Mersenne and Fermat Numbers

The first seventeen even perfect numbers are therefore obtained by substituting these values of ra in the expression 2n_1(2n —1). The first twelve of the Mersenne primes have been known since 1914; the twelfth, 2127 —1, was indeed found by Lucas as early as 1876, and for the next seventy-five years was the largest known prime. More details on the history of the Mersenne numbers may be found in Archibald [l]; see also Kraitchik [4]. The next five Mersenne primes were found in 1952; they are at present the five largest known primes of any form. They were announced in Lehmer [7] and discussed by Uhler [13]. It is clear that 2" —1 can be factored algebraically if ra is composite; hence 2n —1 cannot be prime unless w is prime. Fermat's theorem yields a factor of 2n —1 only when ra + 1 is prime, and hence does not determine any additional cases in which 2"-1 is known to be composite. On the other hand, it follows from Euler's criterion that if ra = 0, 3 (mod 4) and 2ra + l is prime, then 2ra + l is a factor of 2n— 1. Thus, in addition to cases in which ra is composite, we see that 2n— 1 is composite when 2ra + l is prime as well as ra, provided that ra = 3 (mod 4) and ra>3. Aside from this, factors of 2" —1 are known only in individual cases. If no factor is known, the best way to find out whether 2" —1 is prime is to apply a test due essentially to Lucas, but stated in a simplified form by Lehmer [6, Theorem 5.4].