Mersenne and Fermat numbers

The first seventeen even perfect numbers are therefore obtained by substituting these values of n in the expression 2 n- (2n -1). The first twelve of the Mersenne primes have been known since 1914; the twelfth, 21271, 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 1; 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 2n 1 can be factored algebraically if n is composite; hence 2n -1 cannot be prime unless n is prime. Fermat's theorem yields a factor of 2n -1 only when n +1 is prime, and hence does not determine any additional cases in which 2n-1 is known to be composite. On the other hand, it follows from Euler's criterion that if n₀, 3 (mod 4) and 2n+1 is prime, then 2n+1 is a factor of 2n-1. Thus, in addition to cases in which n is composite, we see that 2n 1 is composite when 2n+1 is prime as well as n, provided that n 3 (mod 4) and n >3. Aside from this, factors of 2n -1 are known only in individual cases. If no factor is known, the best way to find out whether 2n -1 is prime is to apply a test due essentially to Lucas, but stated in a simplified form by Lehmer 6, Theorem 5. 4.