# Integer factorization

### 1884

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 .

### 1993

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).

### 1994

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

### 2000

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

### 2001

In 2001, Shor's algorithm was implemented for the first time, by using NMR techniques on molecules that provide 7&nbsp;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 ?).

### 2005

August 2005 version PDF Eric W.

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

### 2019

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.

### 2020

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

### 2021

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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