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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
by pzxc 14y ago
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)
- ctdonath 14y agoAwesome. I'll have to apply the binary notation approach to answering the question I just posted below.
- jnotarstefano 14y agoThis is in fact the consequence of a fact first proven by Euler: every even perfect number is of the form (2^(p-1)) * (2^p - 1) where 2^p - 1 is a prime. That is, it's p ones shifted left by p-1 zeroes.
- john_b 14y agoIn case anyone is curious, the explicit power of two formula for these perfect numbers can be found here: http://primes.utm.edu/mersenne/index.html#theorems http://primes.utm.edu/mersenne/index.html#theorems
- SagelyGuru 14y agoSurely then the third should be: 111110000, not 1111100. Looks like you are two zeroes short of perfect ;)