factor invocation (GNU Coreutils 9.0)

26.1 factor: Print prime factors

factor prints prime factors. Synopses:

factor [number]…
factor option

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

The factor command supports only a small number of options:

‘--help’: Print a short help on standard output, then exit without further processing.
‘--version’: Print the program version on standard output, then exit without further processing.

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.