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prime numbers

Euclid proved there is no biggest prime number. As of 1992, 66 primes with more than 10,000 digits were known, and 4,590 with more than 1,000. The largest prime found to 2018 is 2^77,232,917-1 (which is 23,249,425 digits long in decimal notation). It was discovered by Jonathan Pace on 26 December 2017.

Some special types of primes are:

Mersenne primes

Primes of the form 2^q − 1. For 2^q − 1 to be prime, q must be prime, but not all numbers of this form are prime. Named for Marin Mersenne (1588 – 1648), a French monk. Mersenne primes are the subject of the Great Internet Mersenne Prime Search, with a $100,000 prize for the first to discover a prime with more than 10 million decimal digits. Many participants have used the same software: an algorithm by Richard Crandall implemented in machine language by George Woltman, and networking software by Scott Kurowski. The networking software enables the calculations to be run on a massive distributed computing operation coordinated by Entropia.com, using otherwise unused time on tens of thousands of home and office PCs. See www.mersenne.org for details of new and past discoveries.

Twin primes

A pair of primes that differ by only 2, such as “3” and “5”. The largest known twin primes are (242,206,083 × 2³⁸⁸⁸⁰) ± 1, found by Indlekofer and Ja'rai in November 1995. Each has 11,713 digits.

The previous record, found by H. Dubner, is (1,692,923,232 × 104,020) ± 1. It has 4,030 digits.

Sophie Germain primes

A pair of primes such that one is twice the other plus 1, for example “11” and “23”. Named for a 19ᵗʰ century French mathematician.

resources

C. K. Caldwell maintains a list of known primes (and a lot of other information on primes!) at https://primes.utm.edu/.

S. Torquato, G. Zhang and M. de Courcy-Ireland.
Uncovering multiscale order in the prime numbers via scattering.
Journal of Statistical Mechanics: Theory and Experiment, Volume 2018, (5 September 2018).
doi:10.1088/1742-5468/aad6be

Paulo Ribenboim.
The Little Book of Bigger Primes. Second edition.
Springer-Verlag, 2004.

Richard Crandall and Carl Pomerance.
Prime Numbers. A Computational Perspective.
Springer-Verlag, 2001.

What's the fastest way of computing whether a given large number is a prime? A text at the graduate level.

Vicky Neale.
Closing the Gap: The Quest to Understand Prime Numbers.
Oxford: Oxford University Press, 2017.