A Conversation for Perfect Numbers

Clarifications and additions

Post 1


"the odd perfect number, if it exists, will be expressible in the form i^a.j^b.k^c... where i, j, k... and a are all in the form 4n+1 (that is, one more than a multiple of 4) and b, c... are all even."

Actually, the condition is that i and a are of the form 4n+1 and b, c,... are even. j, k,... can be of the form 4n+1 or 4n+3.

All odd perfect numbers must have their largest prime factor greater than 10,000,000 (Jenkins) and third largest factor at least 100 (Ianucci).

Clarifications and additions

Post 2

Gnomon - time to move on

No, you're wrong there. Euler proved that the condition is as I have stated it.

Your statement, that j, k etc can be of the form 4n+1 or 4n+3 just means that they are odd numbers. Euler's statement imposes a further restriction on them.

I can't verify your statement about the largest prime factor - I've seen it stated that it must be greater than 300,000. You statement about the second largest prime factor is certainly true - in fact it must be greater than 1000.

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