n = 3; while( n < max_lint - 2 ){ YIELD(n); n += 2; while( n < max_lint - 2 && ! is_prime_cached(n) ){ n += 2; } }
}
lint_iterator decompose (lint in_n){
YIELDS(lint); lint n = in_n; /* for p in primes do */ FOR(lint p, primes()){ if( p*p > n ){ BREAK; } else { while( n % p == 0 ){ YIELD(p); n = n / p; } } CONTINUE; } if( n > 1 ){ YIELD(n); }
}
main(){
FOR(lint m, primes()){ lint p = powl(2, m) - 1; printf("2**%lld-1 = %lld, with factors:",m,p); FOR(lint factor, decompose(p)){ printf(" %lld",factor); fflush(stdout); CONTINUE; } printf("\n",m); if( m >= 59 )BREAK; CONTINUE; }
}</lang>
- Output:
2**2-1 = 3, with factors: 3 2**3-1 = 7, with factors: 7 2**5-1 = 31, with factors: 31 2**7-1 = 127, with factors: 127 2**11-1 = 2047, with factors: 23 89 2**13-1 = 8191, with factors: 8191 2**17-1 = 131071, with factors: 131071 2**19-1 = 524287, with factors: 524287 2**23-1 = 8388607, with factors: 47 178481 2**29-1 = 536870911, with factors: 233 1103 2089 2**31-1 = 2147483647, with factors: 2147483647 2**37-1 = 137438953471, with factors: 223 616318177 2**41-1 = 2199023255551, with factors: 13367 164511353 2**43-1 = 8796093022207, with factors: 431 9719 2099863 2**47-1 = 140737488355327, with factors: 2351 4513 13264529 2**53-1 = 9007199254740991, with factors: 6361 69431 20394401 2**59-1 = 576460752303423487, with factors: 179951 3203431780337
Note: gcc took 487,719 BogoMI to complete the example.
To understand what was going on with the above code, pass it through cpp
and read the outcome. Translated into normal C code sans the function call overhead, it's really this (the following uses a adjustable cache, although setting it beyond a few thousands doesn't gain further benefit):<lang c>#include <stdio.h>
- include <stdlib.h>
- include <stdint.h>
typedef uint32_t pint; typedef uint64_t xint; typedef unsigned int uint;
int is_prime(xint);
inline int next_prime(pint p) {
if (p == 2) return 3; for (p += 2; p > 1 && !is_prime(p); p += 2); if (p == 1) return 0; return p;
}
int is_prime(xint n) {
- define NCACHE 256
- define S (sizeof(uint) * 2)
static uint cache[NCACHE] = {0}; pint p = 2; int ofs, bit = -1; if (n < NCACHE * S) { ofs = n / S; bit = 1 << ((n & (S - 1)) >> 1); if (cache[ofs] & bit) return 1; } do { if (n % p == 0) return 0; if (p * p > n) break; } while ((p = next_prime(p))); if (bit != -1) cache[ofs] |= bit; return 1;
}
int decompose(xint n, pint *out) {
int i = 0; pint p = 2; while (n > p * p) { while (n % p == 0) { out[i++] = p; n /= p; } if (!(p = next_prime(p))) break; } if (n > 1) out[i++] = n; return i;
}
int main() {
int i, j, len; xint z; pint out[100]; for (i = 2; i < 64; i = next_prime(i)) { z = (1ULL << i) - 1; printf("2^%d - 1 = %llu = ", i, z); fflush(stdout); len = decompose(z, out); for (j = 0; j < len; j++) printf("%u%s", out[j], j < len - 1 ? " x " : "\n"); } return 0;
}</lang>
Version 2
<lang c> typedef unsigned long long int ulong; // define a type that represent the limit (64-bit)
ulong mod_mul(ulong a, ulong b, const ulong mod) { ulong res = 0, c; // return (a * b) % mod, avoiding overflow errors while doing modular multiplication. for (b %= mod; a; a & 1 ? b >= mod - res ? res -= mod : 0, res += b : 0, a >>= 1, (c = b) >= mod - b ? c -= mod : 0, b += c); return res % mod; }
ulong mod_pow(ulong n, ulong exp, const ulong mod) { ulong res = 1; // return (n ^ exp) % mod for (n %= mod; exp; exp & 1 ? res = mod_mul(res, n, mod) : 0, n = mod_mul(n, n, mod), exp >>= 1); return res; }
ulong square_root(const ulong N) { ulong res = 0, rem = N, c, d; for (c = 1 << 62; c; c >>= 2) { d = res + c; res >>= 1; if (rem >= d) rem -= d, res += c; } // returns the square root of N. return res; }
int is_prime(const ulong N) { ulong i = 1; // return a truthy value about the primality of N. if (N > 1) for (; i < 64 && mod_pow(i, N - 1, N) <= 1; ++i); return i == 64; }
ulong pollard_rho(const ulong N) { // Require : N is a composite number, not a square. // Ensure : res is a non-trivial factor of N. // Option : change the timeout, change the rand function. static const int timeout = 18; static unsigned long long rand_val = 2994439072U; rand_val = (rand_val * 1025416097U + 286824428U) % 4294967291LLU; ulong res = 1, a, b, c, i = 0, j = 1, x = 1, y = 1 + rand_val % (N - 1); for (; res == 1; ++i) { if (i == j) { if (j >> timeout) break; j <<= 1; x = y; } a = y, b = y; // performs y = (y * y) % N for (y = 0; a; a & 1 ? b >= N - y ? y -= N : 0, y += b : 0, a >>= 1, (c = b) >= N - b ? c -= N : 0, b += c); y = (1 + y) % N; for (a = y > x ? y - x : x - y, b = N; (a %= b) && (b %= a);); // compute the gcd(abs(y - x), N); res = a | b; } return res; }
void factor(const ulong N, ulong *array) { // very basic manager that fill the given array (the size of the result is the first array element) // it does not perform initial trial divisions, which is generally highly recommended. if (N < 4 || is_prime(N)) { if (N > 1 || !*array) array[++*array] = N; return; } ulong x = square_root(N); if (x * x != N) x = pollard_rho(N); factor(x, array); factor(N / x, array); }
- include <stdio.h>
int main(void) { // simple test. unsigned long long n = 18446744073709551615U; ulong fac[65] = {0}; factor(n, fac); for (ulong i = 1; i <= *fac; ++i) printf("* %llu\n", fac[i]); }
</lang>
C#
<lang csharp>using System; using System.Collections.Generic;
namespace PrimeDecomposition {
class Program { static void Main(string[] args) { GetPrimes(12); }
static List<int> GetPrimes(decimal n) { List<int> storage = new List<int>(); while (n > 1) { int i = 1; while (true) { if (IsPrime(i)) { if (((decimal)n / i) == Math.Round((decimal) n / i)) { n /= i; storage.Add(i); break; } } i++; } } return storage; }
static bool IsPrime(int n) { if (n <= 1) return false; for (int i = 2; i <= Math.Sqrt(n); i++) if (n % i == 0) return false; return true; } }
}</lang>
Simple trial division
This version a translation from Java of the sample presented by Robert C. Martin during a TDD talk at NDC 2011.
Although this three-line algorithm does not mention anything about primes, the fact that factors are taken out of the number n
in ascending order garantees the list will only contain primes.
<lang csharp>using System.Collections.Generic;
namespace PrimeDecomposition {
public class Primes { public List<int> FactorsOf(int n) { var factors = new List<int>();
for (var divisor = 2; n > 1; divisor++) for (; n % divisor == 0; n /= divisor) factors.Add(divisor);
return factors; }
}</lang>
C++
<lang cpp>#include <iostream>
- include <gmpxx.h>
// This function template works for any type representing integers or // nonnegative integers, and has the standard operator overloads for // arithmetic and comparison operators, as well as explicit conversion // from int. // // OutputIterator must be an output iterator with value_type Integer. // It receives the prime factors. template<typename Integer, typename OutputIterator>
void decompose(Integer n, OutputIterator out)
{
Integer i(2);
while (n != 1) { while (n % i == Integer(0)) { *out++ = i; n /= i; } ++i; }
}
// this is an output iterator similar to std::ostream_iterator, except // that it outputs the separation string *before* the value, but not // before the first value (i.e. it produces an infix notation). template<typename T> class infix_ostream_iterator:
public std::iterator<T, std::output_iterator_tag>
{
class Proxy; friend class Proxy; class Proxy { public: Proxy(infix_ostream_iterator& iter): iterator(iter) {} Proxy& operator=(T const& value) { if (!iterator.first) { iterator.stream << iterator.infix; } iterator.stream << value; } private: infix_ostream_iterator& iterator; };
public:
infix_ostream_iterator(std::ostream& os, char const* inf): stream(os), first(true), infix(inf) { } infix_ostream_iterator& operator++() { first = false; return *this; } infix_ostream_iterator operator++(int) { infix_ostream_iterator prev(*this); ++*this; return prev; } Proxy operator*() { return Proxy(*this); }
private:
std::ostream& stream; bool first; char const* infix;
};
int main() {
std::cout << "please enter a positive number: "; mpz_class number; std::cin >> number; if (number <= 0) std::cout << "this number is not positive!\n;"; else { std::cout << "decomposition: "; decompose(number, infix_ostream_iterator<mpz_class>(std::cout, " * ")); std::cout << "\n"; }
}</lang>
Simple trial division
<lang cpp>// Factorization by trial division in C++11
- include <iostream>
- include <vector>
using long_pair = std::pair<long,long>; using lp_vec = std::vector<long_pair>;
lp_vec factorize(long n) {
lp_vec fs; int cnt = 0; for (;n%2==0; n/=2) cnt++; // optimized by compiler if (cnt > 0) fs.push_back({2, cnt}); for (long i=3; i*i<=n; i+=2) { cnt = 0; for (;n%i==0; n/=i) cnt++; if (cnt>0) fs.push_back({i, cnt}); } if (n>1) fs.push_back({n, 1}); return fs;
}
int main() {
long n; std::cin >> n; auto fs = factorize(n); for (auto fp : fs) { std::cout << fp.first << "^" << fp.second << "\n"; } return 0;
}</lang>
Clojure
<lang clojure>;;; No stack consuming algorithm (defn factors
"Return a list of factors of N." ([n] (factors n 2 ())) ([n k acc] (if (= 1 n) acc (if (= 0 (rem n k)) (recur (quot n k) k (cons k acc)) (recur n (inc k) acc)))))</lang>
Commodore BASIC
It's not easily possible to have arbitrary precision integers in PET basic, so here is at least a version using built-in data types (reals). On return from the subroutine starting at 9000 the global array pf contains the number of factors followed by the factors themselves: <lang zxbasic>9000 REM ----- function generate 9010 REM in ... i ... number 9020 REM out ... pf() ... factors 9030 REM mod ... ca ... pf candidate 9040 pf(0)=0 : ca=2 : REM special case 9050 IF i=1 THEN RETURN 9060 IF INT(i/ca)*ca=i THEN GOSUB 9200 : GOTO 9050 9070 FOR ca=3 TO INT( SQR(i)) STEP 2 9080 IF i=1 THEN RETURN 9090 IF INT(i/ca)*ca=i THEN GOSUB 9200 : GOTO 9080 9100 NEXT 9110 IF i>1 THEN ca=i : GOSUB 9200 9120 RETURN 9200 pf(0)=pf(0)+1 9210 pf(pf(0))=ca 9220 i=i/ca 9230 RETURN</lang>
Common Lisp
<lang Lisp>;;; Recursive algorithm (defun factor (n)
"Return a list of factors of N." (when (> n 1) (loop with max-d = (isqrt n) for d = 2 then (if (evenp d) (+ d 1) (+ d 2)) do (cond ((> d max-d) (return (list n))) ; n is prime ((zerop (rem n d)) (return (cons d (factor (truncate n d)))))))))</lang>
<lang Lisp>;;; Tail-recursive version (defun factor (n &optional (acc '()))
(when (> n 1) (loop with max-d = (isqrt n) for d = 2 then (if (evenp d) (1+ d) (+ d 2)) do (cond ((> d max-d) (return (cons (list n 1) acc))) ((zerop (rem n d)) (return (factor (truncate n d) (if (eq d (caar acc)) (cons (list (caar acc) (1+ (cadar acc))) (cdr acc)) (cons (list d 1) acc)))))))))</lang>
D
<lang d>import std.stdio, std.bigint, std.algorithm, std.traits, std.range;
Unqual!T[] decompose(T)(in T number) pure nothrow in {
assert(number > 1);
} body {
typeof(return) result; Unqual!T n = number;
for (Unqual!T i = 2; n % i == 0; n /= i) result ~= i; for (Unqual!T i = 3; n >= i * i; i += 2) for (; n % i == 0; n /= i) result ~= i;
if (n != 1) result ~= n; return result;
}
void main() {
writefln("%(%s\n%)", iota(2, 10).map!decompose); decompose(1023 * 1024).writeln; BigInt(2 * 3 * 5 * 7 * 11 * 11 * 13 * 17).decompose.writeln; decompose(16860167264933UL.BigInt * 179951).writeln; decompose(2.BigInt ^^ 100_000).group.writeln;
}</lang>
- Output:
[2] [3] [2, 2] [5] [2, 3] [7] [2, 2, 2] [3, 3] [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 11, 31] [2, 3, 5, 7, 11, 11, 13, 17] [179951, 16860167264933] [Tuple!(BigInt, uint)(2, 100000)]
Delphi
<lang Delphi> program Prime_decomposition;
{$APPTYPE CONSOLE}
uses
System.SysUtils;
function IsPrime(n: UInt64): Boolean; var
i: Integer;
begin
if n <= 1 then exit(False);
i := 2; while i < Sqrt(n) do begin if n mod i = 0 then exit(False); inc(i); end;
Result := True;
end;
function GetPrimes(n: UInt64): TArray<UInt64>; var
i: Integer;
begin
while n > 1 do begin i := 1; while True do begin if IsPrime(i) then begin if n / i = (round(n / i)) then begin n := n div i; SetLength(Result, Length(Result) + 1); Result[High(Result)] := i; Break; end; end; inc(i); end; end;
end;
begin
for var v in GetPrimes(12) do write(v, ' '); readln;
end.</lang>
E
This example assumes a function isPrime
and was tested with this one. It could use a self-referential implementation such as the Python task, but the original author of this example did not like the ordering dependency involved.
<lang e>def primes := {
var primesCache := [2] /** A collection of all prime numbers. */ def primes { to iterate(f) { primesCache.iterate(f) for x in (int > primesCache.last()) { if (isPrime(x)) { f(primesCache.size(), x) primesCache with= x } } } }
}
def primeDecomposition(var x :(int > 0)) {
var factors := [] for p in primes { while (x % p <=> 0) { factors with= p x //= p } if (x <=> 1) { break } } return factors
}</lang>
EchoLisp
The built-in prime-factors function performs the task. <lang lisp>(prime-factors 1024)
→ (2 2 2 2 2 2 2 2 2 2)
(lib 'bigint)
- 2^59 - 1
(prime-factors (1- (expt 2 59)))
→ (179951 3203431780337)
(prime-factors 100000000000000000037)
→ (31 821 66590107 59004541)</lang>
Eiffel
Uses the feature prime from the Task Primality by Trial Devision in the contract to check if the Result contains only prime numbers. <lang Eiffel>class
PRIME_DECOMPOSITION
feature
factor (p: INTEGER): ARRAY [INTEGER] -- Prime decomposition of 'p'. require p_positive: p > 0 local div, i, next, rest: INTEGER do create Result.make_empty if p = 1 then Result.force (1, 1) end div := 2 next := 3 rest := p from i := 1 until rest = 1 loop from until rest \\ div /= 0 loop Result.force (div, i) rest := (rest / div).floor i := i + 1 end div := next next := next + 2 end ensure is_divisor: across Result as r all p \\ r.item = 0 end is_prime: across Result as r all prime (r.item) end end</lang>
The test was done in an application class. (Similar as in other Eiffel examples (ex. Selectionsort).)
factor(5000)
- Output:
2x2x2x5x5x5x5
Ela
<lang ela>open integer //arbitrary sized integers
decompose_prime n = loop n 2I
where loop c p | c < (p * p) = [c] | c % p == 0I = p :: (loop (c / p) p) | else = loop c (p + 1I)
decompose_prime 600851475143I</lang>
- Output:
[71,839,1471,6857]
Elixir
<lang elixir>defmodule Prime do
def decomposition(n), do: decomposition(n, 2, []) defp decomposition(n, k, acc) when n < k*k, do: Enum.reverse(acc, [n]) defp decomposition(n, k, acc) when rem(n, k) == 0, do: decomposition(div(n, k), k, [k | acc]) defp decomposition(n, k, acc), do: decomposition(n, k+1, acc)
end
prime = Stream.iterate(2, &(&1+1)) |>
Stream.filter(fn n-> length(Prime.decomposition(n)) == 1 end) |> Enum.take(17)
mersenne = Enum.map(prime, fn n -> {n, round(:math.pow(2,n)) - 1} end) Enum.each(mersenne, fn {n,m} ->
:io.format "~3s :~20w = ~s~n", ["M#{n}", m, Prime.decomposition(m) |> Enum.join(" x ")]
end)</lang>
- Output:
M2 : 3 = 3 M3 : 7 = 7 M5 : 31 = 31 M7 : 127 = 127 M11 : 2047 = 23 x 89 M13 : 8191 = 8191 M17 : 131071 = 131071 M19 : 524287 = 524287 M23 : 8388607 = 47 x 178481 M29 : 536870911 = 233 x 1103 x 2089 M31 : 2147483647 = 2147483647 M37 : 137438953471 = 223 x 616318177 M41 : 2199023255551 = 13367 x 164511353 M43 : 8796093022207 = 431 x 9719 x 2099863 M47 : 140737488355327 = 2351 x 4513 x 13264529 M53 : 9007199254740991 = 6361 x 69431 x 20394401 M59 : 576460752303423487 = 179951 x 3203431780337
Erlang
<lang erlang>% no stack consuming version
factors(N) ->
factors(N,2,[]).
factors(1,_,Acc) -> Acc; factors(N,K,Acc) when N < K*K -> [N|Acc]; factors(N,K,Acc) when N rem K == 0 ->
factors(N div K,K, [K|Acc]);
factors(N,K,Acc) ->
factors(N,K+1,Acc).</lang>
ERRE
<lang ERRE> PROGRAM DECOMPOSE
!
! for rosettacode.org
!
!VAR NUM,J
DIM PF[100]
PROCEDURE STORE_FACTOR
PF[0]=PF[0]+1 PF[PF[0]]=CA I=I/CA
END PROCEDURE
PROCEDURE DECOMP(I)
PF[0]=0 CA=2 ! special case LOOP IF I=1 THEN EXIT PROCEDURE END IF EXIT IF INT(I/CA)*CA<>I STORE_FACTOR END LOOP FOR CA=3 TO INT(SQR(I)) STEP 2 DO LOOP IF I=1 THEN EXIT PROCEDURE END IF EXIT IF INT(I/CA)*CA<>I STORE_FACTOR END LOOP END FOR IF I>1 THEN CA=I STORE_FACTOR END IF
END PROCEDURE
BEGIN
! ----- function generate ! in ... I ... number ! out ... PF[] ... factors ! PF[0] ... # of factors ! mod ... CA ... pr.fact. candidate PRINT(CHR$(12);) !CLS INPUT("Numero ",NUM) DECOMP(NUM) PRINT(NUM;"=";) FOR J=1 TO PF[0] DO PRINT(PF[J];) END FOR PRINT
END PROGRAM </lang> This version is a translation from Commodore BASIC program.
Ezhil
<lang Ezhil>
- இந்த நிரல் தரப்பட்ட எண்ணின் பகாஎண் கூறுகளைக் கண்டறியும்
நிரல்பாகம் பகாஎண்ணா(எண்1)
## இந்த நிரல்பாகம் தரப்பட்ட எண் பகு எண்ணா அல்லது பகா எண்ணா என்று கண்டறிந்து சொல்லும் ## பகுஎண் என்றால் 0 திரும்பத் தரப்படும் ## பகாஎண் என்றால் 1 திரும்பத் தரப்படும்
@(எண்1 < 0) ஆனால்
## எதிர்மறை எண்களை நேராக்குதல்
எண்1 = எண்1 * (-1)
முடி
@(எண்1 < 2) ஆனால்
## பூஜ்ஜியம், ஒன்று ஆகியவை பகா எண்கள் அல்ல
பின்கொடு 0
முடி
@(எண்1 == 2) ஆனால்
## இரண்டு என்ற எண் ஒரு பகா எண்
பின்கொடு 1
முடி
மீதம் = எண்1%2
@(மீதம் == 0) ஆனால்
## இரட்டைப்படை எண், ஆகவே, இது பகா எண் அல்ல
பின்கொடு 0
முடி
எண்1வர்க்கமூலம் = எண்1^0.5
@(எண்2 = 3, எண்2 <= எண்1வர்க்கமூலம், எண்2 = எண்2 + 2) ஆக
மீதம்1 = எண்1%எண்2
@(மீதம்1 == 0) ஆனால்
## ஏதேனும் ஓர் எண்ணால் முழுமையாக வகுபட்டுவிட்டது, ஆகவே அது பகா எண் அல்ல
பின்கொடு 0
முடி
முடி
பின்கொடு 1
முடி
நிரல்பாகம் பகுத்தெடு(எண்1)
## இந்த எண் தரப்பட்ட எண்ணின் பகா எண் கூறுகளைக் கண்டறிந்து பட்டியல் இடும்
கூறுகள் = பட்டியல்()
@(எண்1 < 0) ஆனால்
## எதிர்மறை எண்களை நேராக்குதல்
எண்1 = எண்1 * (-1)
முடி
@(எண்1 <= 1) ஆனால்
## ஒன்று அல்லது அதற்குக் குறைவான எண்களுக்குப் பகா எண் விகிதம் கண்டறியமுடியாது
பின்கொடு கூறுகள்
முடி @(பகாஎண்ணா(எண்1) == 1) ஆனால்
## தரப்பட்ட எண்ணே பகா எண்ணாக அமைந்துவிட்டால், அதற்கு அதுவே பகாஎண் கூறு ஆகும்
பின்இணை(கூறுகள், எண்1) பின்கொடு கூறுகள்
முடி
தாற்காலிகஎண் = எண்1
எண்2 = 2
@(எண்2 <= தாற்காலிகஎண்) வரை
விடை1 = பகாஎண்ணா(எண்2) மீண்டும்தொடங்கு = 0
@(விடை1 == 1) ஆனால்
விடை2 = தாற்காலிகஎண்%எண்2 @(விடை2 == 0) ஆனால்
## பகா எண்ணால் முழுமையாக வகுபட்டுள்ளது, அதனைப் பட்டியலில் இணைக்கிறோம்
பின்இணை(கூறுகள், எண்2) தாற்காலிகஎண் = தாற்காலிகஎண்/எண்2
## மீண்டும் இரண்டில் தொடங்கி இதே கணக்கிடுதலைத் தொடரவேண்டும்
எண்2 = 2 மீண்டும்தொடங்கு = 1
முடி முடி
@(மீண்டும்தொடங்கு == 0) ஆனால்
## அடுத்த எண்ணைத் தேர்ந்தெடுத்துக் கணக்கிடுதலைத் தொடரவேண்டும்
எண்2 = எண்2 + 1
முடி
முடி
பின்கொடு கூறுகள்
முடி
அ = int(உள்ளீடு("உங்களுக்குப் பிடித்த ஓர் எண்ணைத் தாருங்கள்: "))
பகாஎண்கூறுகள் = பட்டியல்()
பகாஎண்கூறுகள் = பகுத்தெடு(அ)
பதிப்பி "நீங்கள் தந்த எண்ணின் பகா எண் கூறுகள் இவை: ", பகாஎண்கூறுகள் </lang>
F#
<lang Fsharp>let decompose_prime n =
let rec loop c p = if c < (p * p) then [c] elif c % p = 0I then p :: (loop (c/p) p) else loop c (p + 1I) loop n 2I
printfn "%A" (decompose_prime 600851475143I)</lang>
- Output:
[71; 839; 1471; 6857]
Factor
factors
from the math.primes.factors
vocabulary converts a number into a sequence of its prime divisors; the rest of the code prints this sequence.
<lang factor>USING: io kernel math math.parser math.primes.factors sequences ;
27720 factors [ number>string ] map " " join print ;</lang>
FALSE
<lang false>[2[\$@$$*@>~][\$@$@$@$@\/*=$[%$." "$@\/\0~]?~[1+1|]?]#%.]d: 27720d;! {2 2 2 3 3 5 7 11}</lang>
Forth
<lang forth>: decomp ( n -- )
2 begin 2dup dup * >= while 2dup /mod swap if drop 1+ 1 or \ next odd number else -rot nip dup . then repeat drop . ;</lang>
Fortran
<lang fortran>module PrimeDecompose
implicit none
integer, parameter :: huge = selected_int_kind(18) ! => integer(8) ... more fails on my 32 bit machine with gfortran(gcc) 4.3.2
contains
subroutine find_factors(n, d) integer(huge), intent(in) :: n integer, dimension(:), intent(out) :: d
integer(huge) :: div, next, rest integer :: i
i = 1 div = 2; next = 3; rest = n do while ( rest /= 1 ) do while ( mod(rest, div) == 0 ) d(i) = div i = i + 1 rest = rest / div end do div = next next = next + 2 end do
end subroutine find_factors
end module PrimeDecompose</lang>
<lang fortran>program Primes
use PrimeDecompose implicit none
integer, dimension(100) :: outprimes integer i
outprimes = 0
call find_factors(12345649494449_huge, outprimes)
do i = 1, 100 if ( outprimes(i) == 0 ) exit print *, outprimes(i) end do
end program Primes</lang>
FreeBASIC
<lang freebasic>' FB 1.05.0 Win64
Function isPrime(n As Integer) As Boolean
If n Mod 2 = 0 Then Return n = 2 If n Mod 3 = 0 Then Return n = 3 Dim d As Integer = 5 While d * d <= n If n Mod d = 0 Then Return False d += 2 If n Mod d = 0 Then Return False d += 4 Wend Return True
End Function
Sub getPrimeFactors(factors() As UInteger, n As UInteger)
If n < 2 Then Return If isPrime(n) Then Redim factors(0 To 0) factors(0) = n Return End If Dim factor As UInteger = 2 Do If n Mod factor = 0 Then Redim Preserve factors(0 To UBound(factors) + 1) factors(UBound(factors)) = factor n \= factor If n = 1 Then Return If isPrime(n) Then factor = n Else factor += 1 End If Loop
End Sub
Dim factors() As UInteger Dim primes(1 To 17) As UInteger = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59} Dim n As UInteger For i As UInteger = 1 To 17
Erase factors n = 1 Shl primes(i) - 1 getPrimeFactors factors(), n Print "2^";Str(primes(i)); Tab(5); " - 1 = "; Str(n); Tab(30);" => "; For j As UInteger = LBound(factors) To UBound(factors) Print factors(j); If j < UBound(factors) Then Print " x "; Next j Print
Next i Print Print "Press any key to quit" Sleep</lang>
- Output:
2^2 - 1 = 3 => 3 2^3 - 1 = 7 => 7 2^5 - 1 = 31 => 31 2^7 - 1 = 127 => 127 2^11 - 1 = 2047 => 23 x 89 2^13 - 1 = 8191 => 8191 2^17 - 1 = 131071 => 131071 2^19 - 1 = 524287 => 524287 2^23 - 1 = 8388607 => 47 x 178481 2^29 - 1 = 536870911 => 233 x 1103 x 2089 2^31 - 1 = 2147483647 => 2147483647 2^37 - 1 = 137438953471 => 223 x 616318177 2^41 - 1 = 2199023255551 => 13367 x 164511353 2^43 - 1 = 8796093022207 => 431 x 9719 x 2099863 2^47 - 1 = 140737488355327 => 2351 x 4513 x 13264529 2^53 - 1 = 9007199254740991 => 6361 x 69431 x 20394401 2^59 - 1 = 576460752303423487 => 179951 x 3203431780337
Frink
Frink has a built-in factoring function which uses wheel factoring, trial division, Pollard p-1 factoring, and Pollard rho factoring. It also recognizes some special forms (e.g. Mersenne numbers) and handles them efficiently. <lang frink>println[factor[2^508-1]]</lang>
- Output:
(total process time including JVM startup = 1.515 s)
[[3, 1], [5, 1], [509, 1], [18797, 1], [26417, 1], [72118729, 1], [140385293, 1], [2792688414613, 1], [8988357880501, 1], [90133566917913517709497, 1], [56713727820156410577229101238628035243, 1], [170141183460469231731687303715884105727, 1]]
Note that this means 31 * 51 * ...
GAP
Built-in function : <lang gap>FactorsInt(2^67-1);
- [ 193707721, 761838257287 ]</lang>
Or using the FactInt package : <lang gap>FactInt(2^67-1);
- [ [ 193707721, 761838257287 ], [ ] ]</lang>
Go
<lang go>package main
import (
"fmt" "math/big"
)
var (
ZERO = big.NewInt(0) ONE = big.NewInt(1)
)
func Primes(n *big.Int) []*big.Int {
res := []*big.Int{} mod, div := new(big.Int), new(big.Int) for i := big.NewInt(2); i.Cmp(n) != 1; { div.DivMod(n, i, mod) for mod.Cmp(ZERO) == 0 { res = append(res, new(big.Int).Set(i)) n.Set(div) div.DivMod(n, i, mod) } i.Add(i, ONE) } return res
}
func main() {
vals := []int64{ 1 << 31, 1234567, 333333, 987653, 2 * 3 * 5 * 7 * 11 * 13 * 17, } for _, v := range vals { fmt.Println(v, "->", Primes(big.NewInt(v))) }
}</lang>
- Output:
2147483648 -> [2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2] 1234567 -> [127 9721] 333333 -> [3 3 7 11 13 37] 987653 -> [29 34057] 510510 -> [2 3 5 7 11 13 17]
Groovy
This solution uses the fact that a given factor must be prime if no smaller factor divides it evenly, so it does not require an "isPrime-like function", assumed or otherwise. <lang groovy>def factorize = { long target ->
if (target == 1) return [1L] if (target < 4) return [1L, target] def targetSqrt = Math.sqrt(target) def lowfactors = (2L..targetSqrt).findAll { (target % it) == 0 } if (lowfactors == []) return [1L, target] def nhalf = lowfactors.size() - ((lowfactors[-1]**2 == target) ? 1 : 0) [1] + lowfactors + (0..<nhalf).collect { target.intdiv(lowfactors[it]) }.reverse() + [target]
}
def decomposePrimes = { target ->
def factors = factorize(target) - [1] def primeFactors = [] factors.eachWithIndex { f, i -> if (i==0 || factors[0..<i].every {f % it != 0}) { primeFactors << f def pfPower = f*f while (target % pfPower == 0) { primeFactors << f pfPower *= f } } } primeFactors
}</lang>
- Test #1:
<lang groovy>((1..30) + [97*4, 1000, 1024, 333333]).each { println ([number:it, primes:decomposePrimes(it)]) }</lang>
- Output #1:
[number:1, primes:[]] [number:2, primes:[2]] [number:3, primes:[3]] [number:4, primes:[2, 2]] [number:5, primes:[5]] [number:6, primes:[2, 3]] [number:7, primes:[7]] [number:8, primes:[2, 2, 2]] [number:9, primes:[3, 3]] [number:10, primes:[2, 5]] [number:11, primes:[11]] [number:12, primes:[2, 2, 3]] [number:13, primes:[13]] [number:14, primes:[2, 7]] [number:15, primes:[3, 5]] [number:16, primes:[2, 2, 2, 2]] [number:17, primes:[17]] [number:18, primes:[2, 3, 3]] [number:19, primes:[19]] [number:20, primes:[2, 2, 5]] [number:21, primes:[3, 7]] [number:22, primes:[2, 11]] [number:23, primes:[23]] [number:24, primes:[2, 2, 2, 3]] [number:25, primes:[5, 5]] [number:26, primes:[2, 13]] [number:27, primes:[3, 3, 3]] [number:28, primes:[2, 2, 7]] [number:29, primes:[29]] [number:30, primes:[2, 3, 5]] [number:388, primes:[2, 2, 97]] [number:1000, primes:[2, 2, 2, 5, 5, 5]] [number:1024, primes:[2, 2, 2, 2, 2, 2, 2, 2, 2, 2]] [number:333333, primes:[3, 3, 7, 11, 13, 37]]
- Test #2:
<lang groovy>def isPrime = {factorize(it).size() == 2} (1..60).step(2).findAll(isPrime).each { println ([number:"2**${it}-1", value:2**it-1, primes:decomposePrimes(2**it-1)]) }</lang>
- Output #2:
[number:2**3-1, value:7, primes:[7]] [number:2**5-1, value:31, primes:[31]] [number:2**7-1, value:127, primes:[127]] [number:2**11-1, value:2047, primes:[23, 89]] [number:2**13-1, value:8191, primes:[8191]] [number:2**17-1, value:131071, primes:[131071]] [number:2**19-1, value:524287, primes:[524287]] [number:2**23-1, value:8388607, primes:[47, 178481]] [number:2**29-1, value:536870911, primes:[233, 1103, 2089]] [number:2**31-1, value:2147483647, primes:[2147483647]] [number:2**37-1, value:137438953471, primes:[223, 616318177]] [number:2**41-1, value:2199023255551, primes:[13367, 164511353]] [number:2**43-1, value:8796093022207, primes:[431, 9719, 2099863]] [number:2**47-1, value:140737488355327, primes:[2351, 4513, 13264529]] [number:2**53-1, value:9007199254740991, primes:[6361, 69431, 20394401]] [number:2**59-1, value:576460752303423487, primes:[179951, 3203431780337]]
Perhaps a more sophisticated algorithm is in order. It took well over 1 hour to calculate the last three decompositions using this solution.
Haskell
The task description hints at using the isPrime
function from the trial division task:
<lang haskell>factorize n = [ d | p <- [2..n], isPrime p, d <- divs n p ]
-- [2..n] >>= (\p-> [p|isPrime p]) >>= divs n where divs n p | rem n p == 0 = p : divs (quot n p) p | otherwise = []</lang>
but it is not very efficient, to put it mildly. Inlining and fusing gets us the progressively more optimized <lang haskell>import Data.Maybe (listToMaybe) import Data.List (unfoldr)
factorize :: Integer -> [Integer] factorize n
= unfoldr (\n -> listToMaybe [(x, div n x) | x <- [2..n], mod n x==0]) n = unfoldr (\(d,n) -> listToMaybe [(x, (x, div n x)) | x <- [d..n], mod n x==0]) (2,n) = unfoldr (\(d,n) -> listToMaybe [(x, (x, div n x)) | x <- takeWhile ((<=n).(^2)) [d..] ++ [n|n>1], mod n x==0]) (2,n) = unfoldr (\(ds,n) -> listToMaybe [(x, (dropWhile (< x) ds, div n x)) | n>1, x <- takeWhile ((<=n).(^2)) ds ++ [n|n>1], mod n x==0]) (primesList,n)</lang>
The library function listToMaybe
gets at most one element from its list argument. The last variant can be written as the optimal
<lang haskell>factorize n = divs n primesList
where divs n ds@(d:t) | d*d > n = [n | n > 1] | r == 0 = d : divs q ds | otherwise = divs n t where (q,r) = quotRem n d</lang>
See Sieve of Eratosthenes or Primality by trial division for a source of primes to use with this function.
Actually as some other entries notice, with any ascending order list containing all primes (e.g. 2:[3,5..]
) used in place of primesList
, the factors found by this function are guaranteed to be prime, so no separate testing for primality is strictly needed; however using just primes is more efficient, if we already have them.
- Output:
λ> mapM_ (print . factorize) $ take 11 [123123451..] [11,41,273001] [2,2,17,53,127,269] [3,229,277,647] [2,61561727] [5,7,13,270601] [2,2,2,2,2,2,2,2,3,3,3,47,379] [37,109,30529] [2,19,97,33403] [3,3167,12959] [2,2,5,6156173] [123123461]
Icon and Unicon
<lang Icon>procedure main() factors := primedecomp(2^43-1) # a big int end
procedure primedecomp(n) #: return a list of factors local F,o,x F := []
every writes(o,n|(x := genfactors(n))) do {
\o := "*" /o := "=" put(F,x) # build a list of factors to satisfy the task }
write() return F end
link factors</lang>
Uses genfactors and prime from factors
Sample Output showing factors of a large integer:
8796093022207=431*9719*2099863
J
<lang j>q:</lang>
- Example use:
<lang j> q: 3684 2 2 3 307</lang>
and, more elaborately:
<lang j> _1+2^128x 340282366920938463463374607431768211455
q: _1+2^128x
3 5 17 257 641 65537 274177 6700417 67280421310721
*/ q: _1+2^128x
340282366920938463463374607431768211455</lang>
Java
This is a version for arbitrary-precision integers which assumes the existence of a function with the signature: <lang java>public boolean prime(BigInteger i);</lang> You will need to import java.util.List, java.util.LinkedList, and java.math.BigInteger. <lang java>public static List<BigInteger> primeFactorBig(BigInteger a){
List<BigInteger> ans = new LinkedList<BigInteger>(); //loop until we test the number itself or the number is 1 for (BigInteger i = BigInteger.valueOf(2); i.compareTo(a) <= 0 && !a.equals(BigInteger.ONE); i = i.add(BigInteger.ONE)){ while (a.remainder(i).equals(BigInteger.ZERO) && prime(i)) { //if we have a prime factor ans.add(i); //put it in the list a = a.divide(i); //factor it out of the number } } return ans;
}</lang>
Alternate version, optimised to be faster. <lang java>private static final BigInteger two = BigInteger.valueOf(2);
public List<BigInteger> primeDecomp(BigInteger a) {
// impossible for values lower than 2 if (a.compareTo(two) < 0) { return null; }
//quickly handle even values List<BigInteger> result = new ArrayList<BigInteger>(); while (a.and(BigInteger.ONE).equals(BigInteger.ZERO)) { a = a.shiftRight(1); result.add(two); }
//left with odd values if (!a.equals(BigInteger.ONE)) { BigInteger b = BigInteger.valueOf(3); while (b.compareTo(a) < 0) { if (b.isProbablePrime(10)) { BigInteger[] dr = a.divideAndRemainder(b); if (dr[1].equals(BigInteger.ZERO)) { result.add(b); a = dr[0]; } } b = b.add(two); } result.add(b); //b will always be prime here... } return result;
}</lang>
Another alternate version designed to make fewer modular calculations: <lang java> private static final BigInteger TWO = BigInteger.valueOf(2); private static final BigInteger THREE = BigInteger.valueOf(3); private static final BigInteger FIVE = BigInteger.valueOf(5);
public static ArrayList<BigInteger> primeDecomp(BigInteger n){
if(n.compareTo(TWO) < 0) return null; ArrayList<BigInteger> factors = new ArrayList<BigInteger>(); // handle even values while(n.and(BigInteger.ONE).equals(BigInteger.ZERO)){ n = n.shiftRight(1); factors.add(TWO); } // handle values divisible by three while(n.mod(THREE).equals(BigInteger.ZERO)){ factors.add(THREE); n = n.divide(THREE); } // handle values divisible by five while(n.mod(FIVE).equals(BigInteger.ZERO)){ factors.add(FIVE); n = n.divide(FIVE); } // much like how we can skip multiples of two, we can also skip // multiples of three and multiples of five. This increment array // helps us to accomplish that int[] pattern = {4,2,4,2,4,6,2,6}; int pattern_index = 0; BigInteger current_test = BigInteger.valueOf(7); while(!n.equals(BigInteger.ONE)){ while(n.mod(current_test).equals(BigInteger.ZERO)){ factors.add(current_test); n = n.divide(current_test); } current_test = current_test.add(BigInteger.valueOf(pattern[pattern_index])); pattern_index = (pattern_index + 1) & 7; } return factors;
} </lang>
Simple but very inefficient method, because it will test divisibility of all numbers from 2 to max prime factor. When decomposing a large prime number this will take O(n) trial divisions instead of more common O(log n). <lang java>public static List<BigInteger> primeFactorBig(BigInteger a){
List<BigInteger> ans = new LinkedList<BigInteger>();
for(BigInteger divisor = BigInteger.valueOf(2); a.compareTo(ONE) > 0; divisor = divisor.add(ONE)) while(a.mod(divisor).equals(ZERO)){ ans.add(divisor); a = a.divide(divisor); } return ans;
}</lang>
JavaScript
This code uses the BigInteger Library jsbn and jsbn2 <lang javascript>function run_factorize(input, output) {
var n = new BigInteger(input.value, 10); var TWO = new BigInteger("2", 10); var divisor = new BigInteger("3", 10); var prod = false;
if (n.compareTo(TWO) < 0) return;
output.value = "";
while (true) { var qr = n.divideAndRemainder(TWO); if (qr[1].equals(BigInteger.ZERO)) { if (prod) output.value += "*"; else prod = true; output.value += "2"; n = qr[0]; } else break; }
while (!n.equals(BigInteger.ONE)) { var qr = n.divideAndRemainder(divisor); if (qr[1].equals(BigInteger.ZERO)) { if (prod) output.value += "*"; else prod = true; output.value += divisor; n = qr[0]; } else divisor = divisor.add(TWO); }
}</lang>
Without any library. <lang javascript>function run_factorize(n) {
if (n <= 3) return [n];
var ans = []; var done = false; while (!done) { if (n % 2 === 0) { ans.push(2); n /= 2; continue; } if (n % 3 === 0) { ans.push(3); n /= 3; continue; } if (n === 1) return ans; var sr = Math.sqrt(n); done = true; // try to divide the checked number by all numbers till its square root. for (var i = 6; i <= (sr + 6); i += 6) { if (n % (i - 1) === 0) { // is n divisible by i-1? ans.push((i - 1)); n /= (i - 1); done = false; break; } if (n % (i + 1) === 0) { // is n divisible by i+1? ans.push((i + 1)); n /= (i + 1); done = false; break; } } } ans.push(n); return ans;
}</lang>
TDD using Jasmine
PrimeFactors.js <lang javascript>function factors(n) {
if (!n || n < 2) return [];
var f = []; for (var i = 2; i <= n; i++){ while (n % i === 0){ f.push(i); n /= i; } }
return f;
}; </lang>
SpecPrimeFactors.js (with tag for Chutzpah) <lang javascript>/// <reference path="PrimeFactors.js" />
describe("Prime Factors", function() {
it("Given nothing, empty is returned", function() { expect(factors()).toEqual([]); });
it("Given 1, empty is returned", function() { expect(factors(1)).toEqual([]); });
it("Given 2, 2 is returned", function() { expect(factors(2)).toEqual([2]); });
it("Given 3, 3 is returned", function() { expect(factors(3)).toEqual([3]); });
it("Given 4, 2 and 2 is returned", function() { expect(factors(4)).toEqual([2, 2]); });
it("Given 5, 5 is returned", function() { expect(factors(5)).toEqual([5]); });
it("Given 6, 2 and 3 is returned", function() { expect(factors(6)).toEqual([2, 3]); });
it("Given 7, 7 is returned", function() { expect(factors(7)).toEqual([7]); });
it("Given 8; 2, 2, and 2 is returned", function() { expect(factors(8)).toEqual([2, 2, 2]); });
it("Given a large number, many primes factors are returned", function() { expect(factors(2*2*2*3*3*7*11*17)) .toEqual([2, 2, 2, 3, 3, 7, 11, 17]); });
it("Given a large prime number, that number is returned", function() { expect(factors(997)).toEqual([997]); });
}); </lang>
jq
Works with gojq, the Go implementation of jq
`factors` as defined below emits a stream of all the prime factors of the input integer. The implementation is compact, fast and space-efficient: no space is required to store the primes or factors already computed, there is no reliance on an "is_prime" function, and square roots are only computed if needed.
The economy comes about through the use of the builtin filter recurse/1, and the use of the state vector: [p, n, valid, sqrt], where p is the candidate factor, n is the number still to be factored, valid is a flag, and sqrt is either null or the square root of n.
gojq supports unlimited-precision integer arithmetic, but the C implementation of jq currently uses IEEE 754 64-bit numbers, so using the latter, the following program will only be reliable for integers up to and including 9,007,199,254,740,992 (2^53). However, "factors" could be easily modified to work with a "BigInt" library for jq, such as BigInt.jq. <lang jq>def factors:
. as $in | [2, $in, false] | recurse( . as [$p, $q, $valid, $s] | if $q == 1 then empty elif $q % $p == 0 then [$p, $q/$p, true] elif $p == 2 then [3, $q, false, $s] else ($s // ($q | sqrt)) as $s | if $p + 2 <= $s then [$p + 2, $q, false, $s] else [$q, 1, true] end end ) | if .[2] then .[0] else empty end ;</lang>
Examples: <lang jq> 24 | factors
- => 2 2 2 3
[9007199254740992 | factors] | length
- => 53
- 2**29-1 is 536870911
[ 536870911 | factors ]
- => [233,1103,2089]</lang>
Julia
using package Primes.jl:
<lang julia>
julia> Pkg.add("Primes")
julia> factor(8796093022207)
[9719=>1,431=>1,2099863=>1]
</lang>
(The factor
function returns a dictionary
whose keys are the factors and whose values are the multiplicity of each factor.)
Kotlin
<lang scala>// version 1.0.6
import java.math.BigInteger
val bigTwo = BigInteger.valueOf(2L) val bigThree = BigInteger.valueOf(3L)
fun getPrimeFactors(n: BigInteger): MutableList<BigInteger> {
val factors = mutableListOf<BigInteger>() if (n < bigTwo) return factors if (n.isProbablePrime(20)) { factors.add(n) return factors } var factor = bigTwo var nn = n while (true) { if (nn % factor == BigInteger.ZERO) { factors.add(factor) nn /= factor if (nn == BigInteger.ONE) return factors if (nn.isProbablePrime(20)) factor = nn } else if (factor >= bigThree) factor += bigTwo else factor = bigThree }
}
fun main(args: Array<String>) {
val primes = intArrayOf(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97) for (prime in primes) { val bigPow2 = bigTwo.pow(prime) - BigInteger.ONE println("2^${"%2d".format(prime)} - 1 = ${bigPow2.toString().padEnd(30)} => ${getPrimeFactors(bigPow2)}") }
}</lang>
- Output:
2^ 2 - 1 = 3 => [3] 2^ 3 - 1 = 7 => [7] 2^ 5 - 1 = 31 => [31] 2^ 7 - 1 = 127 => [127] 2^11 - 1 = 2047 => [23, 89] 2^13 - 1 = 8191 => [8191] 2^17 - 1 = 131071 => [131071] 2^19 - 1 = 524287 => [524287] 2^23 - 1 = 8388607 => [47, 178481] 2^29 - 1 = 536870911 => [233, 1103, 2089] 2^31 - 1 = 2147483647 => [2147483647] 2^37 - 1 = 137438953471 => [223, 616318177] 2^41 - 1 = 2199023255551 => [13367, 164511353] 2^43 - 1 = 8796093022207 => [431, 9719, 2099863] 2^47 - 1 = 140737488355327 => [2351, 4513, 13264529] 2^53 - 1 = 9007199254740991 => [6361, 69431, 20394401] 2^59 - 1 = 576460752303423487 => [179951, 3203431780337] 2^61 - 1 = 2305843009213693951 => [2305843009213693951] 2^67 - 1 = 147573952589676412927 => [193707721, 761838257287] 2^71 - 1 = 2361183241434822606847 => [228479, 48544121, 212885833] 2^73 - 1 = 9444732965739290427391 => [439, 2298041, 9361973132609] 2^79 - 1 = 604462909807314587353087 => [2687, 202029703, 1113491139767] 2^83 - 1 = 9671406556917033397649407 => [167, 57912614113275649087721] 2^89 - 1 = 618970019642690137449562111 => [618970019642690137449562111] 2^97 - 1 = 158456325028528675187087900671 => [11447, 13842607235828485645766393]
Lambdatalk
<lang scheme> {def prime_fact.smallest
{def prime_fact.smallest.r {lambda {:q :r :i} {if {and {> :r 0} {< :i :q}} then {prime_fact.smallest.r :q {% :q {+ :i 1}} {+ :i 1}} else :i}}} {lambda {:q} {prime_fact.smallest.r :q {% :q 2} 2}}}
{def prime_fact
{def prime_fact.r {lambda {:q :d} {if {> :q 1} then {let { {:q :q} {:d :d} {:i {prime_fact.smallest :q}}} {prime_fact.r {floor {/ :q :i}} {#.push! :d :i}} } else {if {= {#.length :d} 1} then {b :d} else :d}}}} {lambda {:n} :n:{prime_fact.r :n {#.new}}}}
{prime_fact {* 2 3 3 3 31 47 173}} -> 13611294:[2,3,3,3,31,47,173]
{map prime_fact {serie 2 101}} -> 2:[2] 3:[3] 4:[2,2] 5:[5] 6:[2,3] 7:[7] 8:[2,2,2] 9:[3,3] 10:[2,5] 11:[11] 12:[2,2,3] 13:[13] 14:[2,7] 15:[3,5] 16:[2,2,2,2] 17:[17] 18:[2,3,3] 19:[19] 20:[2,2,5] 21:[3,7] 22:[2,11] 23:[23] 24:[2,2,2,3] 25:[5,5] 26:[2,13] 27:[3,3,3] 28:[2,2,7] 29:[29] 30:[2,3,5] 31:[31] 32:[2,2,2,2,2] 33:[3,11] 34:[2,17] 35:[5,7] 36:[2,2,3,3] 37:[37] 38:[2,19] 39:[3,13] 40:[2,2,2,5] 41:[41] 42:[2,3,7] 43:[43] 44:[2,2,11] 45:[3,3,5] 46:[2,23] 47:[47] 48:[2,2,2,2,3] 49:[7,7] 50:[2,5,5] 51:[3,17] 52:[2,2,13] 53:[53] 54:[2,3,3,3] 55:[5,11] 56:[2,2,2,7] 57:[3,19] 58:[2,29] 59:[59] 60:[2,2,3,5] 61:[61] 62:[2,31] 63:[3,3,7] 64:[2,2,2,2,2,2] 65:[5,13] 66:[2,3,11] 67:[67] 68:[2,2,17] 69:[3,23] 70:[2,5,7] 71:[71] 72:[2,2,2,3,3] 73:[73] 74:[2,37] 75:[3,5,5] 76:[2,2,19] 77:[7,11] 78:[2,3,13] 79:[79] 80:[2,2,2,2,5] 81:[3,3,3,3] 82:[2,41] 83:[83] 84:[2,2,3,7] 85:[5,17] 86:[2,43] 87:[3,29] 88:[2,2,2,11] 89:[89] 90:[2,3,3,5] 91:[7,13] 92:[2,2,23] 93:[3,31] 94:[2,47] 95:[5,19] 96:[2,2,2,2,2,3] 97:[97] 98:[2,7,7] 99:[3,3,11] 100:[2,2,5,5] 101:[101] </lang>
LFE
<lang lisp> (defun factors (n)
(factors n 2 '()))
(defun factors
((1 _ acc) acc) ((n k acc) (when (== 0 (rem n k))) (factors (div n k) k (cons k acc))) ((n k acc) (factors n (+ k 1) acc)))
</lang>
Lingo
<lang lingo>-- Returns list of prime factors for given number. -- To overcome the limits of integers (signed 32-bit in Lingo), -- the number can be specified as float (which works up to 2^53). -- For the same reason, values in returned list are floats, not integers. on getPrimeFactors (n)
f = [] f.sort() c = sqrt(n) i = 1.0 repeat while TRUE i=i+1 if i>c then exit repeat check = n/i if bitOr(check,0)=check then f.add(i) n = check c = sqrt(n) i = 1.0 end if end repeat f.add(n) return f
end</lang> <lang lingo>put getPrimeFactors(12) -- [2.0000, 2.0000, 3.0000]
-- print floats without fractional digits the floatPrecision=0
put getPrimeFactors(12) -- [2, 2, 3]
put getPrimeFactors(1125899906842623.0) -- [3, 251, 601, 4051, 614141]</lang>
Logo
<lang logo>to decompose :n [:p 2]
if :p*:p > :n [output (list :n)] if less? 0 modulo :n :p [output (decompose :n bitor 1 :p+1)] output fput :p (decompose :n/:p :p)
end</lang>
Lua
The code of the used auxiliary function "IsPrime(n)" is located at Primality by trial division#Lua
<lang lua>function PrimeDecomposition( n )
local f = {} if IsPrime( n ) then f[1] = n return f end
local i = 2 repeat while n % i == 0 do f[#f+1] = i n = n / i end repeat i = i + 1 until IsPrime( i ) until n == 1 return f
end</lang>
M2000 Interpreter
<lang M2000 Interpreter> Module Prime_decomposition {
Inventory Known1=2@, 3@ IsPrime=lambda Known1 (x as decimal) -> { =0=1 if exist(Known1, x) then =1=1 : exit if x<=5 OR frac(x) then {if x == 2 OR x == 3 OR x == 5 then Append Known1, x : =1=1 Break} if frac(x/2) else exit if frac(x/3) else exit x1=sqrt(x):d = 5@ {if frac(x/d ) else exit d += 2: if d>x1 then Append Known1, x : =1=1 : exit if frac(x/d) else exit d += 4: if d<= x1 else Append Known1, x : =1=1: exit loop} } decompose=lambda IsPrime (n as decimal) -> { Inventory queue Factors { k=2@ While frac(n/k)=0 { n/=k Append Factors, k } if n=1 then exit k++ While frac(n/k)=0 { n/=k Append Factors, k } if n=1 then exit { k+=2 while not isprime(k) {k+=2} While frac(n/k)=0 { n/=k Append Factors, k } if n=1 then exit loop } } =Factors } Data 10, 100, 12, 144, 496, 1212454 while not empty { Print Decompose(Number) }
} Prime_decomposition </lang>
Maple
Maple has two commands for integer factorization: ifactor, which returns results in a form resembling textbook presentation and ifactors, which returns a list of two-element lists of prime factors and their multiplicities:
<lang Maple>> ifactor(1337);
(7) (191)
</lang> <lang Maple>> ifactors(1337);
[1, [[7, 1], [191, 1]]]
</lang>
Mathematica/Wolfram Language
Bare built-in function does: <lang Mathematica> FactorInteger[2016] => {{2, 5}, {3, 2}, {7, 1}}</lang>
Read as: 2 to the power 5 times 3 squared times 7 (to the power 1). To show them nicely we could use the following functions: <lang Mathematica>supscript[x_,y_]:=If[y==1,x,Superscript[x,y]] ShowPrimeDecomposition[input_Integer]:=Print@@{input," = ",Sequence@@Riffle[supscript@@@FactorInteger[input]," "]}</lang>
Example for small prime: <lang Mathematica> ShowPrimeDecomposition[1337]</lang> gives: <lang Mathematica> 1337 = 7 191</lang>
Examples for large primes: <lang Mathematica> Table[AbsoluteTiming[ShowPrimeDecomposition[2^a-1]]//Print[#1," sec"]&,{a,50,150,10}];</lang> gives back: <lang Mathematica>1125899906842623 = 3 11 31 251 601 1801 4051 0.000231 sec 1152921504606846975 = 3^2 5^2 7 11 13 31 41 61 151 331 1321 0.000146 sec 1180591620717411303423 = 3 11 31 43 71 127 281 86171 122921 0.001008 sec 1208925819614629174706175 = 3 5^2 11 17 31 41 257 61681 4278255361 0.000340 sec 1237940039285380274899124223 = 3^3 7 11 19 31 73 151 331 631 23311 18837001 0.000192 sec 1267650600228229401496703205375 = 3 5^3 11 31 41 101 251 601 1801 4051 8101 268501 0.000156 sec 1298074214633706907132624082305023 = 3 11^2 23 31 89 683 881 2971 3191 201961 48912491 0.001389 sec 1329227995784915872903807060280344575 = 3^2 5^2 7 11 13 17 31 41 61 151 241 331 1321 61681 4562284561 0.000374 sec 1361129467683753853853498429727072845823 = 3 11 31 131 2731 8191 409891 7623851 145295143558111 0.024249 sec 1393796574908163946345982392040522594123775 = 3 5^2 11 29 31 41 43 71 113 127 281 86171 122921 7416361 47392381 0.009419 sec 1427247692705959881058285969449495136382746623 = 3^2 7 11 31 151 251 331 601 1801 4051 100801 10567201 1133836730401 0.007705 sec</lang>
MATLAB
<lang Matlab>function [outputPrimeDecomposition] = primedecomposition(inputValue)
outputPrimeDecomposition = factor(inputValue);</lang>
Maxima
Using the built-in function: <lang maxima>(%i1) display2d: false$ /* disable rendering exponents as superscripts */ (%i2) factor(2016); (%o2) 2^5*3^2*7 </lang> Using the underlying language: <lang maxima>prime_dec(n) := flatten(create_list(makelist(first(a), second(a)), a, ifactors(n)))$
/* or, slighlty more "functional" */ prime_dec(n) := flatten(map(lambda([a], apply(makelist, a)), ifactors(n)))$
prime_dec(2^4*3^5*5*7^2); /* [2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 7, 7] */</lang>
MUMPS
<lang MUMPS>ERATO1(HI)
SET HI=HI\1 KILL ERATO1 ;Don't make it new - we want it to remain after the quit NEW I,J,P FOR I=2:1:(HI**.5)\1 DO .FOR J=I*I:I:HI DO ..SET P(J)=1 ;$SELECT($DATA(P(J))#10:P(J)+1,1:1) ;WRITE !,"Prime numbers between 2 and ",HI,": " FOR I=2:1:HI DO .S:'$DATA(P(I)) ERATO1(I)=I ;WRITE $SELECT((I<3):"",1:", "),I KILL I,J,P QUIT
PRIMDECO(N)
;Returns its results in the string PRIMDECO ;Kill that before the first call to this recursive function QUIT:N<=1 IF $D(PRIMDECO)=1 SET PRIMDECO="" D ERATO1(N) SET N=N\1,I=0 FOR SET I=$O(ERATO1(I)) Q:+I<1 Q:'(N#I) IF I>1 SET PRIMDECO=$S($L(PRIMDECO)>0:PRIMDECO_"^",1:"")_I D PRIMDECO(N/I) ;that is, if I is a factor of N, add it to the string QUIT</lang>
- Usage:
USER>K ERATO1,PRIMDECO D PRIMDECO^ROSETTA(31415) W PRIMDECO 5^61^103 USER>K ERATO,PRIMDECO D PRIMDECO^ROSETTA(31318) W PRIMDECO 2^7^2237 USER>K ERATO,PRIMDECO D PRIMDECO^ROSETTA(34) W PRIMDECO 2^17 USER>K ERATO,PRIMDECO D PRIMDECO^ROSETTA(68) W PRIMDECO 2^2^17 USER>K ERATO,PRIMDECO D PRIMDECO^ROSETTA(7) W PRIMDECO 7 USER>K ERATO,PRIMDECO D PRIMDECO^ROSETTA(777) W PRIMDECO 3^7^37
Nim
Based on python floating point solution, but using integers rather than floats. <lang nim>import math, sequtils, strformat, strutils, times
proc getStep(n: int64): int64 {.inline.} =
result = 1 + n shl 2 - n shr 1 shl 1
proc primeFac(n: int64): seq[int64] =
var maxq = int64(sqrt(float(n))) var d = 1 var q: int64 = 2 + (n and 1) # Start with 2 or 3 according to oddity. while q <= maxq and n %% q != 0: q = getStep(d) inc d if q <= maxq: let q1 = primeFac(n /% q) let q2 = primeFac(q) result = concat(q2, q1, result) else: result.add(n)
iterator primes(limit: int): int =
var isPrime = newSeq[bool](limit + 1) for n in 2..limit: isPrime[n] = true for n in 2..limit: if isPrime[n]: yield n for i in countup(n *% n, limit, n): isPrime[i] = false
when isMainModule:
# Example: calculate factors of Mersenne numbers from M2 to M59. for m in primes(59): let p = 2i64^m - 1 let s = &"2^{m}-1" stdout.write &"{s:<6} = {p} with factors: " let start = cpuTime() stdout.write primeFac(p).join(", ") echo &" => {(1000 * (cpuTime() - start)).toInt} ms"</lang>
- Output:
Compiled with option -d:release
2^2-1 = 3 with factors: 3 => 0 ms 2^3-1 = 7 with factors: 7 => 0 ms 2^5-1 = 31 with factors: 31 => 0 ms 2^7-1 = 127 with factors: 127 => 0 ms 2^11-1 = 2047 with factors: 23, 89 => 0 ms 2^13-1 = 8191 with factors: 8191 => 0 ms 2^17-1 = 131071 with factors: 131071 => 0 ms 2^19-1 = 524287 with factors: 524287 => 0 ms 2^23-1 = 8388607 with factors: 47, 178481 => 0 ms 2^29-1 = 536870911 with factors: 233, 1103, 2089 => 0 ms 2^31-1 = 2147483647 with factors: 2147483647 => 1 ms 2^37-1 = 137438953471 with factors: 223, 616318177 => 0 ms 2^41-1 = 2199023255551 with factors: 13367, 164511353 => 0 ms 2^43-1 = 8796093022207 with factors: 431, 9719, 2099863 => 0 ms 2^47-1 = 140737488355327 with factors: 2351, 4513, 13264529 => 0 ms 2^53-1 = 9007199254740991 with factors: 6361, 69431, 20394401 => 1 ms 2^59-1 = 576460752303423487 with factors: 179951, 3203431780337 => 6 ms
OCaml
<lang ocaml>open Big_int;;
let prime_decomposition x =
let rec inner c p = if lt_big_int p (square_big_int c) then [p] else if eq_big_int (mod_big_int p c) zero_big_int then c :: inner c (div_big_int p c) else inner (succ_big_int c) p in inner (succ_big_int (succ_big_int zero_big_int)) x;;</lang>
Octave
<lang octave>r = factor(120202039393)</lang>
Oforth
Oforth handles aribitrary precision integers.
<lang Oforth>: factors(n) // ( aInteger -- aList ) | k p |
ListBuffer new 2 ->k n nsqrt ->p while( k p <= ) [ n k /mod swap ifZero: [ dup ->n nsqrt ->p k over add continue ] drop k 1+ ->k ] n 1 > ifTrue: [ n over add ] dup freeze ;</lang>
- Output:
>2 128 pow 1 - dup println factors println 340282366920938463463374607431768211455 [3, 5, 17, 257, 641, 65537, 274177, 6700417, 67280421310721] ok
PARI/GP
GP normally returns factored integers as a matrix
with the first column representing the primes
and the second their exponents.
Thus factor(12)==[2,2;3,1]
is true.
But it's simple enough to convert this to a vector with repetition:
<lang parigp>pd(n)={
my(f=factor(n),v=f[,1]~); for(i=1,#v, while(f[i,2]--, v=concat(v,f[i,1]) ) ); vecsort(v)
};</lang>
Pascal
<lang pascal>Program PrimeDecomposition(output);
type
DynArray = array of integer;
procedure findFactors(n: Int64; var d: DynArray);
var divisor, next, rest: Int64; i: integer; begin i := 0; divisor := 2; next := 3; rest := n; while (rest <> 1) do begin while (rest mod divisor = 0) do begin setlength(d, i+1); d[i] := divisor; inc(i); rest := rest div divisor; end; divisor := next; next := next + 2; end; end;
var
factors: DynArray; j: integer;
begin
setlength(factors, 1); findFactors(1023*1024, factors); for j := low(factors) to high(factors) do writeln (factors[j]);
end.</lang>
- Output:
% ./PrimeDecomposition 2 2 2 2 2 2 2 2 2 2 3 11 31
Optimization:
<lang pascal>Program PrimeDecomposition(output);
type
DynArray = array of integer;
procedure findFactors(n: Int64; var d: DynArray);
var divisor, next, rest: Int64; i: integer; begin i := 0; divisor := 2; next := 3; rest := n; while (rest <> 1) do begin while (rest mod divisor = 0) do begin setlength(d, i+1); d[i] := divisor; inc(i); rest := rest div divisor; end; divisor := next; next := next + 2; // try only odd numbers // cut condition: avoid many useless iterations if (rest < divisor * divisor) then begin setlength(d, i+1); d[i] := rest; rest := 1; end; end; end;
var
factors: DynArray; j: integer;
begin
setlength(factors, 1); findFactors(1023*1024, factors); for j := low(factors) to high(factors) do writeln (factors[j]); readln;
end.</lang>
Perl
These will work for large integers by adding the use bigint; clause.
Trivial trial division (very slow)
<lang perl>sub prime_factors {
my ($n, $d, @out) = (shift, 1); while ($n > 1 && $d++) { $n /= $d, push @out, $d until $n % $d; } @out
}
print "@{[prime_factors(1001)]}\n";</lang>
Better trial division
This is much faster than the trivial version above. <lang perl>sub prime_factors {
my($n, $p, @out) = (shift, 3); return if $n < 1; while (!($n&1)) { $n >>= 1; push @out, 2; } while ($n > 1 && $p*$p <= $n) { while ( ($n % $p) == 0) { $n /= $p; push @out, $p; } $p += 2; } push @out, $n if $n > 1; @out;
}</lang>
Modules
As usual, there are CPAN modules for this that will be much faster. These both take about 1 second to factor all Mersenne numbers from M_1 to M_150.
<lang perl>use ntheory qw/factor forprimes/; use bigint;
forprimes {
my $p = 2 ** $_ - 1; print "2**$_-1: ", join(" ", factor($p)), "\n";
} 100, 150;</lang>
- Output:
2^101-1: 7432339208719 341117531003194129 2^103-1: 2550183799 3976656429941438590393 2^107-1: 162259276829213363391578010288127 2^109-1: 745988807 870035986098720987332873 2^113-1: 3391 23279 65993 1868569 1066818132868207 2^127-1: 170141183460469231731687303715884105727 2^131-1: 263 10350794431055162386718619237468234569 2^137-1: 32032215596496435569 5439042183600204290159 2^139-1: 5625767248687 123876132205208335762278423601 2^149-1: 86656268566282183151 8235109336690846723986161
<lang perl>use Math::Pari qw/:int factorint isprime/;
- Convert Math::Pari's format into simple vector
sub factor {
my ($pn,$pc) = @{Math::Pari::factorint(shift)}; map { ($pn->[$_]) x $pc->[$_] } 0 .. $#$pn;
}
for (100 .. 150) {
next unless isprime($_); my $p = 2 ** $_ - 1; print "2^$_-1: ", join(" ", factor($p)), "\n";
}</lang> With the same output.
Phix
For small numbers less than 253 on 32bit and 264 on 64bit just use prime_factors().
with javascript_semantics requires("1.0.0") include mpfr.e atom t0 = time() mpz z = mpz_init() for i=1 to 25 do integer pi = get_prime(i) mpz_ui_pow_ui(z,2,pi) mpz_sub_ui(z,z,1) string zs = mpz_get_str(z), fs = mpz_factorstring(mpz_pollard_rho(z)) if fs!=zs then zs &= " = "&fs end if printf(1,"2^%d-1 = %s\n",{pi,zs}) end for string s = "600851475143" for i=1 to 2 do mpz_set_str(z,s) printf(1,"%s = %s\n",{s,mpz_factorstring(mpz_pollard_rho(z))}) s = "100000000000000000037" end for ?elapsed(time()-t0)
- Output:
2^2-1 = 3 2^3-1 = 7 2^5-1 = 31 2^7-1 = 127 2^11-1 = 2047 = 23*89 2^13-1 = 8191 2^17-1 = 131071 2^19-1 = 524287 2^23-1 = 8388607 = 47*178481 2^29-1 = 536870911 = 233*1103*2089 2^31-1 = 2147483647 2^37-1 = 137438953471 = 223*616318177 2^41-1 = 2199023255551 = 13367*164511353 2^43-1 = 8796093022207 = 431*9719*2099863 2^47-1 = 140737488355327 = 2351*4513*13264529 2^53-1 = 9007199254740991 = 6361*69431*20394401 2^59-1 = 576460752303423487 = 179951*3203431780337 2^61-1 = 2305843009213693951 2^67-1 = 147573952589676412927 = 193707721*761838257287 2^71-1 = 2361183241434822606847 = 228479*48544121*212885833 2^73-1 = 9444732965739290427391 = 439*2298041*9361973132609 2^79-1 = 604462909807314587353087 = 2687*202029703*1113491139767 2^83-1 = 9671406556917033397649407 = 167*57912614113275649087721 2^89-1 = 618970019642690137449562111 2^97-1 = 158456325028528675187087900671 = 11447*13842607235828485645766393 600851475143 = 71*839*1471*6857 100000000000000000037 = 31*821*59004541*66590107 "1.1s"
Picat
<lang Picat>go =>
% Checking 2**prime-1 foreach(P in primes(60)) Factors = factors(2**P-1), println([n=2**P-1,factors=Factors]) end, nl, % Testing a larger number println(factors(1361129467683753853853498429727072845823)), nl.
% % factors of N % factors(N) = Factors =>
Factors = [], M = N, while (M mod 2 == 0) Factors := Factors ++ [2], M := M div 2 end, T = 3, while (M > 1, T < 1+(sqrt(M))) if M mod T == 0 then [Divisors, NewM] = alldivisorsM(M, T), Factors := Factors ++ Divisors, M := NewM end, T := T + 2 end, if M > 1 then Factors := Factors ++ [M] end.
alldivisorsM(N,Div) = [Divisors,M] =>
M = N, Divisors = [], while (M mod Div == 0) Divisors := Divisors ++ [Div], M := M div Div end.</lang>
- Output:
[n = 3,factors = [3]] [n = 7,factors = [7]] [n = 31,factors = [31]] [n = 127,factors = [127]] [n = 2047,factors = [23,89]] [n = 8191,factors = [8191]] [n = 131071,factors = [131071]] [n = 524287,factors = [524287]] [n = 8388607,factors = [47,178481]] [n = 536870911,factors = [233,1103,2089]] [n = 2147483647,factors = [2147483647]] [n = 137438953471,factors = [223,616318177]] [n = 2199023255551,factors = [13367,164511353]] [n = 8796093022207,factors = [431,9719,2099863]] [n = 140737488355327,factors = [2351,4513,13264529]] [n = 9007199254740991,factors = [6361,69431,20394401]] [n = 576460752303423487,factors = [179951,3203431780337]] [3,11,31,131,2731,8191,409891,7623851,145295143558111]
PicoLisp
The following solution generates a sequence of "trial divisors" (2 3 5 7 11 13 17 19 23 29 31 37 ..), as described by Donald E. Knuth, "The Art of Computer Programming", Vol.2, p.365. <lang PicoLisp>(de factor (N)
(make (let (D 2 L (1 2 2 . (4 2 4 2 4 6 2 6 .)) M (sqrt N)) (while (>= M D) (if (=0 (% N D)) (setq M (sqrt (setq N (/ N (link D))))) (inc 'D (pop 'L)) ) ) (link N) ) ) )
(factor 1361129467683753853853498429727072845823)</lang>
- Output:
-> (3 11 31 131 2731 8191 409891 7623851 145295143558111)
PL/I
<lang pli> test: procedure options (main, reorder);
declare (n, i) fixed binary (31);
get list (n); put edit ( n, '[' ) (x(1), a);
restart:
if is_prime(n) then do; put edit (trim(n), ']' ) (x(1), a); stop; end; do i = n/2 to 2 by -1; if is_prime(i) then if (mod(n, i) = 0) then do; put edit ( trim(i) ) (x(1), a); n = n / i; go to restart; end; end; put edit ( ' ]' ) (a);
is_prime: procedure (n) options (reorder) returns (bit(1));
declare n fixed binary (31); declare i fixed binary (31);
if n < 2 then return ('0'b); if n = 2 then return ('1'b); if mod(n, 2) = 0 then return ('0'b);
do i = 3 to sqrt(n) by 2; if mod(n, i) = 0 then return ('0'b); end; return ('1'b);
end is_prime;
end test; </lang>
- Results from various runs:
1234567 [ 9721 127 ] 32768 [ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ] 99 [ 11 3 3 ] 9876543 [ 14503 227 3 ] 100 [ 5 5 2 2 ] 9999999 [ 4649 239 3 3 ] 5040 [ 7 5 3 3 2 2 2 2 ]
PowerShell
<lang PowerShell> function eratosthenes ($n) {
if($n -gt 1){ $prime = @(1..($n+1) | foreach{$true}) $prime[1] = $false $m = [Math]::Floor([Math]::Sqrt($n)) function multiple($i) { for($j = $i*$i; $j -le $n; $j += $i) { $prime[$j] = $false } } multiple 2 for($i = 3; $i -le $m; $i += 2) { if($prime[$i]) {multiple $i} } 1..$n | where{$prime[$_]} } else { Write-Error "$n is not greater than 1" }
} function prime-decomposition ($n) {
$array = eratosthenes $n $prime = @() foreach($p in $array) { while($n%$p -eq 0) { $n /= $p $prime += @($p) } } $prime
} "$(prime-decomposition 12)" "$(prime-decomposition 100)" </lang> Output:
2 2 3 2 2 5 5
Prolog
<lang Prolog>prime_decomp(N, L) :-
SN is sqrt(N), prime_decomp_1(N, SN, 2, [], L).
prime_decomp_1(1, _, _, L, L) :- !.
% Special case for 2, increment 1 prime_decomp_1(N, SN, D, L, LF) :-
( 0 is N mod D -> Q is N / D, SQ is sqrt(Q), prime_decomp_1(Q, SQ, D, [D |L], LF) ; D1 is D+1, ( D1 > SN -> LF = [N |L] ; prime_decomp_2(N, SN, D1, L, LF) ) ).
% General case, increment 2 prime_decomp_2(1, _, _, L, L) :- !.
prime_decomp_2(N, SN, D, L, LF) :-
( 0 is N mod D -> Q is N / D, SQ is sqrt(Q), prime_decomp_2(Q, SQ, D, [D |L], LF); D1 is D+2, ( D1 > SN -> LF = [N |L] ; prime_decomp_2(N, SN, D1, L, LF) ) ).</lang>
- Output:
<lang Prolog> ?- time(prime_decomp(9007199254740991, L)). % 138,882 inferences, 0.344 CPU in 0.357 seconds (96% CPU, 404020 Lips) L = [20394401,69431,6361].
?- time(prime_decomp(576460752303423487, L)).
% 2,684,734 inferences, 0.672 CPU in 0.671 seconds (100% CPU, 3995883 Lips) L = [3203431780337,179951].
?- time(prime_decomp(1361129467683753853853498429727072845823, L)).
% 18,080,807 inferences, 7.953 CPU in 7.973 seconds (100% CPU, 2273422 Lips) L = [145295143558111,7623851,409891,8191,2731,131,31,11,3].</lang>
Simple version
Optimized to stop on square root, and count by +2 on odds, above 2.
<lang Prolog>factors( N, FS):-
factors2( N, FS).
factors2( N, FS):-
( N < 2 -> FS = [] ; 4 > N -> FS = [N] ; 0 is N rem 2 -> FS = [K|FS2], N2 is N div 2, factors2( N2, FS2) ; factors( N, 3, FS) ).
factors( N, K, FS):-
( N < 2 -> FS = [] ; K*K > N -> FS = [N] ; 0 is N rem K -> FS = [K|FS2], N2 is N div K, factors( N2, K, FS2) ; K2 is K+2, factors( N, K2, FS) ).</lang>
Expression Tree version
Uses a 2*3*5*7 factor wheel, but the main feature is that it returns the decomposition as a fully simplified expression tree. <lang Prolog> wheel2357(L) :-
W = [2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 2, 10 | W], L = [1, 2, 2, 4 | W].
factor(1, 1) :- !. factor(N, Fac) :-
N > 1, wheel2357(W), factor(N, 2, W, 1, Fac0), reverse_factors(Fac0, Fac).
factor(N, F, _, Fac1, Fac2) :- F*F > N, !, add_factor(N, Fac1, Fac2). factor(N, F, W, Fac1, Fac) :-
divmod(N, F, Q, 0), !, add_factor(F, Fac1, Fac2), factor(Q, F, W, Fac2, Fac).
factor(N, F1, [A|As], Fac1, Fac) :-
F2 is F1 + A, factor(N, F2, As, Fac1, Fac).
add_factor(F, 1, F) :- !. add_factor(F, F, F**2) :- !. add_factor(F, F**Ex1, F**Ex2) :- succ(Ex1, Ex2), !.
add_factor(F, F*A, F**2*A) :- !. add_factor(F, F**Ex1*Rest, F**Ex2*Rest) :- succ(Ex1, Ex2), !. add_factor(F, Fac, F*Fac).
reverse_factors(A*B, C*A) :- reverse_factors(B, C), !. reverse_factors(A, A). </lang>
- Output:
?- factor(277,X). X = 277. ?- factor(1003,X). X = 17*59. ?- factor(1024,X). X = 2**10. ?- factor(768,X). X = 2**8*3. ?- factor(1361129467683753853853498429727072845823,X). X = 3*11*31*131*2731*8191*409891*7623851*145295143558111. ?- factor(360,X). X = 2**3*3**2*5.
Pure
<lang pure>factor n = factor 2 n with
factor k n = k : factor k (n div k) if n mod k == 0; = if n>1 then [n] else [] if k*k>n; = factor (k+1) n if k==2; = factor (k+2) n otherwise;
end;</lang>
PureBasic
<lang PureBasic> CompilerIf #PB_Compiler_Debugger
CompilerError "Turn off the debugger if you want reasonable speed in this example."
CompilerEndIf
Define.q
Procedure Factor(Number, List Factors())
Protected I = 3 While Number % 2 = 0 AddElement(Factors()) Factors() = 2 Number / 2 Wend Protected Max = Number While I <= Max And Number > 1 While Number % I = 0 AddElement(Factors()) Factors() = I Number/I Wend I + 2 Wend
EndProcedure
Number = 9007199254740991 NewList Factors() time = ElapsedMilliseconds() Factor(Number, Factors()) time = ElapsedMilliseconds()-time S.s = "Factored " + Str(Number) + " in " + StrD(time/1000, 2) + " seconds." ForEach Factors()
S + #CRLF$ + Str(Factors())
Next MessageRequester("", S)</lang>
- Output:
Factored 9007199254740991 in 0.27 seconds. 6361 69431 20394401
Python
Python: Using Croft Spiral sieve
Note: the program below is saved to file prime_decomposition.py
and imported as a library here, here, here, here and here.
<lang python>from __future__ import print_function
import sys from itertools import islice, cycle, count
try:
from itertools import compress
except ImportError:
def compress(data, selectors): """compress('ABCDEF', [1,0,1,0,1,1]) --> A C E F""" return (d for d, s in zip(data, selectors) if s)
def is_prime(n):
return list(zip((True, False), decompose(n)))[-1][0]
class IsPrimeCached(dict):
def __missing__(self, n): r = is_prime(n) self[n] = r return r
is_prime_cached = IsPrimeCached()
def croft():
"""Yield prime integers using the Croft Spiral sieve.
This is a variant of wheel factorisation modulo 30. """ # Copied from: # https://code.google.com/p/pyprimes/source/browse/src/pyprimes.py # Implementation is based on erat3 from here: # http://stackoverflow.com/q/2211990 # and this website: # http://www.primesdemystified.com/ # Memory usage increases roughly linearly with the number of primes seen. # dict ``roots`` stores an entry x:p for every prime p. for p in (2, 3, 5): yield p roots = {9: 3, 25: 5} # Map d**2 -> d. primeroots = frozenset((1, 7, 11, 13, 17, 19, 23, 29)) selectors = (1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0) for q in compress( # Iterate over prime candidates 7, 9, 11, 13, ... islice(count(7), 0, None, 2), # Mask out those that can't possibly be prime. cycle(selectors) ): # Using dict membership testing instead of pop gives a # 5-10% speedup over the first three million primes. if q in roots: p = roots[q] del roots[q] x = q + 2*p while x in roots or (x % 30) not in primeroots: x += 2*p roots[x] = p else: roots[q*q] = q yield q
primes = croft
def decompose(n):
for p in primes(): if p*p > n: break while n % p == 0: yield p n //=p if n > 1: yield n
if __name__ == '__main__':
# Example: calculate factors of Mersenne numbers to M59 # import time for m in primes(): p = 2 ** m - 1 print( "2**{0:d}-1 = {1:d}, with factors:".format(m, p) ) start = time.time() for factor in decompose(p): print(factor, end=' ') sys.stdout.flush() print( "=> {0:.2f}s".format( time.time()-start ) ) if m >= 59: break</lang>
- Output:
2**2-1 = 3, with factors: 3 => 0.00s 2**3-1 = 7, with factors: 7 => 0.01s 2**5-1 = 31, with factors: 31 => 0.00s 2**7-1 = 127, with factors: 127 => 0.00s 2**11-1 = 2047, with factors: 23 89 => 0.00s 2**13-1 = 8191, with factors: 8191 => 0.00s 2**17-1 = 131071, with factors: 131071 => 0.00s 2**19-1 = 524287, with factors: 524287 => 0.00s 2**23-1 = 8388607, with factors: 47 178481 => 0.01s 2**29-1 = 536870911, with factors: 233 1103 2089 => 0.01s 2**31-1 = 2147483647, with factors: 2147483647 => 0.03s 2**37-1 = 137438953471, with factors: 223 616318177 => 0.02s 2**41-1 = 2199023255551, with factors: 13367 164511353 => 0.01s 2**43-1 = 8796093022207, with factors: 431 9719 2099863 => 0.01s 2**47-1 = 140737488355327, with factors: 2351 4513 13264529 => 0.01s 2**53-1 = 9007199254740991, with factors: 6361 69431 20394401 => 0.04s 2**59-1 = 576460752303423487, with factors: 179951 3203431780337 => 1.22s
Python: Using floating point
Here a shorter and marginally faster algorithm:
<lang python>from math import floor, sqrt try:
long
except NameError:
long = int
def fac(n):
step = lambda x: 1 + (x<<2) - ((x>>1)<<1) maxq = long(floor(sqrt(n))) d = 1 q = 2 if n % 2 == 0 else 3 while q <= maxq and n % q != 0: q = step(d) d += 1 return [q] + fac(n // q) if q <= maxq else [n]
if __name__ == '__main__':
import time start = time.time() tocalc = 2**59-1 print("%s = %s" % (tocalc, fac(tocalc))) print("Needed %ss" % (time.time() - start))</lang>
- Output:
576460752303423487 = [3203431780337, 179951] Needed 0.9240529537200928s
Quackery
<lang Quackery> [ [] swap
dup times [ [ dup i^ 2 + /mod 0 = while nip dip [ i^ 2 + join ] again ] drop dup 1 = if conclude ] drop ] is primefactors ( n --> [ )
1047552 primefactors echo</lang>
- Output:
[ 2 2 2 2 2 2 2 2 2 2 3 11 31 ]
R
<lang R>findfactors <- function(num) {
x <- NULL firstprime<- 2; secondprime <- 3; everyprime <- num while( everyprime != 1 ) { while( everyprime%%firstprime == 0 ) { x <- c(x, firstprime) everyprime <- floor(everyprime/ firstprime) } firstprime <- secondprime secondprime <- secondprime + 2 } x
}
print(findfactors(1027*4))</lang>
Or a more explicit (but less efficient) recursive approach:
Recursive Approach (Less efficient for large numbers)
<lang R> primes <- as.integer(c())
max_prime_checker <- function(n){
divisor <<- NULL
primes <- primes[primes <= n]
for(i in 1:length(primes)){ if((n/primes[i]) %% 1 == 0){ divisor[i]<<-1 } else { divisor[i]<<-0 } } num_find <<- primes*as.integer(divisor) return(max(num_find))
}
- recursive prime finder
prime_factors <- function(n){
factors <- NULL large <- max_prime_checker(n) n1 <- n/large if(max_prime_checker(n1) == n1){ factors <- c(large,n1) return(factors) } else { factors <- c(large, prime_factors(n1)) return(factors) }
} </lang>
Alternate solution
<lang R>findfactors <- function(n) {
a <- NULL if (n > 1) { while (n %% 2 == 0) { a <- c(a, 2) n <- n %/% 2 } k <- 3 while (k * k <= n) { while (n %% k == 0) { a <- c(a, k) n <- n %/% k } k <- k + 2 } if (n > 1) a <- c(a, n) } a
}</lang>
Racket
<lang Racket>
- lang racket
(require math) (define (factors n)
(append-map (λ (x) (make-list (cadr x) (car x))) (factorize n)))
</lang>
Or, an explicit (and less efficient) computation: <lang Racket>
- lang racket
(define (factors number)
(let loop ([n number] [i 2]) (if (= n 1) '() (let-values ([(q r) (quotient/remainder n i)]) (if (zero? r) (cons i (loop q i)) (loop n (add1 i)))))))
</lang>
Raku
(formerly Perl 6)
Pure Raku
This is a pure Raku version that uses no outside libraries. It uses a variant of Pollard's rho factoring algorithm and is fairly performent when factoring numbers < 2⁸⁰; typically taking well under a second on an i7. It starts to slow down with larger numbers, but really bogs down factoring numbers that have more than 1 factor larger than about 2⁴⁰.
<lang perl6>sub prime-factors ( Int $n where * > 0 ) {
return $n if $n.is-prime; return () if $n == 1; my $factor = find-factor( $n ); sort flat ( $factor, $n div $factor ).map: &prime-factors;
}
sub find-factor ( Int $n, $constant = 1 ) {
return 2 unless $n +& 1; if (my $gcd = $n gcd 6541380665835015) > 1 { # magic number: [*] primes 3 .. 43 return $gcd if $gcd != $n } my $x = 2; my $rho = 1; my $factor = 1; while $factor == 1 { $rho = $rho +< 1; my $fixed = $x; my int $i = 0; while $i < $rho { $x = ( $x * $x + $constant ) % $n; $factor = ( $x - $fixed ) gcd $n; last if 1 < $factor; $i = $i + 1; } } $factor = find-factor( $n, $constant + 1 ) if $n == $factor; $factor;
}
.put for (2²⁹-1, 2⁴¹-1, 2⁵⁹-1, 2⁷¹-1, 2⁷⁹-1, 2⁹⁷-1, 2¹¹⁷-1, 2²⁴¹-1, 5465610891074107968111136514192945634873647594456118359804135903459867604844945580205745718497)\ .hyper(:1batch).map: -> $n {
my $start = now; "factors of $n: ", prime-factors($n).join(' × '), " \t in ", (now - $start).fmt("%0.3f"), " sec."
}</lang>
- Output:
factors of 536870911: 233 × 1103 × 2089 in 0.004 sec. factors of 2199023255551: 13367 × 164511353 in 0.011 sec. factors of 576460752303423487: 179951 × 3203431780337 in 0.023 sec. factors of 2361183241434822606847: 228479 × 48544121 × 212885833 in 0.190 sec. factors of 604462909807314587353087: 2687 × 202029703 × 1113491139767 in 0.294 sec. factors of 158456325028528675187087900671: 11447 × 13842607235828485645766393 in 0.005 sec. factors of 166153499473114484112975882535043071: 7 × 73 × 79 × 937 × 6553 × 8191 × 86113 × 121369 × 7830118297 in 0.022 sec. factors of 3533694129556768659166595001485837031654967793751237916243212402585239551: 22000409 × 160619474372352289412737508720216839225805656328990879953332340439 in 0.085 sec. factors of 5465610891074107968111136514192945634873647594456118359804135903459867604844945580205745718497: 165901 × 10424087 × 18830281 × 53204737 × 56402249 × 59663291 × 91931221 × 95174413 × 305293727939 × 444161842339 × 790130065009 in 28.427 sec.
There is a Raku module available: Prime::Factor, that uses essentially this algorithm with some minor performance tweaks.
External library
If you really need a speed boost, load the highly optimized Perl 5 ntheory module. It needs a little extra plumbing to deal with the lack of built-in big integer support, but for large number factoring the interface overhead is worth it. <lang perl6>use Inline::Perl5; my $p5 = Inline::Perl5.new(); $p5.use( 'ntheory' );
sub prime-factors ($i) {
my &primes = $p5.run('sub { map { ntheory::todigitstring $_ } sort {$a <=> $b} ntheory::factor $_[0] }'); primes("$i");
}
for 2²⁹-1, 2⁴¹-1, 2⁵⁹-1, 2⁷¹-1, 2⁷⁹-1, 2⁹⁷-1, 2¹¹⁷-1, 5465610891074107968111136514192945634873647594456118359804135903459867604844945580205745718497
-> $n { my $start = now; say "factors of $n: ", prime-factors($n).join(' × '), " \t in ", (now - $start).fmt("%0.3f"), " sec."
}</lang>
- Output:
factors of 536870911: 233 × 1103 × 2089 in 0.001 sec. factors of 2199023255551: 13367 × 164511353 in 0.001 sec. factors of 576460752303423487: 179951 × 3203431780337 in 0.001 sec. factors of 2361183241434822606847: 228479 × 48544121 × 212885833 in 0.012 sec. factors of 604462909807314587353087: 2687 × 202029703 × 1113491139767 in 0.003 sec. factors of 158456325028528675187087900671: 11447 × 13842607235828485645766393 in 0.001 sec. factors of 166153499473114484112975882535043071: 7 × 73 × 79 × 937 × 6553 × 8191 × 86113 × 121369 × 7830118297 in 0.001 sec. factors of 5465610891074107968111136514192945634873647594456118359804135903459867604844945580205745718497: 165901 × 10424087 × 18830281 × 53204737 × 56402249 × 59663291 × 91931221 × 95174413 × 305293727939 × 444161842339 × 790130065009 in 0.064 sec.
REXX
optimized slightly
No (error) checking was done for the input arguments to test their validity.
The number of decimal digits is adjusted to match the size of the top-of-the-range (top).
Also, a count of primes found is shown.
If the top number is negative, only the number of primes up to abs(top) is shown.
A method exists in this REXX program to also test Mersenne-type numbers (2n - 1).
Since the majority of computing time is spent looking for primes, that part of the program was
optimized somewhat (but could be extended if more optimization is wanted).
<lang rexx>/*REXX pgm does prime decomposition of a range of positive integers (with a prime count)*/
numeric digits 1000 /*handle thousand digits for the powers*/
parse arg bot top step base add /*get optional arguments from the C.L. */
if bot== then do; bot=1; top=100; end /*no BOT given? Then use the default.*/
if top== then top=bot /* " TOP? " " " " " */
if step== then step= 1 /* " STEP? " " " " " */
if add == then add= -1 /* " ADD? " " " " " */
tell= top>0; top=abs(top) /*if TOP is negative, suppress displays*/
w=length(top) /*get maximum width for aligned display*/
if base\== then w=length(base**top) /*will be testing powers of two later? */
@.=left(, 7); @.0="{unity}"; @.1='[prime]' /*some literals: pad; prime (or not).*/
numeric digits max(9, w+1) /*maybe increase the digits precision. */
- =0 /*#: is the number of primes found. */
do n=bot to top by step /*process a single number or a range.*/ ?=n; if base\== then ?=base**n + add /*should we perform a "Mercenne" test? */ pf=factr(?); f=words(pf) /*get prime factors; number of factors.*/ if f==1 then #=#+1 /*Is N prime? Then bump prime counter.*/ if tell then say right(?,w) right('('f")",9) 'prime factors: ' @.f pf end /*n*/
say ps= 'primes'; if p==1 then ps= "prime" /*setup for proper English in sentence.*/ say right(#, w+9+1) ps 'found.' /*display the number of primes found. */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ factr: procedure; parse arg x 1 d,$ /*set X, D to argument 1; $ to null.*/ if x==1 then return /*handle the special case of X = 1. */
do while x//2==0; $=$ 2; x=x%2; end /*append all the 2 factors of new X.*/ do while x//3==0; $=$ 3; x=x%3; end /* " " " 3 " " " " */ do while x//5==0; $=$ 5; x=x%5; end /* " " " 5 " " " " */ do while x//7==0; $=$ 7; x=x%7; end /* " " " 7 " " " " */ /* ___*/
q=1; do while q<=x; q=q*4; end /*these two lines compute integer √ X */ r=0; do while q>1; q=q%4; _=d-r-q; r=r%2; if _>=0 then do; d=_; r=r+q; end; end
do j=11 by 6 to r /*insure that J isn't divisible by 3.*/ parse var j -1 _ /*obtain the last decimal digit of J. */ if _\==5 then do while x//j==0; $=$ j; x=x%j; end /*maybe reduce by J. */ if _ ==3 then iterate /*Is next Y is divisible by 5? Skip.*/ y=j+2; do while x//y==0; $=$ y; x=x%y; end /*maybe reduce by J. */ end /*j*/ /* [↓] The $ list has a leading blank.*/
if x==1 then return $ /*Is residual=unity? Then don't append.*/
return $ x /*return $ with appended residual. */</lang>
output when using the default input of: 1 100
(Shown at three-quarter size.)
1 (0) prime factors: {unity} 2 (1) prime factors: [prime] 2 3 (1) prime factors: [prime] 3 4 (2) prime factors: 2 2 5 (1) prime factors: [prime] 5 6 (2) prime factors: 2 3 7 (1) prime factors: [prime] 7 8 (3) prime factors: 2 2 2 9 (2) prime factors: 3 3 10 (2) prime factors: 2 5 11 (1) prime factors: [prime] 11 12 (3) prime factors: 2 2 3 13 (1) prime factors: [prime] 13 14 (2) prime factors: 2 7 15 (2) prime factors: 3 5 16 (4) prime factors: 2 2 2 2 17 (1) prime factors: [prime] 17 18 (3) prime factors: 2 3 3 19 (1) prime factors: [prime] 19 20 (3) prime factors: 2 2 5 21 (2) prime factors: 3 7 22 (2) prime factors: 2 11 23 (1) prime factors: [prime] 23 24 (4) prime factors: 2 2 2 3 25 (2) prime factors: 5 5 26 (2) prime factors: 2 13 27 (3) prime factors: 3 3 3 28 (3) prime factors: 2 2 7 29 (1) prime factors: [prime] 29 30 (3) prime factors: 2 3 5 31 (1) prime factors: [prime] 31 32 (5) prime factors: 2 2 2 2 2 33 (2) prime factors: 3 11 34 (2) prime factors: 2 17 35 (2) prime factors: 5 7 36 (4) prime factors: 2 2 3 3 37 (1) prime factors: [prime] 37 38 (2) prime factors: 2 19 39 (2) prime factors: 3 13 40 (4) prime factors: 2 2 2 5 41 (1) prime factors: [prime] 41 42 (3) prime factors: 2 3 7 43 (1) prime factors: [prime] 43 44 (3) prime factors: 2 2 11 45 (3) prime factors: 3 3 5 46 (2) prime factors: 2 23 47 (1) prime factors: [prime] 47 48 (5) prime factors: 2 2 2 2 3 49 (2) prime factors: 7 7 50 (3) prime factors: 2 5 5 51 (2) prime factors: 3 17 52 (3) prime factors: 2 2 13 53 (1) prime factors: [prime] 53 54 (4) prime factors: 2 3 3 3 55 (2) prime factors: 5 11 56 (4) prime factors: 2 2 2 7 57 (2) prime factors: 3 19 58 (2) prime factors: 2 29 59 (1) prime factors: [prime] 59 60 (4) prime factors: 2 2 3 5 61 (1) prime factors: [prime] 61 62 (2) prime factors: 2 31 63 (3) prime factors: 3 3 7 64 (6) prime factors: 2 2 2 2 2 2 65 (2) prime factors: 5 13 66 (3) prime factors: 2 3 11 67 (1) prime factors: [prime] 67 68 (3) prime factors: 2 2 17 69 (2) prime factors: 3 23 70 (3) prime factors: 2 5 7 71 (1) prime factors: [prime] 71 72 (5) prime factors: 2 2 2 3 3 73 (1) prime factors: [prime] 73 74 (2) prime factors: 2 37 75 (3) prime factors: 3 5 5 76 (3) prime factors: 2 2 19 77 (2) prime factors: 7 11 78 (3) prime factors: 2 3 13 79 (1) prime factors: [prime] 79 80 (5) prime factors: 2 2 2 2 5 81 (4) prime factors: 3 3 3 3 82 (2) prime factors: 2 41 83 (1) prime factors: [prime] 83 84 (4) prime factors: 2 2 3 7 85 (2) prime factors: 5 17 86 (2) prime factors: 2 43 87 (2) prime factors: 3 29 88 (4) prime factors: 2 2 2 11 89 (1) prime factors: [prime] 89 90 (4) prime factors: 2 3 3 5 91 (2) prime factors: 7 13 92 (3) prime factors: 2 2 23 93 (2) prime factors: 3 31 94 (2) prime factors: 2 47 95 (2) prime factors: 5 19 96 (6) prime factors: 2 2 2 2 2 3 97 (1) prime factors: [prime] 97 98 (3) prime factors: 2 7 7 99 (3) prime factors: 3 3 11 100 (4) prime factors: 2 2 5 5 25 primes found.
output when using the input of: 9007199254740991
9007199254740991 (3) prime factors: 6361 69431 20394401 0 primes found.
output when using the input of: 2543821448263974486045199
2543821448263974486045199 (6) prime factors: 701 1123 1123 2411 1092461 1092461 0 primes found.
output when using the input of: 1 -1000000
78498 primes found.
output when using the input of: 2 50 1 2 -1
(essentially testing for Mersenne primes: 2n -1)
(Shown at three-quarter size.)
3 (1) prime factors: [prime] 3 7 (1) prime factors: [prime] 7 15 (2) prime factors: 3 5 31 (1) prime factors: [prime] 31 63 (3) prime factors: 3 3 7 127 (1) prime factors: [prime] 127 255 (3) prime factors: 3 5 17 511 (2) prime factors: 7 73 1023 (3) prime factors: 3 11 31 2047 (2) prime factors: 23 89 4095 (5) prime factors: 3 3 5 7 13 8191 (1) prime factors: [prime] 8191 16383 (2) prime factors: 3 5461 32767 (2) prime factors: 7 4681 65535 (4) prime factors: 3 5 17 257 131071 (1) prime factors: [prime] 131071 262143 (5) prime factors: 3 3 3 7 1387 524287 (1) prime factors: [prime] 524287 1048575 (6) prime factors: 3 5 5 11 41 31 2097151 (3) prime factors: 7 7 42799 4194303 (4) prime factors: 3 23 89 683 8388607 (2) prime factors: 47 178481 16777215 (7) prime factors: 3 3 5 7 13 17 241 33554431 (1) prime factors: [prime] 33554431 67108863 (2) prime factors: 3 22369621 134217727 (2) prime factors: 7 19173961 268435455 (5) prime factors: 3 5 29 113 5461 536870911 (3) prime factors: 233 1103 2089 1073741823 (5) prime factors: 3 3 7 11 1549411 2147483647 (1) prime factors: [prime] 2147483647 4294967295 (5) prime factors: 3 5 17 257 65537 8589934591 (4) prime factors: 7 23 89 599479 17179869183 (3) prime factors: 3 43691 131071 34359738367 (3) prime factors: 71 122921 3937 68719476735 (7) prime factors: 3 3 3 5 7 13 5593771 137438953471 (1) prime factors: [prime] 137438953471 274877906943 (2) prime factors: 3 91625968981 549755813887 (2) prime factors: 7 78536544841 1099511627775 (7) prime factors: 3 5 5 11 17 41 1912111 2199023255551 (2) prime factors: 13367 164511353 4398046511103 (5) prime factors: 3 3 7 7 9972894583 8796093022207 (3) prime factors: 431 9719 2099863 17592186044415 (6) prime factors: 3 5 23 89 683 838861 35184372088831 (2) prime factors: 7 5026338869833 70368744177663 (4) prime factors: 3 47 178481 2796203 140737488355327 (2) prime factors: 2351 59862819377 281474976710655 (8) prime factors: 3 3 5 7 13 17 257 15732721 562949953421311 (1) prime factors: [prime] 562949953421311 1125899906842623 (4) prime factors: 3 11 251 135928999981 11 primes found.
output when using the input of: 1 50 1 2 +1
(essentially testing for 2n +1)
(Shown at three-quarter size.)
3 (1) prime factors: [prime] 3 5 (1) prime factors: [prime] 5 9 (2) prime factors: 3 3 17 (1) prime factors: [prime] 17 33 (2) prime factors: 3 11 65 (2) prime factors: 5 13 129 (2) prime factors: 3 43 257 (1) prime factors: [prime] 257 513 (4) prime factors: 3 3 3 19 1025 (3) prime factors: 5 5 41 2049 (2) prime factors: 3 683 4097 (2) prime factors: 17 241 8193 (2) prime factors: 3 2731 16385 (3) prime factors: 5 29 113 32769 (4) prime factors: 3 3 11 331 65537 (1) prime factors: [prime] 65537 131073 (2) prime factors: 3 43691 262145 (3) prime factors: 5 13 4033 524289 (2) prime factors: 3 174763 1048577 (2) prime factors: 17 61681 2097153 (3) prime factors: 3 3 233017 4194305 (2) prime factors: 5 838861 8388609 (2) prime factors: 3 2796203 16777217 (2) prime factors: 257 65281 33554433 (4) prime factors: 3 11 251 4051 67108865 (4) prime factors: 5 53 1613 157 134217729 (5) prime factors: 3 3 3 3 1657009 268435457 (2) prime factors: 17 15790321 536870913 (3) prime factors: 3 59 3033169 1073741825 (5) prime factors: 5 5 13 41 80581 2147483649 (2) prime factors: 3 715827883 4294967297 (2) prime factors: 641 6700417 8589934593 (4) prime factors: 3 3 683 1397419 17179869185 (4) prime factors: 5 137 953 26317 34359738369 (5) prime factors: 3 11 281 86171 43 68719476737 (2) prime factors: 17 4042322161 137438953473 (2) prime factors: 3 45812984491 274877906945 (2) prime factors: 5 54975581389 549755813889 (3) prime factors: 3 3 61083979321 1099511627777 (2) prime factors: 257 4278255361 2199023255553 (3) prime factors: 3 83 8831418697 4398046511105 (5) prime factors: 5 13 29 113 20647621 8796093022209 (2) prime factors: 3 2932031007403 17592186044417 (3) prime factors: 17 353 2931542417 35184372088833 (5) prime factors: 3 3 3 11 118465899289 70368744177665 (4) prime factors: 5 1013 30269 458989 140737488355329 (2) prime factors: 3 46912496118443 281474976710657 (2) prime factors: 65537 4294901761 562949953421313 (2) prime factors: 3 187649984473771 1125899906842625 (6) prime factors: 5 5 5 41 101 2175126601 5 primes found.
optimized more
This REXX version is about 20% faster than the 1st REXX version when factoring one million numbers. <lang rexx>/*REXX pgm does prime decomposition of a range of positive integers (with a prime count)*/ numeric digits 1000 /*handle thousand digits for the powers*/ parse arg bot top step base add /*get optional arguments from the C.L. */ if bot== then do; bot=1; top=100; end /*no BOT given? Then use the default.*/ if top== then top=bot /* " TOP? " " " " " */ if step== then step= 1 /* " STEP? " " " " " */ if add == then add= -1 /* " ADD? " " " " " */ tell= top>0; top=abs(top) /*if TOP is negative, suppress displays*/ w=length(top) /*get maximum width for aligned display*/ if base\== then w=length(base**top) /*will be testing powers of two later? */ @.=left(, 7); @.0="{unity}"; @.1='[prime]' /*some literals: pad; prime (or not).*/ numeric digits max(9, w+1) /*maybe increase the digits precision. */
- =0 /*#: is the number of primes found. */
do n=bot to top by step /*process a single number or a range.*/ ?=n; if base\== then ?=base**n + add /*should we perform a "Mercenne" test? */ pf=factr(?); f=words(pf) /*get prime factors; number of factors.*/ if f==1 then #=#+1 /*Is N prime? Then bump prime counter.*/ if tell then say right(?,w) right('('f")",9) 'prime factors: ' @.f pf end /*n*/
say ps= 'primes'; if p==1 then ps= "prime" /*setup for proper English in sentence.*/ say right(#, w+9+1) ps 'found.' /*display the number of primes found. */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ factr: procedure; parse arg x 1 d,$ /*set X, D to argument 1; $ to null.*/ if x==1 then return /*handle the special case of X = 1. */
do while x// 2==0; $=$ 2; x=x%2; end /*append all the 2 factors of new X.*/ do while x// 3==0; $=$ 3; x=x%3; end /* " " " 3 " " " " */ do while x// 5==0; $=$ 5; x=x%5; end /* " " " 5 " " " " */ do while x// 7==0; $=$ 7; x=x%7; end /* " " " 7 " " " " */ do while x//11==0; $=$ 11; x=x%11; end /* " " " 11 " " " " */ /* ◄■■■■ added.*/ do while x//13==0; $=$ 13; x=x%13; end /* " " " 13 " " " " */ /* ◄■■■■ added.*/ do while x//17==0; $=$ 17; x=x%17; end /* " " " 17 " " " " */ /* ◄■■■■ added.*/ do while x//19==0; $=$ 19; x=x%19; end /* " " " 19 " " " " */ /* ◄■■■■ added.*/ do while x//23==0; $=$ 23; x=x%23; end /* " " " 23 " " " " */ /* ◄■■■■ added.*/ /* ___*/
q=1; do while q<=x; q=q*4; end /*these two lines compute integer √ X */ r=0; do while q>1; q=q%4; _=d-r-q; r=r%2; if _>=0 then do; d=_; r=r+q; end; end
do j=29 by 6 to r /*insure that J isn't divisible by 3.*/ /* ◄■■■■ changed.*/ parse var j -1 _ /*obtain the last decimal digit of J. */ if _\==5 then do while x//j==0; $=$ j; x=x%j; end /*maybe reduce by J. */ if _ ==3 then iterate /*Is next Y is divisible by 5? Skip.*/ y=j+2; do while x//y==0; $=$ y; x=x%y; end /*maybe reduce by J. */ end /*j*/ /* [↓] The $ list has a leading blank.*/
if x==1 then return $ /*Is residual=unity? Then don't append.*/
return $ x /*return $ with appended residual. */</lang>
output is identical to the 1st REXX version.
Ring
<lang ring> prime = 18705 decomp(prime)
func decomp nr x = "" for i = 1 to nr
if isPrime(i) and nr % i = 0 x = x + string(i) + " * " ok if i = nr x2 = substr(x,1,(len(x)-2)) see string(nr) + " = " + x2 + nl ok
next
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>
Ruby
Built in
<lang ruby>irb(main):001:0> require 'prime' => true irb(main):003:0> 2543821448263974486045199.prime_division => [[701, 1], [1123, 2], [2411, 1], [1092461, 2]]</lang>
Simple algorithm
<lang ruby># Get prime decomposition of integer _i_.
- This routine is terribly inefficient, but elegance rules.
def prime_factors(i)
v = (2..i-1).detect{|j| i % j == 0} v ? ([v] + prime_factors(i/v)) : [i]
end
- Example: Decompose all possible Mersenne primes up to 2**31-1.
- This may take several minutes to show that 2**31-1 is prime.
(2..31).each do |i|
factors = prime_factors(2**i-1) puts "2**#{i}-1 = #{2**i-1} = #{factors.join(' * ')}"
end</lang>
- Output:
... 2**28-1 = 268435455 = 3 * 5 * 29 * 43 * 113 * 127 2**29-1 = 536870911 = 233 * 1103 * 2089 2**30-1 = 1073741823 = 3 * 3 * 7 * 11 * 31 * 151 * 331 2**31-1 = 2147483647 = 2147483647
Faster algorithm
<lang ruby># Get prime decomposition of integer _i_.
- This routine is more efficient than prime_factors,
- and quite similar to Integer#prime_division of MRI 1.9.
def prime_factors_faster(i)
factors = [] check = proc do |p| while(q, r = i.divmod(p) r.zero?) factors << p i = q end end check[2] check[3] p = 5 while p * p <= i check[p] p += 2 check[p] p += 4 # skip multiples of 2 and 3 end factors << i if i > 1 factors
end
- Example: Decompose all possible Mersenne primes up to 2**70-1.
- This may take several minutes to show that 2**61-1 is prime,
- but 2**62-1 and 2**67-1 are not prime.
(2..70).each do |i|
factors = prime_factors_faster(2**i-1) puts "2**#{i}-1 = #{2**i-1} = #{factors.join(' * ')}"
end</lang>
- Output:
... 2**67-1 = 147573952589676412927 = 193707721 * 761838257287 2**68-1 = 295147905179352825855 = 3 * 5 * 137 * 953 * 26317 * 43691 * 131071 2**69-1 = 590295810358705651711 = 7 * 47 * 178481 * 10052678938039 2**70-1 = 1180591620717411303423 = 3 * 11 * 31 * 43 * 71 * 127 * 281 * 86171 * 122921
This benchmark compares the different implementations.
<lang ruby>require 'benchmark' require 'mathn' Benchmark.bm(24) do |x|
[2**25 - 6, 2**35 - 7].each do |i| puts "#{i} = #{prime_factors_faster(i).join(' * ')}" x.report(" prime_factors") { prime_factors(i) } x.report(" prime_factors_faster") { prime_factors_faster(i) } x.report(" Integer#prime_division") { i.prime_division } end
end</lang>
With MRI 1.8, prime_factors is slow, Integer#prime_division is fast, and prime_factors_faster is very fast. With MRI 1.9, Integer#prime_division is also very fast.
Rust
Rust's largest built-in integer type is u128
(128-bit unsigned integer) which is pretty large, but not unlimited.
The solution therefore uses external crates for big integers.
Add the dependencies in Cargo.toml
:
[package] name = "prime_decomposition" version = "0.1.1" edition = "2018" [dependencies] num-bigint = "0.3.0" num-traits = "0.2.12"
The implementation:
<lang Rust>use num_bigint::BigUint; use num_traits::{One, Zero}; use std::fmt::{Display, Formatter};
- [derive(Clone, Debug)]
pub struct Factors {
pub number: BigUint, pub result: Vec<BigUint>,
}
impl Factors {
pub fn of(number: BigUint) -> Factors { let mut factors = Self { number: number.clone(), result: Vec::new(), };
let big_2 = BigUint::from(2u8); let big_4 = BigUint::from(4u8);
factors.check(&big_2); factors.check(&BigUint::from(3u8));
let mut divisor = BigUint::from(5u8); while &divisor * &divisor <= factors.number { factors.check(&divisor); divisor += &big_2; factors.check(&divisor); divisor += &big_4; }
if factors.number > BigUint::one() { factors.result.push(factors.number); }
factors.number = number; // Restore the number factors }
pub fn is_prime(&self) -> bool { self.result.len() == 1 }
fn check(&mut self, divisor: &BigUint) { while (&self.number % divisor).is_zero() { self.result.push(divisor.clone()); self.number /= divisor; } }
}
impl Display for Factors {
fn fmt(&self, f: &mut Formatter) -> std::fmt::Result { let mut iter = self.result.iter();
match iter.next() { None => write!(f, "[]"),
Some(first) => { write!(f, "[{}", first)?; for next in iter { write!(f, ", {}", next)?; }
write!(f, "]") } } }
}
fn print_factors(number: BigUint) {
let factors = Factors::of(number);
if factors.is_prime() { println!("{} -> {} (prime)", factors.number, factors); } else { println!("{} -> {}", factors.number, factors); }
}
fn main() {
print_factors(24u32.into()); print_factors(32u32.into()); print_factors(37u32.into());
// Find Mersenne primes
for n in 2..70 { print!("2**{} - 1: ", n); print_factors((BigUint::from(2u8) << n) - BigUint::one()); }
} </lang>
S-BASIC
<lang S-BASIC> rem - return p mod q function mod(p, q = integer) = integer end = p - q * (p/q)
dim integer factors(16) rem log2(maxint) is sufficiently large
comment
Find the prime factors of n and store in global array factors (arrays cannot be passed as parameters) and return the number found. If n is prime, it will be stored as the only factor.
end function primefactors(n = integer) = integer
var p, count = integer p = 2 count = 1 while n >= (p * p) do begin if mod(n, p) = 0 then begin factors(count) = p count = count + 1 n = n / p end else p = p + 1 end factors(count) = n
end = count
rem -- exercise the routine by checking odd numbers from 77 to 99
var i, k, nfound = integer
for i = 77 to 99 step 2
nfound = primefactors(i) print i;"; "; for k = 1 to nfound print factors(k); next k print
next i
end </lang>
- Output:
77: 7 11 79: 79 81: 3 3 3 3 83: 83 85: 5 17 87: 3 29 89: 89 91: 7 13 93: 3 31 95: 5 19 97: 97 99: 3 3 11
Scala
<lang Scala>import annotation.tailrec import collection.parallel.mutable.ParSeq
object PrimeFactors extends App {
def factorize(n: Long): List[Long] = { @tailrec def factors(tuple: (Long, Long, List[Long], Int)): List[Long] = { tuple match { case (1, _, acc, _) => acc case (n, k, acc, _) if (n % k == 0) => factors((n / k, k, acc ++ ParSeq(k), Math.sqrt(n / k).toInt)) case (n, k, acc, sqr) if (k < sqr) => factors(n, k + 1, acc, sqr) case (n, k, acc, sqr) if (k >= sqr) => factors((1, k, acc ++ ParSeq(n), 0)) } } factors((n, 2, List[Long](), Math.sqrt(n).toInt)) }
def mersenne(p: Int): BigInt = (BigInt(2) pow p) - 1
def sieve(nums: Stream[Int]): Stream[Int] = Stream.cons(nums.head, sieve((nums.tail) filter (_ % nums.head != 0))) // An infinite stream of primes, lazy evaluation and memo-ized val oddPrimes = sieve(Stream.from(3, 2)) def primes = sieve(2 #:: oddPrimes)
oddPrimes takeWhile (_ <= 59) foreach { p => { // Needs some intermediate results for nice formatting val numM = s"M${p}" val nMersenne = mersenne(p).toLong val lit = f"${nMersenne}%30d"
val datum = System.nanoTime val result = factorize(nMersenne) val mSec = ((System.nanoTime - datum) / 1.0e+6).round
def decStr = { if (lit.length > 30) f"(M has ${lit.length}%3d dec)" else "" } def sPrime = { if (result.isEmpty) " is a prime number." else "" }
println( f"$numM%4s = 2^$p%03d - 1 = ${lit}%s${sPrime} ($mSec%,4d msec) composed of ${result.mkString(" × ")}") } }
}</lang>
- Output:
M3 = 2^003 - 1 = 7 ( 23 msec) composed of 7 M5 = 2^005 - 1 = 31 ( 0 msec) composed of 31 M7 = 2^007 - 1 = 127 ( 0 msec) composed of 127 M11 = 2^011 - 1 = 2047 ( 0 msec) composed of 23 × 89 M13 = 2^013 - 1 = 8191 ( 0 msec) composed of 8191 M17 = 2^017 - 1 = 131071 ( 1 msec) composed of 131071 M19 = 2^019 - 1 = 524287 ( 1 msec) composed of 524287 M23 = 2^023 - 1 = 8388607 ( 1 msec) composed of 47 × 178481 M29 = 2^029 - 1 = 536870911 ( 2 msec) composed of 233 × 1103 × 2089 M31 = 2^031 - 1 = 2147483647 ( 39 msec) composed of 2147483647 M37 = 2^037 - 1 = 137438953471 ( 8 msec) composed of 223 × 616318177 M41 = 2^041 - 1 = 2199023255551 ( 2 msec) composed of 13367 × 164511353 M43 = 2^043 - 1 = 8796093022207 ( 2 msec) composed of 431 × 9719 × 2099863 M47 = 2^047 - 1 = 140737488355327 ( 2 msec) composed of 2351 × 4513 × 13264529 M53 = 2^053 - 1 = 9007199254740991 ( 7 msec) composed of 6361 × 69431 × 20394401 M59 = 2^059 - 1 = 576460752303423487 ( 152 msec) composed of 179951 × 3203431780337
Getting the prime factors does not require identifying prime numbers. Since the problems seems to ask for it, here is one version that does it:
<lang Scala>class PrimeFactors(n: BigInt) extends Iterator[BigInt] {
val zero = BigInt(0) val one = BigInt(1) val two = BigInt(2) def isPrime(n: BigInt) = n.isProbablePrime(10) var currentN = n var prime = two
def nextPrime = if (prime == two) { prime += one } else { prime += two while (!isPrime(prime)) { prime += two if (prime * prime > currentN) prime = currentN } }
def next = { if (!hasNext) throw new NoSuchElementException("next on empty iterator") while(currentN % prime != zero) { nextPrime } currentN /= prime prime }
def hasNext = currentN != one && currentN > zero
}</lang>
The method isProbablePrime(n) has a chance of 1 - 1/(2^n) of correctly identifying a prime. Next is a version that does not depend on identifying primes, and works with arbitrary integral numbers: <lang Scala>class PrimeFactors[N](n: N)(implicit num: Integral[N]) extends Iterator[N] {
import num._ val two = one + one var currentN = n var divisor = two
def next = { if (!hasNext) throw new NoSuchElementException("next on empty iterator") while(currentN % divisor != zero) { if (divisor == two) divisor += one else divisor += two if (divisor * divisor > currentN) divisor = currentN } currentN /= divisor divisor }
def hasNext = currentN != one && currentN > zero
}</lang>
- Output:
Both versions can be rather slow, as they accept arbitrarily big numbers, as requested.
- Test:
scala> BigInt(2) to BigInt(30) filter (_ isProbablePrime 10) map (p => (p, BigInt(2).pow(p.toInt) - 1)) foreach { | case (prime, n) => println("2**"+prime+"-1 = "+n+", with factors: "+new PrimeFactors(n).mkString(", ")) | } 2**2-1 = 3, with factors: 3 2**3-1 = 7, with factors: 7 2**5-1 = 31, with factors: 31 2**7-1 = 127, with factors: 127 2**11-1 = 2047, with factors: 23, 89 2**13-1 = 8191, with factors: 8191 2**17-1 = 131071, with factors: 131071 2**19-1 = 524287, with factors: 524287 2**23-1 = 8388607, with factors: 47, 178481 2**29-1 = 536870911, with factors: 233, 1103, 2089 2**31-1 = 2147483647, with factors: 2147483647 2**37-1 = 137438953471, with factors: 223, 616318177 2**41-1 = 2199023255551, with factors: 13367, 164511353 2**43-1 = 8796093022207, with factors: 431, 9719, 2099863 2**47-1 = 140737488355327, with factors: 2351, 4513, 13264529 2**53-1 = 9007199254740991, with factors: 6361, 69431, 20394401 2**59-1 = 576460752303423487, with factors: 179951, 3203431780337
Alternatively, Scala LazyLists and Iterators support quite elegant one-line encodings of iterative/recursive algorithms, allowing us to to define the prime factorization like so: <lang scala>import spire.math.SafeLong import spire.implicits._ def pFactors(num: SafeLong): Vector[SafeLong] = Iterator.iterate((Vector[SafeLong](), num, SafeLong(2))){case (ac, n, f) => if(n%f == 0) (ac :+ f, n/f, f) else (ac, n, f + 1)}.dropWhile(_._2 != 1).next._1</lang>
Scheme
<lang scheme>(define (factor number)
(define (*factor divisor number) (if (> (* divisor divisor) number) (list number) (if (= (modulo number divisor) 0) (cons divisor (*factor divisor (/ number divisor))) (*factor (+ divisor 1) number)))) (*factor 2 number))
(display (factor 111111111111)) (newline)</lang>
- Output:
(3 7 11 13 37 101 9901)
Seed7
<lang seed7>const func array integer: factorise (in var integer: number) is func
result var array integer: result is 0 times 0; local var integer: checker is 2; begin while checker * checker <= number do if number rem checker = 0 then result &:= [](checker); number := number div checker; else incr(checker); end if; end while; if number <> 1 then result &:= [](number); end if; end func;</lang>
Original source: [1]
SequenceL
Recursive Using isPrime
<lang sequencel>isPrime(n) := n = 2 or (n > 1 and none(n mod ([2]++((1...floor(sqrt(n)/2))*2+1)) = 0));
primeFactorization(num) := primeFactorizationHelp(num, []);
primeFactorizationHelp(num, current(1)) :=
let primeFactors[i] := i when num mod i = 0 and isPrime(i) foreach i within 2 ... num; in current when size(primeFactors) = 0 else primeFactorizationHelp(num / product(primeFactors), current ++ primeFactors);</lang>
Using isPrime Based On: [2]
Recursive Trial Division
<lang sequencel>primeFactorization(num) := primeFactorizationHelp(num, 2, []);
primeFactorizationHelp(num, divisor, factors(1)) :=
factors when num <= 1 else primeFactorizationHelp(num, divisor + 1, factors) when num mod divisor /= 0 else primeFactorizationHelp(num / divisor, divisor, factors ++ [divisor]);</lang>
Sidef
Built-in: <lang ruby>say factor(536870911) #=> [233, 1103, 2089] say factor_exp(536870911) #=> [[233, 1], [1103, 1], [2089, 1]]</lang>
Trial division: <lang ruby>func prime_factors(n) {
return [] if (n < 1) gather { while (!(n & 1)) { n >>= 1 take(2) } var p = 3 while ((n > 1) && (p*p <= n)) { while (n %% p) { n //= p take(p) } p += 2 } take(n) if (n > 1) }
}</lang>
Calling the function: <lang ruby>say prime_factors(536870911) #=> [233, 1103, 2089]</lang>
Simula
Simula has no built-in function to test for prime numbers.
Code for class bignum can be found here: https://rosettacode.org/wiki/Pi#Simula
<lang simula>
EXTERNAL CLASS BIGNUM;
BIGNUM
BEGIN
CLASS TEXTLIST; BEGIN CLASS TEXTARRAY(N); INTEGER N; BEGIN TEXT ARRAY DATA(1:N); END TEXTARRAY; PROCEDURE EXPAND(N); INTEGER N; BEGIN REF(TEXTARRAY) NEWARR; INTEGER I; NEWARR :- NEW TEXTARRAY(N); FOR I := 1 STEP 1 UNTIL SIZE DO BEGIN NEWARR.DATA(I) :- ARR.DATA(I); END; ARR :- NEWARR; END EXPAND; PROCEDURE APPEND(T); TEXT T; BEGIN IF SIZE = ARR.N THEN EXPAND(2*ARR.N); SIZE := SIZE+1; ARR.DATA(SIZE) :- T; END EXPAND; TEXT PROCEDURE GET(I); INTEGER I; GET :- ARR.DATA(I); REF(TEXTARRAY) ARR; INTEGER SIZE; EXPAND(20); END TEXTLIST;
REF(TEXTLIST) PROCEDURE PRIME_FACTORS(N); TEXT N; BEGIN REF(TEXTLIST) FACTORS; REF(DIVMOD) DM; TEXT P; FACTORS :- NEW TEXTLIST; IF TCMP(N, "1") < 0 THEN GOTO RETURN; P :- "2"; FOR DM :- TDIVMOD(N,P) WHILE TISZERO(DM.MOD) DO BEGIN N :- DM.DIV; FACTORS.APPEND(P); END; P :- "3"; WHILE TCMP(N,"1") > 0 AND THEN TCMP(TMUL(P,P),N) <= 0 DO BEGIN FOR DM :- TDIVMOD(N, P) WHILE TISZERO(DM.MOD) DO BEGIN N :- DM.DIV; FACTORS.APPEND(P); END; P :- TADD(P,"2"); END; IF TCMP(N,"1") > 0 THEN FACTORS.APPEND(N); RETURN: PRIME_FACTORS :- FACTORS; END PRIME_FACTORS;
REF(TEXTLIST) FACTORS; TEXT INP; INTEGER I;
FOR INP :- "536870911", "6768768", "1957", "64865899369365843" DO BEGIN FACTORS :- PRIME_FACTORS(INP); OUTTEXT("PRIME FACTORS OF "); OUTTEXT(INP); OUTTEXT(" => ["); FOR I := 1 STEP 1 UNTIL FACTORS.SIZE DO BEGIN IF I > 1 THEN OUTTEXT(", "); OUTTEXT(FACTORS.GET(I)); END; OUTTEXT("]"); OUTIMAGE; END;
END; </lang>
- Output:
PRIME FACTORS OF 536870911 => [233, 1103, 2089] PRIME FACTORS OF 6768768 => [2, 2, 2, 2, 2, 2, 2, 3, 17627] PRIME FACTORS OF 1957 => [19, 103] PRIME FACTORS OF 64865899369365843 => [3, 7, 397, 276229, 28166791] 5320 garbage collection(s) in 1.9 seconds.
Slate
Admittedly, this is just based on the Smalltalk entry below: <lang slate>n@(Integer traits) primesDo: block "Decomposes the Integer into primes, applying the block to each (in increasing order)." [| div next remaining |
div: 2. next: 3. remaining: n. [[(remaining \\ div) isZero] whileTrue: [block applyTo: {div}. remaining: remaining // div]. remaining = 1] whileFalse: [div: next. next: next + 2] "Just look at the next odd integer."
].</lang>
Smalltalk
<lang smalltalk>Integer extend [
primesDo: aBlock [ | div next rest | div := 2. next := 3. rest := self. [ [ rest \\ div == 0 ] whileTrue: [ aBlock value: div. rest := rest // div ]. rest = 1] whileFalse: [ div := next. next := next + 2 ] ]
] 123456 primesDo: [ :each | each printNl ]</lang>
SPAD
<lang SPAD>
(1) -> factor 102400
12 2 (1) 2 5 Type: Factored(Integer)
(2) -> factor 23193931893819371
(2) 83 3469 71341 1129153 Type: Factored(Integer)
</lang>
Domain:Factored(R)
Standard ML
Trial division <lang SML> val factor = fn n :IntInf.int => let
val unfactored = fn (u,_,_) => u val factors = fn (_,f,_) => f val try = fn (_,_,i) => i fun getresult t = unfactored t::(factors t) fun until done change x = if done x then getresult x else until done change (change x); (* iteration *)
fun lastprime t = unfactored t < (try t)*(try t) fun trymore t = if unfactored t mod (try t) = 0 then (unfactored t div (try t) , try t::(factors t) , try t ) else (unfactored t , factors t , try t + 1)
in
until lastprime trymore (n,[],2)
end; </lang>
- Array.fromList(factor 122489234920000001278234798233);; val it = fromList[658601127263, 41259943, 34942753, 43, 3]: IntInf.int array
Stata
The following Mata function will factor any representable positive integer (that is, between 1 and 2^53).
<lang stata>function factor(n_) {
n = n_ a = J(0,2,.) if (n<2) { return(a) } else if (n<4) { return((n,1)) } else { if (mod(n,2)==0) { for (i=0; mod(n,2)==0; i++) n = floor(n/2) a = a\(2,i) } for (k=3; k*k<=n; k=k+2) { if (mod(n,k)==0) { for (i=0; mod(n,k)==0; i++) n = floor(n/k) a = a\(k,i) } } if (n>1) a = a\(n,1) return(a) }
}</lang>
Swift
Uses the sieve of Eratosthenes. This is generic on any type that conforms to BinaryInteger. So in theory any BigInteger library should work with it.
<lang swift>func primeDecomposition<T: BinaryInteger>(of n: T) -> [T] {
guard n > 2 else { return [] }
func step(_ x: T) -> T { return 1 + (x << 2) - ((x >> 1) << 1) }
let maxQ = T(Double(n).squareRoot()) var d: T = 1 var q: T = n % 2 == 0 ? 2 : 3
while q <= maxQ && n % q != 0 { q = step(d) d += 1 }
return q <= maxQ ? [q] + primeDecomposition(of: n / q) : [n]
}
for prime in Eratosthenes(upTo: 60) {
let m = Int(pow(2, Double(prime))) - 1 let decom = primeDecomposition(of: m)
print("2^\(prime) - 1 = \(m) => \(decom)")
}</lang>
- Output:
2^2 - 1 = 3 => [3] 2^3 - 1 = 7 => [7] 2^5 - 1 = 31 => [31] 2^7 - 1 = 127 => [127] 2^11 - 1 = 2047 => [23, 89] 2^13 - 1 = 8191 => [8191] 2^17 - 1 = 131071 => [131071] 2^19 - 1 = 524287 => [524287] 2^23 - 1 = 8388607 => [47, 178481] 2^29 - 1 = 536870911 => [233, 1103, 2089] 2^31 - 1 = 2147483647 => [2147483647] 2^37 - 1 = 137438953471 => [223, 616318177] 2^41 - 1 = 2199023255551 => [13367, 164511353] 2^43 - 1 = 8796093022207 => [431, 9719, 2099863] 2^47 - 1 = 140737488355327 => [2351, 4513, 13264529] 2^53 - 1 = 9007199254740991 => [6361, 69431, 20394401] 2^59 - 1 = 576460752303423487 => [179951, 3203431780337]
Tcl
<lang tcl>proc factors {x} {
# list the prime factors of x in ascending order set result [list] while {$x % 2 == 0} { lappend result 2 set x [expr {$x / 2}] } for {set i 3} {$i*$i <= $x} {incr i 2} { while {$x % $i == 0} { lappend result $i set x [expr {$x / $i}] } } if {$x != 1} {lappend result $x} return $result
} </lang> Testing <lang tcl>foreach m {2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59} {
set n [expr {2**$m - 1}] catch {time {set primes [factors $n]} 1} tm puts [format "2**%02d-1 = %-18s = %-22s => %s" $m $n [join $primes *] $tm]
}</lang>
- Output:
2**02-1 = 3 = 3 => 184 microseconds per iteration 2**03-1 = 7 = 7 => 8 microseconds per iteration 2**05-1 = 31 = 31 => 8 microseconds per iteration 2**07-1 = 127 = 127 => 23 microseconds per iteration 2**11-1 = 2047 = 23*89 => 12 microseconds per iteration 2**13-1 = 8191 = 8191 => 22 microseconds per iteration 2**17-1 = 131071 = 131071 => 69 microseconds per iteration 2**19-1 = 524287 = 524287 => 131 microseconds per iteration 2**23-1 = 8388607 = 47*178481 => 81 microseconds per iteration 2**29-1 = 536870911 = 233*1103*2089 => 199 microseconds per iteration 2**31-1 = 2147483647 = 2147483647 => 9509 microseconds per iteration 2**37-1 = 137438953471 = 223*616318177 => 4377 microseconds per iteration 2**41-1 = 2199023255551 = 13367*164511353 => 2389 microseconds per iteration 2**43-1 = 8796093022207 = 431*9719*2099863 => 1711 microseconds per iteration 2**47-1 = 140737488355327 = 2351*4513*13264529 => 802 microseconds per iteration 2**53-1 = 9007199254740991 = 6361*69431*20394401 => 13109 microseconds per iteration 2**59-1 = 576460752303423487 = 179951*3203431780337 => 316009 microseconds per iteration
TI-83 BASIC
<lang ti83b>::prgmPREMIER Disp "FACTEURS PREMIER" Prompt N If N<1:Stop ClrList L1�,L2 0→K iPart(√(N))→L N→M For(I,2,L) 0→J While fPart(M/I)=0 J+1→J M/I→M End If J≠0 Then K+1→K I→L�1(K) J→L2(K) I→Z:prgmVSTR " "+Str0→Str1 If J≠1 Then J→Z:prgmVSTR Str1+"^"+Str0→Str1 End Disp Str1 End If M=1:Stop End If M≠1 Then If M≠N Then M→Z:prgmVSTR " "+Str0→Str1 Disp Str1 Else Disp "PREMIER" End End
- prgmVSTR
{Z,Z}→L5 {1,2}→L6 LinReg(ax+b)L6,L5,Y�₀ Equ►String(Y₀,Str0) length(Str0)→O sub(Str0,4,O-3)→Str0 ClrList L5,L6 DelVar Y�</lang>
- Output:
FACTEURS PREMIER N=?1047552 2^10 3 11 31
Tiny BASIC
<lang Tiny BASIC>10 PRINT "Enter a number." 20 INPUT N 25 PRINT "------" 30 IF N<0 THEN LET N = -N 40 IF N<2 THEN END 50 LET I = 2 60 IF I*I > N THEN GOTO 200 70 IF (N/I)*I = N THEN GOTO 300 80 LET I = I + 1 90 GOTO 60 200 PRINT N 210 END 300 LET N = N / I 310 PRINT I 320 GOTO 50</lang>
- Output:
Enter a number. 32 ------ 2 2 2 2 2 Enter a number. 2520 ------ 2 2 2 3 3 5 7 Enter a number. 13 ------ 13
TXR
<lang txr>@(next :args) @(do
(defun factor (n) (if (> n 1) (for ((max-d (isqrt n)) (d 2)) () ((inc d (if (evenp d) 1 2))) (cond ((> d max-d) (return (list n))) ((zerop (mod n d)) (return (cons d (factor (trunc n d))))))))))
@{num /[0-9]+/} @(bind factors @(factor (int-str num 10))) @(output) @num -> {@(rep)@factors, @(last)@factors@(end)} @(end)</lang>
- Output:
$ txr factor.txr 1139423842450982345 1139423842450982345 -> {5, 19, 37, 12782467, 25359769} $ txr factor.txr 1 1 -> {} $ txr factor.txr 2 2 -> {2} $ txr factor.txr 3 3 -> {3} $ txr factor.txr 2 2 -> {2} $ txr factor.txr 3 3 -> {3} $ txr factor.txr 4 4 -> {2, 2} $ txr factor.txr 5 5 -> {5} $ txr factor.txr 6 6 -> {2, 3}
V
like in scheme (using variables) <lang v>[prime-decomposition
[inner [c p] let [c c * p >] [p unit] [ [p c % zero?] [c c p c / inner cons] [c 1 + p inner] ifte] ifte]. 2 swap inner].</lang>
(mostly) the same thing using stack (with out variables) <lang v>[prime-decomposition
[inner [dup * <] [pop unit] [ [% zero?] [ [p c : [c p c / c]] view i inner cons] [succ inner] ifte] ifte]. 2 inner].</lang>
Using it <lang v>|1221 prime-decomposition puts</lang>
=[3 11 37]
VBScript
<lang vb>Function PrimeFactors(n)
arrP = Split(ListPrimes(n)," ") divnum = n Do Until divnum = 1 'The -1 is to account for the null element of arrP For i = 0 To UBound(arrP)-1 If divnum = 1 Then Exit For ElseIf divnum Mod arrP(i) = 0 Then divnum = divnum/arrP(i) PrimeFactors = PrimeFactors & arrP(i) & " " End If Next Loop
End Function
Function IsPrime(n)
If n = 2 Then IsPrime = True ElseIf n <= 1 Or n Mod 2 = 0 Then IsPrime = False Else IsPrime = True For i = 3 To Int(Sqr(n)) Step 2 If n Mod i = 0 Then IsPrime = False Exit For End If Next End If
End Function
Function ListPrimes(n)
ListPrimes = "" For i = 1 To n If IsPrime(i) Then ListPrimes = ListPrimes & i & " " End If Next
End Function
WScript.StdOut.Write PrimeFactors(CInt(WScript.Arguments(0))) WScript.StdOut.WriteLine</lang>
- Output:
C:\>cscript /nologo primefactors.vbs 12 2 3 2 C:\>cscript /nologo primefactors.vbs 50 2 5 5
Wren
The examples are borrowed from the Go solution. <lang ecmascript>import "/big" for BigInt import "/fmt" for Fmt
var vals = [1 << 31, 1234567, 333333, 987653, 2 * 3 * 5 * 7 * 11 * 13 * 17] for (val in vals) {
Fmt.print("$10d -> $n", val, BigInt.primeFactors(val))
}</lang>
- Output:
2147483648 -> [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 1234567 -> [127, 9721] 333333 -> [3, 3, 7, 11, 13, 37] 987653 -> [29, 34057] 510510 -> [2, 3, 5, 7, 11, 13, 17]
XSLT
Let's assume that in XSLT the application of a template is similar to the invocation of a function. So when the following template <lang xml><xsl:stylesheet xmlns:xsl="http://www.w3.org/1999/XSL/Transform" version="1.0">
<xsl:template match="/numbers"> <html> <body>
-
<xsl:apply-templates />
</body> </html> </xsl:template>
<xsl:template match="number">
</xsl:template> <xsl:template match="value" mode="value"> <xsl:apply-templates /> </xsl:template> <xsl:template match="value" mode="factors"> <xsl:call-template name="generate"> <xsl:with-param name="number" select="number(current())" /> <xsl:with-param name="candidate" select="number(2)" /> </xsl:call-template> </xsl:template> <xsl:template name="generate"> <xsl:param name="number" /> <xsl:param name="candidate" /> <xsl:choose> <xsl:when test="$number = 1"></xsl:when> <xsl:when test="$candidate * $candidate > $number"> <xsl:value-of select="$number" /> </xsl:when> <xsl:when test="$number mod $candidate = 0"> <xsl:value-of select="$candidate" /> <xsl:text> </xsl:text> <xsl:call-template name="generate"> <xsl:with-param name="number" select="$number div $candidate" /> <xsl:with-param name="candidate" select="$candidate" /> </xsl:call-template> </xsl:when> <xsl:otherwise> <xsl:choose> <xsl:when test="$candidate = 2"> <xsl:call-template name="generate"> <xsl:with-param name="number" select="$number" /> <xsl:with-param name="candidate" select="$candidate + 1" /> </xsl:call-template> </xsl:when> <xsl:otherwise> <xsl:call-template name="generate"> <xsl:with-param name="number" select="$number" /> <xsl:with-param name="candidate" select="$candidate + 2" /> </xsl:call-template> </xsl:otherwise> </xsl:choose> </xsl:otherwise> </xsl:choose> </xsl:template> </xsl:stylesheet></lang> is applied against the document <lang xml><numbers> <number><value>1</value></number> <number><value>2</value></number> <number><value>4</value></number> <number><value>8</value></number> <number><value>9</value></number> <number><value>255</value></number> </numbers></lang> then the output contains the prime decomposition of each number: <lang html><html> <body>
- Number: 1 Factors:
- Number: 2 Factors: 2
- Number: 4 Factors: 2 2
- Number: 8 Factors: 2 2 2
- Number: 9 Factors: 3 3
- Number: 255 Factors: 3 5 17
</body> </html></lang>
zkl
With 64 bit ints: <lang zkl>fcn primeFactors(n){ // Return a list of factors of n
acc:=fcn(n,k,acc,maxD){ // k is 2,3,5,7,9,... not optimum if(n==1 or k>maxD) acc.close(); else{ q,r:=n.divr(k); // divr-->(quotient,remainder) if(r==0) return(self.fcn(q,k,acc.write(k),q.toFloat().sqrt())); return(self.fcn(n,k+1+k.isOdd,acc,maxD)) } }(n,2,Sink(List),n.toFloat().sqrt()); m:=acc.reduce('*,1); // mulitply factors if(n!=m) acc.append(n/m); // opps, missed last factor else acc;
}</lang> <lang zkl>foreach n in (T(5,12, 2147483648, 2199023255551, 8796093022207,
9007199254740991, 576460752303423487)){ println(n,": ",primeFactors(n).concat(", "))
}</lang>
- Output:
5: 5 12: 2, 2, 3 2147483648: 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 2199023255551: 13367, 164511353 8796093022207: 431, 9719, 2099863 9007199254740991: 6361, 69431, 20394401 576460752303423487: 179951, 3203431780337
Unfortunately, big ints (GMP) don't have (quite) the same interface as ints (since there is no big float, BI.toFloat() truncates to a double so BI.toFloat().sqrt() is wrong). So mostly duplicate code is needed: <lang zkl>fcn factorsBI(n){ // Return a list of factors of n
acc:=fcn(n,k,acc,maxD){ // k is 2,3,5,7,9,... not optimum if(n==1 or k>maxD) acc.close(); else{ q,r:=n.div2(k); // divr-->(quotient,remainder) if(r==0) return(self.fcn(q,k,acc.write(k),q.root(2))); return(self.fcn(n,k+1+k.isOdd,acc,maxD)) } }(n,2,Sink(List),n.root(2)); m:=acc.reduce('*,BN(1)); // mulitply factors if(n!=m) acc.append(n/m); // opps, missed last factor else acc;
}</lang> <lang zkl>var BN=Import("zklBigNum"); foreach n in (T(BN("12"),
BN("340282366920938463463374607431768211455"))){ println(n,": ",factorsBI(n).concat(", "))
}</lang>
- Output:
12: 2, 2, 3 340282366920938463463374607431768211455: 3, 5, 17, 257, 641, 65537, 274177, 6700417, 67280421310721
- Programming Tasks
- Prime Numbers
- Arbitrary precision
- GUISS/Omit
- 11l
- 360 Assembly
- AArch64 Assembly
- ABAP
- ACL2
- Ada
- ALGOL 68
- ALGOL-M
- Applesoft BASIC
- Arturo
- AutoHotkey
- AWK
- Batch file
- Befunge
- BQN
- Burlesque
- C
- C sharp
- C++
- GMP
- Clojure
- Commodore BASIC
- Common Lisp
- D
- Delphi
- System.SysUtils
- E
- EchoLisp
- Eiffel
- Ela
- Elixir
- Erlang
- ERRE
- Ezhil
- F Sharp
- Factor
- FALSE
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