Factorizing integers is a well-researched field, and there are several complicated algorithms which are used in order to factorize integers of over 100 digits. These include the Quadratic Sieve, the Elliptic Curve Method, and the General Number Field Sieve. Programs which can factorize numbers over 100 digits may take minutes, hours or days to run, depending on the factors of the number. The algorithm on this page uses three sub-algorithms: trial division, Pollard's Rho, and the Miller-Rabin algorithm. Trial division simply tries integers of increasing size up to the square root of the number to be factorized, to see if they are factors. This can be very inefficient. Pollard's Rho algorithm quickly finds factors (which may be non-prime), allowing the integer to be partially factorized. Using Pollard's Rho repeatedly can be more efficient than trial division. The Miller-Rabin algorithm quickly identifies if an integer is prime or not, and so is very useful to check factors found using Pollard's Rho. The version of Pollard's Rho used here involves random quadratics. Therefore it is possible that the program may factorize an integer on one occasion but not on another.

Strong pseudoprimes are harder for prime factorization algorithms to factorize. You can find a more technical definition here at Wikipedia. You can find lists of strong pseudoprimes in the Online Encyclopaedia of Integer Sequences (OEIS).

Here are some strong pseudoprimes to factorize!

3215031751, 118670087467, 307768373641, 315962312077, 354864744877, 457453568161, 528929554561, 546348519181, 602248359169, 1362242655901, 1871186716981, 2152302898747, 2273312197621, 2366338900801, 3343433905957