Mersenne and Fermat Primes

Number Theory · Axiom Academy

LESSON Mersenne and Fermat Primes Exploring two remarkable families of prime numbers with unique mathematical properties A Mersenne number is a number of the form M p = 2 p - 1, where p is a prime number. When M p is itself prime, we call it a Mersenne prime . Important: Even when p is prime, M p is not always prime! For example, M 11 = 2 11 - 1 = 2047 = 23 × 89, which is composite. The search for Mersenne primes continues today through the GIMPS (Great Internet Mersenne Prime Search) project, a distributed computing effort. The largest known prime number is always a Mersenne prime! A Fermat number is a number of the form F n = 2 2 n + 1. Pierre de Fermat conjectured that all Fermat numbers are prime, but this turned out to be spectacularly wrong! These five primes led Fermat to believe all F n would be prime. However, in 1732, Leonhard Euler discovered that F 5 = 2 32 + 1 = 4,294,967,297 = 641 × 6,700,417, which is composite! In fact, no Fermat primes beyond F 4 have ever been found , and many Fermat numbers are known to be composite. The Fermat primes have a beautiful connection to classical geometry through constructible polygons . This means we can construct regular polygons with 3, 5, 17, 257, and 65,537 sides, but not with 7, 9, 11, 13, or 14 sides (unless combined with powers of 2)! Carl Friedrich Gauss was so proud of his discovery of the constructibility of the 17-gon that he requested it be inscribed on his tombstone (though this was never done).

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