[Dr Sardonicus]

Originally Posted by bhelmes

If f | Mp and p is prime then

a) f | 2n²-1

b) but not f=2n²-1


a) is clear for me but b) is a guess (I checked some numbers)


The first and fastest counterexample is welcome (Or a proof ?)

First example: Taking n = 1, f = 1 divides 2^p - 1 for any p.

First example with prime f: p = 3, n = 2; f = 2*2² - 1 = 7 divides 2³ - 1 = 7.

Second example with prime f: p = 5, n = 4; f = 2*4² - 1 = 31 divides 2⁵ - 1 = 31

48 other examples known where f = 2^p - 1 is prime.

Hey, you didn't say f had to be a proper divisor -- or prime.

I checked the factor table of 2ⁿ - 1, odd n < 1200, and did not find any examples with f prime other than f = M_p

I didn't check for composite proper factors of the form 2*n² - 1.