Prime decomposition
From Rosetta Code
You are encouraged to solve this task according to the task description, using any language you may know.
Write a function which returns an array or collection which contains the prime decomposition of a given number, n, 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).
[edit] Ada
with Ada.Text_IO; use Ada.Text_IO;
procedure Test_Prime is
generic
type Number is private;
Zero : Number;
One : Number;
Two : Number;
with function Image (X : Number) return String 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;
procedure Put (List : Number_List);
end Prime_Numbers;
package body Prime_Numbers is
function Decompose (N : Number) return Number_List is
Size : Natural := 0;
M : Number := N;
K : Number := Two;
begin
-- Estimation of the result length from above
while M >= Two loop
M := (M + One) / Two;
Size := Size + 1;
end loop;
M := N;
-- Filling the result with prime numbers
declare
Result : Number_List (1..Size);
Index : Positive := 1;
begin
while N >= K loop -- Divisors loop
while Zero = (M mod K) loop -- While divides
Result (Index) := K;
Index := Index + 1;
M := M / K;
end loop;
K := K + One;
end loop;
return Result (1..Index - 1);
end;
end Decompose;
procedure Put (List : Number_List) is
begin
for Index in List'Range loop
Put (Image (List (Index)));
end loop;
end Put;
end Prime_Numbers;
package Integer_Numbers is new Prime_Numbers (Natural, 0, 1, 2, Positive'Image);
use Integer_Numbers;
begin
Put (Decompose (12));
end Test_Prime;
The solution is generic. The package 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 package also provides a procedure to output an array of prime numbers and a function to convert a number to string (as a parameter). 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. In the example provided, the package is instantiated with plain integer type. Sample output:
2 2 3
[edit] ALGOL 68
Translation of: Python - note: This specimen retains the original Python coding style.
Works with: ALGOL 68 version Revision 1 - no extensions to language used
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny
Works with: ELLA ALGOL 68 version Any (with appropriate job cards) - tested with release 1.8-8d
#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
)
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.
[edit] 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)
factors = %f%`n%next%
Return factors
}
f++
}
}
[edit] AWK
As the examples show, pretty large numbers can be factored in tolerable time:
$ awk 'func pfac(n){r="";f=2;while(f<=n){while(!(n%f)){n=n/f;r=r" "f};f=f+2-(f==2)};return r}{print pfac($1)}'
36
2 2 3 3
77
7 11
536870911
233 1103 2089
8796093022207
431 9719 2099863
[edit] C
Library: GMP primedecompose.h
#ifndef _PRIMEDECOMPOSE_H_
#define _PRIMEDECOMPOSE_H_
#include <gmp.h>
int decompose(mpz_t n, mpz_t *o);
#endif
primedecompose.c
#include "primedecompose.h"
int decompose(mpz_t n, mpz_t *o) {
int i;
mpz_t tmp, d;
i = 0;
mpz_init(tmp);
mpz_init(d);
while(mpz_cmp_si(n, 1)) {
mpz_set_ui(d, 1);
do {
mpz_add_ui(tmp, d, 1);
mpz_swap(tmp, d);
} while(!mpz_divisible_p(n, d));
mpz_divexact(tmp, n, d);
mpz_swap(tmp, n);
mpz_init(o[i]);
mpz_set(o[i], d);
i++;
}
return i;
}
Testing
#include <stdio.h>
#include <stdlib.h>
#include <gmp.h>
#include "primedecompose.h"
mpz_t dest[100]; /* must be big enough to hold all the factors! */
int main(int argc, char **argv) {
mpz_t n;
int i, l;
if(argc != 2) {
puts("Pass a parameter");
return EXIT_SUCCESS;
}
mpz_init_set_str(n, argv[1], 10);
l = decompose(n, dest);
for(i=0; i < l; i++) {
gmp_printf("%s%Zd", i?" * ":"", dest[i]);
mpz_clear(dest[i]);
}
printf("\n");
return EXIT_SUCCESS;
}
Using GNU Compiler Collection gcc extensions
Translation of: ALGOL 68
Works with: gcc version 4.3.0 20080428 (Red Hat 4.3.0-8)
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.
#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 */
#ifdef NEED_GOTO
#include <setjmp.h>
/* declare label otherwise it is not visible in sub-scope */
#define LABEL(label) jmp_buf label; if(setjmp(label))goto label;
#define GOTO(label) longjmp(label, true)
#endif
/* 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;
}
}
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.
[edit] C++
Works with: g++ version 4.1.2 20061115 (prerelease) (Debian 4.1.1-21) Library: GMP
#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";
}
}
[edit] C#
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;
}
}
}
[edit] Clojure
(use '[clojure.contrib.lazy-seqs :only [primes])
(defn prime-factors [arg]
(assert (and (integer? arg) (>= arg 2)))
(loop [pfs [], n arg] ; pfs is the vector of prime factors already determined
(if (= n 1)
pfs
(let [dps (for [p primes :while (<= (* p p) n) :when (zero? (rem n p))] p)
ps (for [p dps, q (rest (iterate #(/ % p) n)) :while (integer? q)] p)]
(if (empty? dps)
(recur (conj pfs n), 1)
(recur (into pfs ps), (apply / n ps)))))))
[edit] Common 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)))))))))
[edit] 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.
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
}
[edit] Erlang
factors(N) ->
factors(N,2).
factors(1,_) -> [];
factors(N,K) when N rem K == 0 ->
[K|factors(N div K,K)];
factors(N,K) ->
factors(N,K+1).
[edit] Factor
Factor already offers this functionality in its standard library. Example use:
USING: math.primes.factors ;
12 factors .
{ 2 2 3 }
576460752303423487 factors .
{ 179951 3203431780337 }
[edit] FALSE
[2[\$@$$*@>~][\$@$@$@$@\/*=$[%$." "$@\/\0~]?~[1+1|]?]#%.]d:
27720d;! {2 2 2 3 3 5 7 11}
[edit] 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 . ;
[edit] Fortran
Works with: Fortran version 90 and later
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
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
[edit] Haskell
primes = sieve [2..]
where
sieve (p:xs) = p : sieve [x|x <- xs, x `mod` p > 0]
factorize n pps@(p:ps) = case n `divMod` p of
(0,1) -> []
(remainder,0) -> p : factorize remainder pps
_ -> factorize n ps
[edit] Icon and Unicon
[edit] 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
Using the Library: Icon Programming Library genfactors and prime
Sample Output showing factors of a large integer:
8796093022207=431*9719*2099863
[edit] Unicon
The Icon solution works in Unicon.
[edit] J
q:
Example use:
q: 3684
2 2 3 307
_1+2^128x
340282366920938463463374607431768211455
q: _1+2^128x
3 5 17 257 641 65537 274177 6700417 67280421310721
*/ q: _1+2^128x
340282366920938463463374607431768211455
[edit] Java
Works with: Java version 1.5+ This is a version for arbitrary-precision integers which assumes the existence of a function with the signature:
public boolean prime(BigInteger i);
You will need to import java.util.List, java.util.LinkedList, and java.math.BigInteger.
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;
}
Alternate version, optimised to be faster. Uses java.util.List, java.util.LinkedList and java.math.BigInteger.
public boolean isPrime(BigInteger i);
private static final BigInteger two = BigInteger.valueOf(2);
public List<BigInteger> primeDecomp(BigInteger a) {
//Is it even possible
if(a.compareTo(two)<0) {
return null; //impossible for values lower than 2
}
//quickly handle even values
List<BigInteger> result = new LinkedList<BigInteger>();
while(a.and(BigInteger.ONE).equals(BigInteger.ONE)) {
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(isPrime(b)) {
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;
}
[edit] JavaScript
This code uses the BigInteger Library jsbn and jsbn2
http://xenon.stanford.edu/~tjw/jsbn/jsbn.js
http://xenon.stanford.edu/~tjw/jsbn/jsbn2.js
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].compareTo(BigInteger.ZERO) == 0) {
if (prod)
output.value += "*";
else
prod = true;
output.value += "2";
n = qr[0];
}
else
break;
}
while (n.compareTo(BigInteger.ONE) != 0) {
var qr = n.divideAndRemainder(divisor);
if (qr[1].compareTo(BigInteger.ZERO) == 0) {
if (prod)
output.value += "*";
else
prod = true;
output.value += divisor;
n = qr[0];
}
else
divisor = divisor.add(TWO);
}
}
[edit] 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
[edit] Mathematica
Bare built-in function does:
FactorInteger[2016] => {{2, 5}, {3, 2}, {7, 1}}
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:
supscript[x_,y_]:=If[y==1,x,Superscript[x,y]]
ShowPrimeDecomposition[input_Integer]:=Print@@{input," = ",Sequence@@Riffle[supscript@@@FactorInteger[input]," "]}
Example for small prime:
ShowPrimeDecomposition[1337]
gives:
1337 = 7 191
Examples for large primes:
Table[AbsoluteTiming[ShowPrimeDecomposition[2^a-1]]//Print[#[[1]]," sec"]&,{a,50,150,10}];
gives back:
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
[edit] MATLAB
function [outputPrimeDecomposition] = primedecomposition(inputValue)
outputPrimeDecomposition = factor(inputValue);
[edit] Maxima
prime_decomposition(x) := map(first, ifactors(x))
The builtin function factor(integer) also returns the prime decomposition of an integer, but it returns it as a product expression rather than a collection.
(%i62) prime_decomposition(10);
(%o62) [2,5]
[edit] MUMPS
ERATO1(HI)Usage:
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
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
[edit] 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;;
[edit] Octave
r = factor(120202039393);
disp(r);
[edit] PicoLisp
(de factor (N)
(let (D 2 L (1 2 2) M (sqrt N) Res)
(while (>= M D)
(cond
((=0 (% N D))
(push 'Res D)
(setq M (sqrt (setq N (/ N D)))) )
(L (inc 'D (pop 'L)))
(T (inc 'D 4)
(setq L (2 4 2 4 6 2 6)) ) ) )
(if (= 1 N) Res (cons N Res)) ) )
(factor 1361129467683753853853498429727072845823)
Output:
-> (145295143558111 7623851 409891 8191 2731 131 31 11 3)
[edit] PureBasic
Works with: PureBasic version 4.41
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)
Factored 9007199254740991 in 0.27 seconds. 6361 69431 20394401
[edit] Python
Works with: Python version 2.5.1
import sys, time
def is_prime(n):
return 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 primes():
yield 2
n = 3
while n < sys.maxint - 2:
yield n
n += 2
while n < sys.maxint - 2 and not is_prime_cached[n]:
n += 2
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
# Example: calculate factors of Mersenne numbers to M59 #
for m in primes():
p = 2 ** m - 1
print "2**%d-1 = %d, with factors:"%(m, p),
start = time.time()
for factor in decompose(p):
print factor,
sys.stdout.flush()
print "=> %.2fs"%(time.time()-start)
if m >= 59: break
Output:
2**2-1 = 3, with factors: 3 => 0.00s 2**3-1 = 7, with factors: 7 => 0.00s 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 => 0.01s 2**23-1 = 8388607, with factors: 47 178481 => 0.00s 2**29-1 = 536870911, with factors: 233 1103 2089 => 0.01s 2**31-1 = 2147483647, with factors: 2147483647 => 1.67s 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 => 0.00s 2**53-1 = 9007199254740991, with factors: 6361 69431 20394401 => 1.17s 2**59-1 = 576460752303423487, with factors: 179951 3203431780337 => 211.07s
Note: Python took 740,238 BogoMI to complete the example.
Modifying the primes() and is_prime() functions as below increases performance.
def is_prime(n):
return all( y==n for y in decompose(n))
primelist = [2,3]
max_tested = 3
def primes():
global max_tested
for n in primelist:
yield n
n = max_tested
pmax = sys.maxint-2
while n < pmax:
n += 2
while not is_prime(n) and n < pmax:
n += 2
if n < pmax:
primelist.append(n)
max_tested = n
yield n
[edit] R
findfactors <- function(n) {
d <- c()
div <- 2; nxt <- 3; rest <- n
while( rest != 1 ) {
while( rest%%div == 0 ) {
d <- c(d, div)
rest <- floor(rest / div)
}
div <- nxt
nxt <- nxt + 2
}
d
}
print(findfactors(1005025))
[edit] 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
(2..31).each do |i|
puts "prime_factors(#{2**i-1}): #{prime_factors(2**i-1).join(' ')}"
end
A more efficient version, and quite similar to the Integer#prime_division method added by the Library: mathn.rbpackage in the Ruby stdlib:
require 'mathn'
def prime_factors(n)
factors = []
prime_number_generator = Prime.new
p = prime_number_generator.next
while p <= n
q, r = n.divmod(p)
if r == 0
factors << p
n = q
elsif p**2 >= n
break
else
p = prime_number_generator.next
end
end
factors << n if n > 1
factors
end
# example: decompose all possible Mersenne primes up to 2**31-1
results = []
png = Prime.new
require 'benchmark'
Benchmark.bm(7) do |x|
begin
i = png.next
n = 2**i- 1
f = prime_factors(n)
results << "%2d : %-20d : %s\n" % [i, n, f.inspect]
x.report("new-#{i}") {prime_factors(n)}
x.report("ruby-#{i}") {n.prime_division}
end while i < 53
end
puts results
user system total real new-2 0.000000 0.000000 0.000000 ( 0.000000) ruby-2 0.000000 0.000000 0.000000 ( 0.000000) new-3 0.000000 0.000000 0.000000 ( 0.000000) ruby-3 0.000000 0.000000 0.000000 ( 0.000000) new-5 0.000000 0.000000 0.000000 ( 0.000000) ruby-5 0.000000 0.000000 0.000000 ( 0.000000) new-7 0.000000 0.000000 0.000000 ( 0.000000) ruby-7 0.000000 0.000000 0.000000 ( 0.000000) new-11 0.000000 0.000000 0.000000 ( 0.000000) ruby-11 0.000000 0.000000 0.000000 ( 0.000000) new-13 0.000000 0.000000 0.000000 ( 0.000000) ruby-13 0.000000 0.000000 0.000000 ( 0.002000) new-17 0.000000 0.000000 0.000000 ( 0.005000) ruby-17 0.015000 0.000000 0.015000 ( 0.007000) new-19 0.016000 0.000000 0.016000 ( 0.013000) ruby-19 0.015000 0.000000 0.015000 ( 0.013000) new-23 0.000000 0.000000 0.000000 ( 0.006000) ruby-23 0.000000 0.000000 0.000000 ( 0.006000) new-29 0.047000 0.000000 0.047000 ( 0.024000) ruby-29 0.031000 0.000000 0.031000 ( 0.025000) new-31 13.016000 0.000000 13.016000 ( 13.390000) ruby-31 13.437000 0.000000 13.437000 ( 13.578000) new-37 5.031000 0.000000 5.031000 ( 4.639000) ruby-37 4.750000 0.000000 4.750000 ( 4.648000) new-41 1.594000 0.000000 1.594000 ( 1.668000) ruby-41 1.546000 0.000000 1.546000 ( 1.568000) new-43 0.844000 0.000000 0.844000 ( 0.879000) ruby-43 0.906000 0.000000 0.906000 ( 0.914000) new-47 0.265000 0.000000 0.265000 ( 0.256000) ruby-47 0.266000 0.000000 0.266000 ( 0.240000) new-53 27.938000 0.000000 27.938000 ( 28.369000) ruby-53 28.562000 0.000000 28.562000 ( 28.227000) 2 : 3 : [3] 3 : 7 : [7] 5 : 31 : [31] 7 : 127 : [127] 11 : 2047 : [23, 89] 13 : 8191 : [8191] 17 : 131071 : [131071] 19 : 524287 : [524287] 23 : 8388607 : [47, 178481] 29 : 536870911 : [233, 1103, 2089] 31 : 2147483647 : [2147483647] 37 : 137438953471 : [223, 616318177] 41 : 2199023255551 : [13367, 164511353] 43 : 8796093022207 : [431, 9719, 2099863] 47 : 140737488355327 : [2351, 4513, 13264529] 53 : 9007199254740991 : [6361, 69431, 20394401]
[edit] Scala
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:
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
}
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:
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
}
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
[edit] 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)
Output:
(3 7 11 13 37 101 9901)
[edit] 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;
Original source: [1]
[edit] Slate
Admittedly, this is just based on the Smalltalk entry below:
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."
].
[edit] 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 ]
[edit] Tcl
namespace eval primes {}
proc primes::reset {} {
variable list [list]
variable current_index end
}
namespace eval primes {reset}
proc primes::restart {} {
variable list
variable current_index
if {[llength $list] > 0} {
set current_index 0
}
}
proc primes::is_prime {candidate} {
variable list
if {$candidate in $list} {return true}
foreach prime $list {
if {$candidate % $prime == 0} {
return false
}
if {$prime * $prime > $candidate} {
return true
}
}
while true {
set largest [get_next_prime]
if {$largest * $largest >= $candidate} {
return [is_prime $candidate]
}
}
}
proc primes::get_next_prime {} {
variable list
variable current_index
if {$current_index ne "end"} {
set p [lindex $list $current_index]
if {[incr current_index] == [llength $list]} {
set current_index end
}
return $p
}
switch -exact -- [llength $list] {
0 {set candidate 2}
1 {set candidate 3}
default {
set candidate [lindex $list end]
while true {
incr candidate 2
if {[is_prime $candidate]} break
}
}
}
lappend list $candidate
return $candidate
}
# return the prime factors of a number in a dictionary.
# The keys will be the factors, the value will be the number
# of times the factor divides the given number
#
# example: 120 = 2**3 * 3 * 5, so
# [primes::factors 120] returns 2 3 3 1 5 1
# so: set prod 1
# dict for {p e} [primes::factors 120] {
# set prod [expr {$prod * $p**$e}]
# }
# expr {$prod == 120} ;# ==> true
#
proc primes::factors {num} {
restart
set factors [dict create]
for {set i [get_next_prime]} {$i <= $num} {} {
if {$num % $i == 0} {
dict incr factors $i
set num [expr {$num / $i}]
continue
} elseif {$i*$i > $num} {
dict incr factors $num
break
} else {
set i [get_next_prime]
}
}
return $factors
}
Testing
primes::reset
foreach m {2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59} {
set n [expr {2**$m - 1}]
catch {time {set f [dict create {*}[primes::factors $n]]} 1} tm
set primes [list]
dict for {p e} $f {lappend primes {*}[lrepeat $e $p]}
puts [format "2**%02d-1 = %-18s = %-22s => %s" $m $n [join $primes *] $tm]
}
Outputs
2**02-1 = 3 = 3 => 20 microseconds per iteration 2**03-1 = 7 = 7 => 16 microseconds per iteration 2**05-1 = 31 = 31 => 27 microseconds per iteration 2**07-1 = 127 = 127 => 33 microseconds per iteration 2**11-1 = 2047 = 23*89 => 43 microseconds per iteration 2**13-1 = 8191 = 8191 => 159 microseconds per iteration 2**17-1 = 131071 = 131071 => 535 microseconds per iteration 2**19-1 = 524287 = 524287 => 911 microseconds per iteration 2**23-1 = 8388607 = 47*178481 => 162 microseconds per iteration 2**29-1 = 536870911 = 233*1103*2089 => 982 microseconds per iteration 2**31-1 = 2147483647 = 2147483647 => 138831 microseconds per iteration 2**37-1 = 137438953471 = 223*616318177 => 5154 microseconds per iteration 2**41-1 = 2199023255551 = 13367*164511353 => 2901 microseconds per iteration 2**43-1 = 8796093022207 = 431*9719*2099863 => 2141 microseconds per iteration 2**47-1 = 140737488355327 = 2351*4513*13264529 => 1102 microseconds per iteration 2**53-1 = 9007199254740991 = 6361*69431*20394401 => 97472 microseconds per iteration 2**59-1 = 576460752303423487 = 179951*3203431780337 => 12664437 microseconds per iteration
[edit] V
like in scheme (using variables)
[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].
(mostly) the same thing using stack (with out variables)
[prime-decomposition
[inner
[dup * <]
[pop unit]
[ [% zero?]
[ [p c : [c p c / c]] view i inner cons]
[succ inner]
ifte]
ifte].
2 inner].
Using it
|1221 prime-decomposition puts
=[3 11 37]
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