26.1 factor: Print prime factors

factor prints prime factors. Synopsis:

factor [option]… [number]…

If no number is specified on the command line, factor reads numbers from standard input, delimited by newlines, tabs, or spaces.

The program accepts the following options. Also see Common options.

‘-h’ ¶

‘--exponents’

print factors in the form p^e, rather than repeating the prime ‘p’, ‘e’ times. If the exponent ‘e’ is 1, then it is omitted.

$ factor --exponents 3000
3000: 2^3 3 5^3

If the number to be factored is small (less than 2^{127} on typical machines), factor uses a faster algorithm. For example, on a circa-2017 Intel Xeon Silver 4116, factoring the product of the eighth and ninth Mersenne primes (approximately 2^{92}) takes about 4 ms of CPU time:

$ M8=$(echo 2^31-1 | bc)
$ M9=$(echo 2^61-1 | bc)
$ n=$(echo "$M8 * $M9" | bc)
$ bash -c "time factor $n"
4951760154835678088235319297: 2147483647 2305843009213693951

real	0m0.004s
user	0m0.004s
sys	0m0.000s

For larger numbers, factor uses a slower algorithm. On the same platform, factoring the eighth Fermat number 2^{256} + 1 takes about 14 seconds, and the slower algorithm would have taken about 750 ms to factor 2^{127} - 3 instead of the 50 ms needed by the faster algorithm.

Factoring large numbers is, in general, hard. The Pollard-Brent rho algorithm used by factor is particularly effective for numbers with relatively small factors. If you wish to factor large numbers which do not have small factors (for example, numbers which are the product of two large primes), other methods are far better.

An exit status of zero indicates success, and a nonzero value indicates failure.