- #1

MathematicalPhysicist

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We all know that 3^3+4^3+5^3=216=6^3

This and the well known pythogrean triplet 3^2+4^2=5^2.

So I thought of toying a little bit with powers of three and two, and I found by coincidence that:

3^3+5^3+7^3=495= 496 -1 , where we all know that 496 is a perfect number.

Then I thought ridicuosly that this can happen also for other perfect numbers, but to no a veil, for 6 we can't have powers of three which are distinct from each other, but we do have powers of 2, 6-1=2^2+1^2.

So I thought to myself, maybe every perfect number minus 1 can be represented as powers of 3 or 2 of distinct natural numbers.

For 28 we have 28-1=3^3=27.

For 8128 we have 8128-1=19^3+8^3+7^3+6^3+5^3+4^3+2^3.

All the above is sheer luck and coincidence, but this raises the conjecture:

**Every (even) perfect number minus one can be represented as a sum of distinct powers of 2 or 3**

I don't have enough time to check for the next perfect number.

I did the last calculation via google, check me that I don't have mistakes.

Is this already known?