Mersenne numbers

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1: 27.18 Methods of Computation: Primes

… ►For small values of $n$, primality is proven by showing that $n$ is not divisible by any prime not exceeding $\sqrt{n}$. … ►These algorithms are used for testing primality of Mersenne numbers, $2^n-1$, and Fermat numbers, $2^{2^n}+1$. …

2: 27.12 Asymptotic Formulas: Primes

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

3: 27.22 Software

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Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below ${10}^{16}$. Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard $p-1$, and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

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GIMPS. This includes updates of the largest known Mersenne prime.

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Number Theory Web. References and links to software for factorization and primality testing.

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4: Errata

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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).

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5: Bibliography G

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Notes:

Information on Mersenne primes and the source code for the Great Internet Mersenne Prime Search.

External Links:

Home Page

Referenced by:

5th item.

Encodings:

BibTeX

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Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.

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External Links:

ISSN 0026-9255, MathReview (Peter Stevenhagen), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Number-Divisor Tables. British Association Mathematical Tables, Vol. VIII, Cambridge University Press, Cambridge, England.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

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Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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External Links:

FullText, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

Encodings:

BibTeX

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Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

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External Links:

FullText, MathReview (B. M. Stewart), ZentralBlatt

Referenced by:

§24.15(iii), §24.6.

Encodings:

BibTeX

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