Mersenne Primes

[Enola_Straight]

A Mersenne prime has the form 2^n-1.

What do you call a prime of the form 2^n+1?

[Q.E.D]

Fermat primes:

In honour of the inspired pioneers, the numbers of the form 2[sup]n[/sup]-1 are now called the Mersenne numbers and the numbers of the form 2[sup]n[/sup]+1 the Fermat numbers.

[Earthling]

I don’t know, QED. My number theory textbook gives this definition:

A Fermat number is an integer in the form[ul]F[sub]n[/sub] = 2[sup]2[sup]n[/sup][/sup] + 1,[/ul]If F[sub]n[/sub] is a prime, it is said to be a Fermat prime.

Other than that, it only says “these may be considered as a special case of the integers of the form 2[sup]m[/sup] +1.” No other name was given.

. . .

Well, further research leads me to MathWorld, which says that 2[sup]n[/sup] + 1 is the less common of the two definitions for Fermat numbers. So, that makes everybody happy, I suppose.

[Q.E.D]

Fermat numbers are generally given in the form b•2[sup]n[/sup]+1, so 1•2[sup]n[/sup]+1 = 2[sup]n[/sup]+1 would be a special case.

[Jabba]

Earthling: Note that if 2[sup]n[/sup] + 1 is prime, then n is of the form 2[sup]m[/sup]. Thus any prime of the form 2[sup]n[/sup] + 1 is of the form 2[sup]2[sup]n[/sup][/sup] + 1, so either of the two definitions of Fermat prime will do.