[QUOTE=353085177;457639]My discovery about Mersenne prime
I find there are a lot of numbers similar to Mersenne prime and Fermat number.
(a+1)^n-a^n，(a+b)^(2^n)+a^(2^n)
Looking at [url]http://weibo.com/ttarticle/p/show?id=2309404101129943270060[/url][/QUOTE]

There are already plenty of generations to Mersenne numbers only using polynomials P(2^x).

Let n be any odd number, and the sequence 2^n-1 = 1, 7, 31, 127, 511, 2047, 8191, 32767, 131071, 524287,......

1) switch to (2^n+1)/3 for all odd n is another generalization:

1, 3, 11, 43, 171, 683, 2731, 10923, 43691, 174763,......

2) 2^n+-2^((n+1)/2)+1 defines a similar sequence depending on n = (1, 7) or (3, 5) (mod 8):

1, 13, 41, 113, 481, 2113, 8321, 32513, 130561, 525313,......

3) (2^(d*n)-1)/((2^d-1)*(2^n-1)) is another generalize prime exponent form.

[QUOTE=carpetpool;457852]There are already plenty of generations to Mersenne numbers only using polynomials P(2^x).

Let n be any odd number, and the sequence 2^n-1 = 1, 7, 31, 127, 511, 2047, 8191, 32767, 131071, 524287,......

1) switch to (2^n+1)/3 for all odd n is another generalization:

1, 3, 11, 43, 171, 683, 2731, 10923, 43691, 174763,......

2) 2^n+-2^((n+1)/2)+1 defines a similar sequence depending on n = (1, 7) or (3, 5) (mod 8):

1, 13, 41, 113, 481, 2113, 8321, 32513, 130561, 525313,......

3) (2^(d*n)-1)/((2^d-1)*(2^n-1)) is another generalize prime exponent form.[/QUOTE]

as for the actual primes themselves they also have generalizations in forms:

[url]https://en.wikipedia.org/wiki/Mersenne_prime#Generalizations[/url]