# largest known

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5 matching pages ♦

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## 5 matching pages

##### 1: 27.22 Software

##### 2: 27.12 Asymptotic Formulas: Primes

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►where $\lambda (\alpha )$ depends only on $\alpha $, and $\varphi \left(m\right)$ is the Euler totient function (§27.2).
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►The largest known prime (2018) is the Mersenne prime ${2}^{82,589,933}-1$.
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##### 3: Errata

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Paragraph Prime Number Theorem (in §27.12)
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The largest known prime, which is a Mersenne prime, was updated from ${2}^{43,112,609}-1$ (2009) to ${2}^{82,589,933}-1$ (2018).

##### 4: 27.19 Methods of Computation: Factorization

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►Deterministic algorithms are slow but are guaranteed to find the factorization within a known period of time.
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►As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard $p-1$, and a 67-digit prime for ecm.
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►As of January 2009 the snfs holds the record for the largest integer that has been factored by a Type II probabilistic algorithm, a 307-digit composite integer.
…The largest composite numbers that have been factored by other Type II probabilistic algorithms are a 63-digit integer by cfrac, a 135-digit integer by mpqs, and a 182-digit integer by gnfs.
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