If n 6 is an even perfect number prove that n 4 mod Solutio

If n > 6 is an even perfect number, prove that n 4 (mod

Solution

For any odd prime number p

2^p mod(6) = 2

For any even prime number p, it will leave a residue of 4

2^(p-1) mod (6) = 4

Every even perfect number can be written as mersenne prime form

[2^(p-1)][2^p - 1] = (4)(2-1) mod(6) = 4 mod 6

Hence every even perfect number will be congruent to 4 mod 5, which is greater than 6