In my freshman year of college, I discovered something neat about perfect numbers and Mersenne primes (verified by my math professor):

The first perfect number, 6, written in binary is 110 The second, 28, in binary is 11100 The third, 496, in binary is 1111100 The fourth, 8128, in binary is 1111111000000

See the pattern? Its a number of ones equal to the mersenne prime that generates the perfect number, followed by one less number of zeroes.

Just an interesting tidbit I discovered completely by accident in college that I always thought was cool. :)

Of course it derives directly from the fact that mersenne primes (and perfect numbers) are based on powers of two, but I still thought it was cool (probably because I discovered it accidentally and independently without trying to)