# largest known

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## 5 matching pages

##### 1: 27.22 Software
• GIMPS. This includes updates of the largest known Mersenne prime.

• ##### 2: 27.12 Asymptotic Formulas: Primes
where $\lambda(\alpha)$ depends only on $\alpha$, and $\phi\left(m\right)$ is the Euler totient function (§27.2). … The largest known prime (2018) is the Mersenne prime $2^{82,589,933}-1$. …
##### 3: Errata
• Paragraph Prime Number Theorem (in §27.12)

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
Deterministic algorithms are slow but are guaranteed to find the factorization within a known period of time. … 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. … 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. …
##### 5: 3.8 Nonlinear Equations
(More precisely, $p$ is the largest of the possible set of indices for (3.8.3).) … For multiple zeros the convergence is linear, but if the multiplicity $m$ is known then quadratic convergence can be restored by multiplying the ratio $f(z_{n})/f^{\prime}(z_{n})$ in (3.8.4) by $m$. …