As a contrasting example, if n is the product of the primes 13729, 1372933, and 18848997161, where , Fermat's factorization method will begin with which immediately yields and hence the factors and .

As of 2021-03-12, the algorithm with best theoretical asymptotic running time is the general number field sieve (GNFS), first published in 1993, running on a b-bit number n in time. \exp\left( \left(\sqrt[3]{\frac{64}{9}} + o(1)\right)(\ln n)^{\frac{1}{3}}(\ln \ln n)^{\frac{2}{3}}\right). For current computers, GNFS is the best published algorithm for large n (more than about 400 bits).

For a quantum computer, however, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time.

Addison-Wesley, 1997.

Brent, "Recent Progress and Prospects for Integer Factorisation Algorithms", Computing and Combinatorics", 2000, pp. 3–22.

In 2001, Shor's algorithm was implemented for the first time, by using NMR techniques on molecules that provide 7 qubits. It is not known exactly which complexity classes contain the decision version of the integer factorization problem (that is: does have a factor smaller than ?).

August 2005 version PDF Eric W.

Weisstein, “RSA-640 Factored” MathWorld Headline News, November 8, 2005 Computational hardness assumptions Unsolved problems in computer science

In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number (RSA-240) utilizing approximately 900 core-years of computing power.

The largest such semiprime yet factored was RSA-250, a 829-bit number with 250 decimal digits, in February 2020.

As of 2021-03-12, the algorithm with best theoretical asymptotic running time is the general number field sieve (GNFS), first published in 1993, running on a b-bit number n in time. \exp\left( \left(\sqrt[3]{\frac{64}{9}} + o(1)\right)(\ln n)^{\frac{1}{3}}(\ln \ln n)^{\frac{2}{3}}\right). For current computers, GNFS is the best published algorithm for large n (more than about 400 bits).

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