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

In my last blog I noticed that

28 = 1

^{3}+ 3

^{3}

496= 1

^{3}+ 3

^{3}+ 5

^{3}+ 7

^{3}

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 = 1

^{3}+ 3

^{3}+ ... + 15

^{3}

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 (2

^{p-1})(2

^{p}- 1) where 2

^{p}-1 is prime.

Known Theorem 2: 1

^{3}+ 3

^{3}+ 5

^{3}+ ... + (2(m-1)+1)

^{3}= m

^{2}(2m

^{2}-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 2

^{p-1}(2

^{p}-1) where p is odd.

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

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