Tuesday, April 05, 2016

Are Perfect Numbers Bigger than Six initial sums of odd cubes (answered)

(NONE of this is my work. In fact some of it is on Wikipedia.)

In my last blog I noticed that

28 = 13  + 33

496= 13 + 33 + 53 + 73

noting that 28 and 496 are the 2nd and 3rd perfect numbers.

I asked if 8128, the next perfect number is also an initial sum of odd cubes. It is!

8128 = 13 + 33 + ... + 153

I also asked if there was something interesting going on .The answer is YES but not that interesting.

All of the math with proofs are  here. I sketch below.

Known Theorem  1: n is an even perfect number iff n is of the form (2p-1)(2p- 1) where 2p-1 is prime.

Known Theorem  2: 13 + 33 + 53 + ... + (2(m-1)+1)3 = m2(2m2-1).

Interesting theorem: if n is an even perfect number larger than 6 and p is the p from Known Theorem 1 then n is the sum of the first  2(p-1)/2 odd cubes.

Why this is less interesting: The proof does not use that n is perfect. It holds for any number of the form 2p-1(2p-1) where p is odd.

So the theorem has nothing to do with perfect numbers. Oh well.

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