fermat number and mersenne number.?

Question

Fermat number is of the form 2^(n)+1 when a special square of the form [2^(n-1)]^(2) is added to this fermat number then it will produce a square whose root would be consecutive to the number 2^(n-1).
Mersenne numbers are 2^(n)-1 when a special square of the form [2^(n-1)-1]^(2)is added to mersenne numbers then it will produce a square whose root would be consecutive to the number [2^(n-1)-1].

Answer 1

(a + b)^2 = a^2 + 2ab + b^2

so

(a + 1)^2 = a^2 + 2a + 1
(a - 1)^2 = a^2 - 2a + 1



Fermat:

If a = 2^(n-1) then
(a + 1)^2 = (2^(n-1) + 1)^2
= (2^(n-1))^2 + 2(2^(n-1)) + 1
=( 2^(n-1)^2 ) + (2^n + 1)



Mersenne:

(2^n - 1) + (2^(n-1) - 1)^2 =
= 2^n - 1 + (2^(n-1))^2 - 2(2^(n-1)) + 1)
= 2^n - 1 + (2^(n-1))^2 - 2^n + 1
= (2^(n-1))^2
= ( (2^(n-1) - 1) + 1 )^2

Answer 2

And the question is?

http://en.wikipedia.org/wiki/Mersenne_prime