By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization. (By convention, 1 is the empty product.) Testing whether the integer is prime can be done in polynomial time, for example, by the See more

In number theory, there are many integer factoring algorithms that heuristically have expected running time
$${\displaystyle L_{n}\left[{\tfrac {1}{2}},1+o(1)\right]=e^{(1+o(1)){\sqrt {(\log n)(\log \log n)}}}}$$
in See more

• Aurifeuillean factorization
• Bach's algorithm for generating random numbers with their factorizations See more

• msieve – SIQS and NFS – has helped complete some of the largest public factorizations known
• Richard P. Brent, "Recent Progress and Prospects for Integer … See more

Among the b-bit numbers, the most difficult to factor in practice using existing algorithms are those semiprimes whose factors are of similar size. For this reason, these are the … See more

Special-purpose
A special-purpose factoring algorithm's running time depends on the properties of the number to be factored or on one of its unknown factors: size, special form, etc. The parameters which determine the running time vary … See more

The Schnorr–Seysen–Lenstra probabilistic algorithm has been rigorously proven by Lenstra and Pomerance to have expected running time Ln[1/2, … See more