Fermat prime and Mersenne prime

Fermat prime

In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form. F_n=2²ⁿ+1, where n is a nonnegative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617. And he is the first to investigate numbers of the form 2²ⁿ.

There is the Pépin test which gives sufficient and necessary condition for the primality of the Fermat prime and this can only be implemented by use of modern computers. Pépin’s test is a primality test, which can be used to determine whether a Fermat number is prime. It is a variant of Proth’s test. The test is named for a French mathematician, Théophile Pépin.Let F_n be a Fermat number. F_n is prime if and only if 3(F_n-1)/2 = -1 (mod F_n).Here 3 can be replaced by any positive integer k for which the Jacobi symbol (k|F_n) is -1. These include k=3, 5, and 10.If F_n is prime, this primality can be shown by Pepin’s test, but when F_n is composite, Pepin’s test does not tell us what the factors will be (only that it is composite). For example, Selfridge and Hurwitz showed that F₁₄ was composite in 1963, but we still do not know any of its divisors. (Chris K. Caldwell)

Mersenne numbers, which take the form of 2ⁿ-1, were named after Marin Mersenne, a French monk from the early 17 century, who corresponded with Fermat. We are particularly interested in the case when Mersenne number are prime. It is not doubt that the first 17 primes of the form 2ⁿ-1 match the following n values: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281 (Zegarelli 287).

The first 12 Mersenne primes were known since 1914 and the 12^th, 2¹²⁷-1, was established by Anatole Lucas in 1876. It was one of the largest-known prime number for over 75 years. The next 5 Mersenne primes (p=13 to 17)were discovered in 1952. It was in 1952 when the testing program for Mersenne numbers was began. This led to the establishment of three other primes. More testing has been carried out using modern day computers and the smallest Mersenne number that is untested is 2²³⁰⁹-1 (~2013), and this has not been a case of great interest. There is a conjecture that 2ⁿ-1 is always prime when n is a Mersenne number. And the more interesting case is the 2⁸¹⁹¹-1 because the 8191 also is the Mersenne number. (Křížek, Florian and Lawrence 214).

Works Cited

Chris K. Caldwell (2014). Pepin’s test . [ONLINE] Available at: http://primes.utm.edu/glossary/page.php?sort=PepinsTest.

Crandall, Richard E, and Carl Pomerance. Prime Numbers: A Computational Perspective. New York: Springer, 2005. Internet resource.

Zegarelli, Mark. Basic Math & Pre-Algebra Workbook for Dummies. Hoboken, N.J: Wiley, 2008. Internet resource.

Křížek, Michal, Florian Luca, and Lawrence Somer. 17 Lectures on Fermat Numbers:    From Number Theory to Geometry. New York, NY [u.a.: Springer, 2001. Print.