# Largest prime factor

Largest prime factor is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

The task description is taken for Project Euler (https://projecteuler.net/problem=3)
What is the largest prime factor of the number 600851475143 ?

## ALGOL 68

Based on the Wren and Go samples.

```BEGIN # find the largest prime factor of a number #
# returns the largest prime factor of n #
PROC max prime factor = ( LONG INT n )LONG INT:
IF   n < 2
THEN 1
ELSE
LONG INT max factor := n;
# even factors - only 2 is prime #
LONG INT v := n;
WHILE NOT ODD v DO
max factor := 2;
v OVERAB 2
OD;
# odd factors #
LONG INT k := 3;
LONG INT root n = ENTIER long sqrt( n );
WHILE k <= root n AND v > 1 DO
WHILE v MOD k = 0 AND v > 1 DO
max factor := k;
v OVERAB k
OD;
k +:= 2
OD;
IF v > 1 THEN v ELSE max factor FI
FI # max prime factor # ;
# test the max prime factor routine #
PROC test = ( LONG INT n )VOID:
print( ( "Largest prime factor of ", whole( n, 0 ), " is ", whole( max prime factor( n ), 0 ), newline ) );
# test cases #
test( 600 851 475 143 );
test(           6 008 );
test(             751 )
END```
Output:
```Largest prime factor of 600851475143 is 6857
Largest prime factor of 6008 is 751
Largest prime factor of 751 is 751
```

## AutoHotkey

```MsgBox % result := max(prime_numbers(600851475143)*)

prime_numbers(n) {
if (n <= 3)
return [n]
ans := [], done := false
while !done {
if !Mod(n,2)
ans.push(2), n /= 2
else if !Mod(n,3)
ans.push(3), n /= 3
else if (n = 1)
return ans
else {
sr := sqrt(n), done := true, 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
if !Mod(n, i+1)        ; is n divisible by i+1?
ans.push(i+1), n /= i+1, done := false
if !done
break
i+=6
}}}
ans.push(Format("{:d}", n))
return ans
}
```
Output:
`6857`

## AWK

```# syntax: GAWK -f LARGEST_PRIME_FACTOR.AWK
# converted from FreeBASIC
BEGIN {
N = n = "600851475143"
j = 3
while (!is_prime(n)) {
if (n % j == 0) {
n /= j
}
j += 2
}
printf("The largest prime factor of %s is %d\n",N,n)
exit(0)
}
function is_prime(x,  i) {
if (x <= 1) {
return(0)
}
for (i=2; i<=int(sqrt(x)); i++) {
if (x % i == 0) {
return(0)
}
}
return(1)
}
```
Output:
```The largest prime factor of 600851475143 is 6857
```

## BASIC

### FreeBASIC

```#include"isprime.bas"
dim as ulongint n = 600851475143, j = 3
while not isprime(n)
if n mod j = 0 then n/=j
j+=2
wend
print n
```
Output:
`6857`

### GW-BASIC

No primality testing is even required.

```10 N#=600851475143#
20 J#=3
30 IF J#=N# THEN GOTO 100
40 IF INT(N#/J#) = N#/J# THEN N# = N#/J#
50 J#=J#+2
60 GOTO 30
100 PRINT N#
```
Output:
`6857`

## BCPL

This version creates a 2,3,5 wheel object, which is instantiated by the factorization routine.

```GET "libhdr"

LET new_235wheel() = VALOF {
LET w = getvec(1)
w!0 := 1   // accumulator
w!1 := -3  // index (negative => first few primes)
RESULTIS w
}

LET next235(w) = VALOF {
LET p3 = TABLE 2, 3, 5
LET wheel235 = TABLE 6, 4, 2, 4, 2, 4, 6, 2
LET a, i = w!0, w!1

TEST i < 0
THEN {
a := p3[i + 3]
i +:= 1
}
ELSE {
a +:= wheel235[i]
i := (i + 1) & 7
w!0 := a
}
w!1 := i
RESULTIS a
}

LET gpf(n) = VALOF {
LET w = new_235wheel()
LET d = next235(w)

UNTIL d*d > n {
TEST n MOD d = 0
THEN n /:= d
ELSE d := next235(w)
}

freevec(w)
RESULTIS n
}

LET start() = VALOF {
writef("The largest prime factor of 600,851,475,143 is %d *n", gpf(600_851_475_143))
RESULTIS 0
}```
Output:
```BCPL 64-bit Cintcode System (13 Jan 2020)
0.000> The largest prime factor of 600,851,475,143 is 6857
0.001>
```

## C

```#include <stdio.h>
#include <stdlib.h>

int isprime( long int n ) {
int i=3;
if(!(n%2)) return 0;
while( i*i < n ) {
if(!(n%i)) return 0;
i+=2;
}
return 1;
}

int main(void) {
long int n=600851475143, j=3;

while(!isprime(n)) {
if(!(n%j)) n/=j;
j+=2;
}
printf( "%ld\n", n );
return 0;
}
```
Output:
`6857`

## CoffeeScript

```wheel235 = () ->
yield 2
yield 3
yield 5
a = 1
i = 0
wheel = [6, 4, 2, 4, 2, 4, 6, 2]
loop
a += wheel[i]
yield a
i = (i + 1) & 7

gpf = (n) ->
w = wheel235()
d = w.next().value
until d*d > n
if n % d is 0
n //= d
else
d = w.next().value
n

console.log "The largest prime factor of 600,851,475,143 is #{gpf(600_851_475_143)}"
```
Output:
```The largest prime factor of 600,851,475,143 is 6857
```

## Elixir

```defmodule Factor do
def wheel235(), do:
Stream.concat(
[2, 3, 5],
Stream.scan(Stream.cycle([6, 4, 2, 4, 2, 4, 6, 2]), 1, &+/2)
)

def gpf(n), do: gpf n, wheel235()
defp gpf(n, divs) do
[d] = Enum.take divs, 1
cond do
d*d > n -> n
rem(n, d) === 0 -> gpf div(n, d), divs
true -> gpf n, Stream.drop(divs, 1)
end
end
end

IO.puts "The largest prime factor of 600,851,475,143 is #{Factor.gpf(600_851_475_143)}"
```
Output:
```The largest prime factor of 600,851,475,143 is 6857
```

## Erlang

Uses a factorization wheel, but without builtin lazy lists, it's rather awkward for a functional language.

```main(_) ->
test(),
io:format("The largest prime factor of 600,851,475,143 is ~w~n", [gpf(600851475143)]).

gpf(N) -> gpf(N, 2, 0, <<1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6>>).
gpf(N, D, J, Wheel) when J =:= byte_size(Wheel) -> gpf(N, D, 3, Wheel);
gpf(N, D, _, _) when D*D > N -> N;
gpf(N, D, J, Wheel) when N rem D =:= 0 -> gpf(N div D, D, J, Wheel);
gpf(N, D, J, Wheel) -> gpf(N, D + binary:at(Wheel, J), J + 1, Wheel).

test() ->
3 = gpf(27),
5 = gpf(125),
7 = gpf(98),
101 = gpf(101),
23 = gpf(23 * 13).
```
Output:
```The largest prime factor of 600,851,475,143 is 6857
```

## F#

```printfn "%d" (Seq.last<|Open.Numeric.Primes.Prime.Factors 600851475143L)
```
Output:
```6857
```

## Factor

```USING: math.primes.factors prettyprint sequences ;

600851475143 factors last .
```

## Fermat

```n:=600851475143;
j:=3;
while Isprime(n)<>1 do
if Divides(j, n) then n:=n/j fi;
j:=j+2;
od;
!!n;```
Output:
`6857`

## Go

Translation of: Wren
```package main

import "fmt"

func largestPrimeFactor(n uint64) uint64 {
if n < 2 {
return 1
}
inc := uint64{4, 2, 4, 2, 4, 6, 2, 6}
max := uint64(1)
for n%2 == 0 {
max = 2
n /= 2
}
for n%3 == 0 {
max = 3
n /= 3
}
for n%5 == 0 {
max = 5
n /= 5
}
k := uint64(7)
i := 0
for k*k <= n {
if n%k == 0 {
max = k
n /= k
} else {
k += inc[i]
i = (i + 1) % 8
}
}
if n > 1 {
return n
}
return max
}

func main() {
n := uint64(600851475143)
fmt.Println("The largest prime factor of", n, "is", largestPrimeFactor(n), "\b.")
}
```
Output:
```The largest prime factor of 600851475143 is 6857.
```

## J

Translation of: jq
```   {:q:600851475143
6857
```

## jq

Works with: jq

Works with gojq, the Go implementation of jq

Using `factors` as defined at Prime_decomposition#jq:

`600851475143 | last(factors)`
Output:
```6857
```

## Julia

``` using Primes

@show first(last(factor(600851475143).pe))
```
Output:
`first(last((factor(600851475143)).pe)) = 6857`

## Mathematica / Wolfram Language

```Max[FactorInteger[[All, 1]]]
```
Output:
```
6857

```

## PARI/GP

```A=factor(600851475143)
print(A[matsize(A),1])```
Output:
`6857`

## Perl

```use strict;
use warnings;
use feature 'say';

sub f {
my(\$n) = @_;
\$n % \$_ or return \$_, f(\$n/\$_) for 2..\$n
}

say +(f 600851475143)[-2]
```
Output:
`6857`

## Phix

```with javascript_semantics
?prime_factors(600851475143,false,-1)[\$]
```
Output:
```6857
```

## PL/I

Based on the Wren, Go and Algol 68 samples.

```/* find the largest prime factor of 600851475143 */
largestPrimeFactor: procedure options( main );
declare ( n, maxFactor, v, k, rootN ) decimal( 12 )fixed;
n         = 600851475143;
maxFactor = n;
/* even factors */
v         = n;
do while( mod( v, 2 ) = 0 );
maxFactor = 2;
v = v / 2;
end;
/* odd factors */
k     = 3;
rootN = sqrt( n );
do while( k <= rootN & v > 1 );
do while( mod( v, k ) = 0 & v > 1 );
maxFactor = k;
v = v / k;
end;
k = k + 2;
end;
if v > 1 then maxFactor = v;
put skip list( 'Largest prime factor of ', n, ' is ', maxFactor );
end largestPrimeFactor;```
Output:
```Largest prime factor of     600851475143  is             6857
```

## 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].

gpf(N, P) :-  % greatest prime factor
wheel2357(W),
gpf(N, 2, W, P).

gpf(N, D, _, N) :- D*D > N, !.
gpf(N, D, W, X) :-
N mod D =:= 0, !,
N2 is N/D,
gpf(N2, D, W, X).
gpf(N, D, [S|Ss], X) :-
plus(D, S, D2),
gpf(N, D2, Ss, X).

main :-
gpf(600_851_475_143, Euler003),
format("The largest prime factor of 600,851,475,143 is ~p~n", [Euler003]),
halt.

?- main.
```
Output:
```The largest prime factor of 600,851,475,143 is 6857
```

## Python

```#!/usr/bin/python

def isPrime(n):
for i in range(2, int(n**0.5) + 1):
if n % i == 0:
return False
return True

if __name__ == '__main__':
n = 600851475143
j = 3
while not isPrime(n):
if n % j == 0:
n /= j
j += 2
print(n);
```

## R

First uses the Sieve of Eratosthenes to find possible factors then tests each possible prime p for divisibility and also n/p.

```sieve <- function(n) {
if (n < 2)
return (NULL)

primes <- rep(TRUE, n)
primes <- FALSE

for (i in 1:floor(sqrt(n)))
if (primes[i])
primes[seq(i*i, n, by = i)] <- FALSE

which(primes)
}

prime.factors <- function(n) {
primes <- sieve(floor(sqrt(n)))
factors <- primes[n %% primes == 0]
if (length(factors) == 0)
n
else {
for (p in factors) { # add all elements of n/p that are also prime
d <- n / p
if (d != p && all(d %% primes[primes <= floor(sqrt(d))] != 0))
factors <- c(factors, d)
}
factors
}
}

cat("The prime factors of 600,851,475,143 are", paste(prime.factors(600851475143), collapse = ", "), "\n")
```
Output:
```The prime factors of 600,851,475,143 are 71, 839, 1471, 6857
```

## Raku

Note: These are both extreme overkill for the task requirements.

### Not particularly fast

Pure Raku. Using Prime::Factor from the Raku ecosystem. Makes it to 2^95 - 1 in 1 second on my system.

```use Prime::Factor;

my \$start = now;

for flat 600851475143, (1..∞).map: { 2 +< \$_ - 1 } {
say "Largest prime factor of \$_: ", max prime-factors \$_;
last if now - \$start > 1; # quit after one second of total run time
}
```
```Largest prime factor of 600851475143: 6857
Largest prime factor of 3: 3
Largest prime factor of 7: 7
Largest prime factor of 15: 5
Largest prime factor of 31: 31
Largest prime factor of 63: 7
Largest prime factor of 127: 127
Largest prime factor of 255: 17
Largest prime factor of 511: 73
Largest prime factor of 1023: 31
Largest prime factor of 2047: 89
Largest prime factor of 4095: 13
Largest prime factor of 8191: 8191
Largest prime factor of 16383: 127
Largest prime factor of 32767: 151
Largest prime factor of 65535: 257
Largest prime factor of 131071: 131071
Largest prime factor of 262143: 73
Largest prime factor of 524287: 524287
Largest prime factor of 1048575: 41
Largest prime factor of 2097151: 337
Largest prime factor of 4194303: 683
Largest prime factor of 8388607: 178481
Largest prime factor of 16777215: 241
Largest prime factor of 33554431: 1801
Largest prime factor of 67108863: 8191
Largest prime factor of 134217727: 262657
Largest prime factor of 268435455: 127
Largest prime factor of 536870911: 2089
Largest prime factor of 1073741823: 331
Largest prime factor of 2147483647: 2147483647
Largest prime factor of 4294967295: 65537
Largest prime factor of 8589934591: 599479
Largest prime factor of 17179869183: 131071
Largest prime factor of 34359738367: 122921
Largest prime factor of 68719476735: 109
Largest prime factor of 137438953471: 616318177
Largest prime factor of 274877906943: 524287
Largest prime factor of 549755813887: 121369
Largest prime factor of 1099511627775: 61681
Largest prime factor of 2199023255551: 164511353
Largest prime factor of 4398046511103: 5419
Largest prime factor of 8796093022207: 2099863
Largest prime factor of 17592186044415: 2113
Largest prime factor of 35184372088831: 23311
Largest prime factor of 70368744177663: 2796203
Largest prime factor of 140737488355327: 13264529
Largest prime factor of 281474976710655: 673
Largest prime factor of 562949953421311: 4432676798593
Largest prime factor of 1125899906842623: 4051
Largest prime factor of 2251799813685247: 131071
Largest prime factor of 4503599627370495: 8191
Largest prime factor of 9007199254740991: 20394401
Largest prime factor of 18014398509481983: 262657
Largest prime factor of 36028797018963967: 201961
Largest prime factor of 72057594037927935: 15790321
Largest prime factor of 144115188075855871: 1212847
Largest prime factor of 288230376151711743: 3033169
Largest prime factor of 576460752303423487: 3203431780337
Largest prime factor of 1152921504606846975: 1321
Largest prime factor of 2305843009213693951: 2305843009213693951
Largest prime factor of 4611686018427387903: 2147483647
Largest prime factor of 9223372036854775807: 649657
Largest prime factor of 18446744073709551615: 6700417
Largest prime factor of 36893488147419103231: 145295143558111
Largest prime factor of 73786976294838206463: 599479
Largest prime factor of 147573952589676412927: 761838257287
Largest prime factor of 295147905179352825855: 131071
Largest prime factor of 590295810358705651711: 10052678938039
Largest prime factor of 1180591620717411303423: 122921
Largest prime factor of 2361183241434822606847: 212885833
Largest prime factor of 4722366482869645213695: 38737
Largest prime factor of 9444732965739290427391: 9361973132609
Largest prime factor of 18889465931478580854783: 616318177
Largest prime factor of 37778931862957161709567: 10567201
Largest prime factor of 75557863725914323419135: 525313
Largest prime factor of 151115727451828646838271: 581283643249112959
Largest prime factor of 302231454903657293676543: 22366891
Largest prime factor of 604462909807314587353087: 1113491139767
Largest prime factor of 1208925819614629174706175: 4278255361
Largest prime factor of 2417851639229258349412351: 97685839
Largest prime factor of 4835703278458516698824703: 8831418697
Largest prime factor of 9671406556917033397649407: 57912614113275649087721
Largest prime factor of 19342813113834066795298815: 14449
Largest prime factor of 38685626227668133590597631: 9520972806333758431
Largest prime factor of 77371252455336267181195263: 2932031007403
Largest prime factor of 154742504910672534362390527: 9857737155463
Largest prime factor of 309485009821345068724781055: 2931542417
Largest prime factor of 618970019642690137449562111: 618970019642690137449562111
Largest prime factor of 1237940039285380274899124223: 18837001
Largest prime factor of 2475880078570760549798248447: 23140471537
Largest prime factor of 4951760157141521099596496895: 2796203
Largest prime factor of 9903520314283042199192993791: 658812288653553079
Largest prime factor of 19807040628566084398385987583: 165768537521
Largest prime factor of 39614081257132168796771975167: 30327152671```

### Particularly fast

Using Perl 5 ntheory library via Inline::Perl5. Makes it to about 2^155 - 1 in 1 second on my system. Varies from 2^145-1 (lowest seen) to 2^168-1 (highest seen).

```use Inline::Perl5;
my \$p5 = Inline::Perl5.new();
\$p5.use: 'ntheory';
my &lpf = \$p5.run('sub { ntheory::todigitstring ntheory::vecmax ntheory::factor \$_ }');

my \$start = now;

for flat 600851475143, (1..∞).map: { 2 +< \$_ - 1 } {
say "Largest prime factor of \$_: ", lpf "\$_";
last if now - \$start > 1; # quit after one second of total run time
}
```

Same output only much longer.

## Ring

```load "stdlib.ring"
see "working..." + nl
see "The largest prime factor of the number 600851475143 is:" + nl
num = 600851475143
numSqrt = sqrt(num)
numSqrt = floor(numSqrt)
if numSqrt%2 = 0
numSqrt++
ok

for n = numSqrt to 3 step -2
if isprime(n) and num%n = 0
exit
ok
next

see "" + n + nl
see "done..." + nl```
Output:
```working...
The largest prime factor of the number 600851475143 is:
6857
done...
```

## Sidef

```var n = 600851475143

say gpf(n)             #=> 6857
say factor(n).last     #=> 6857
```

## Wren

Without using any library functions at all (it's a one-liner otherwise):

```var largestPrimeFactor = Fn.new { |n|
if (!n.isInteger || n < 2) return 1
var inc = [4, 2, 4, 2, 4, 6, 2, 6]
var max = 1
while (n%2 == 0) {
max = 2
n = (n/2).floor
}
while (n%3 == 0) {
max = 3
n = (n/3).floor
}
while (n%5 == 0) {
max = 5
n = (n/5).floor
}
var k = 7
var i = 0
while (k * k <= n) {
if (n%k == 0) {
max = k
n = (n/k).floor
} else {
k = k + inc[i]
i = (i + 1) % 8
}
}
return (n > 1) ? n : max
}

var  n = 600851475143
System.print("The largest prime factor of %(n) is %(largestPrimeFactor.call(n)).")
```
Output:
```The largest prime factor of 600851475143 is 6857.
```

## XPL0

```real Num, Max, Div, Quot;
[Num:= 600851475143.;
Max:= 0.;
Div:= 2.;
repeat  loop    [Quot:= Num / Div;
if Mod(Quot, 1.) < 1E-10 then \evenly divisible
[Num:= Quot;
Max:= Div;
]
else    quit;
if Div > Num then quit;
];
Div:= Div + 1.;
until   Div > Num;
Format(1, 0);
RlOut(0, Max);
]```
Output:
```6857
```