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 … See more

In number theory, there are many integer factoring algorithms that heuristically have expected running time
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• Aurifeuillean factorization
• Bach's algorithm for generating random numbers with their factorizations
• Canonical representation of a positive integer
• Factorization See more

Among the b-bit numbers, the most difficult to factor in practice using existing algorithms are those semiprimes whose factors are of … See more

Special-purpose
A special-purpose factoring algorithm's running time depends on the properties of the number to be … See more

The Schnorr–Seysen–Lenstra probabilistic algorithm has been rigorously proven by Lenstra and Pomerance to have expected running time Ln[1/2, 1+o(1)] by replacing the GRH assumption … 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 Factorisation Algorithms", Computing and Combinatorics", … See more