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What is the latest progress in the research on Odd Perfect numbers? I may be wrong, but I found a little on Perfect numbers in the latest issues of SCI journals. Is it really so? I would like to have the latest update on Perfect numbers.

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closed as too broad by Eric Wofsey, Andy Putman, Andrey Rekalo, Ramiro de la Vega, Kevin P. Costello Sep 26 '13 at 20:29

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Did you look at – Lucia Sep 26 '13 at 18:23
You can likewise try looking at… – user11235813 Oct 1 '13 at 16:50
up vote 4 down vote accepted

I am not claiming that I do know the latest update about perfect numbers. However, I believe that still nobody has proved that there are no odd perfect numbers (despite of some doubtful papers on the web). There are heuristics (e.g., by C. Pomerance) that there should be no odd perfect numbers. People have proved a number of restrictive properties that odd perfect number must have, if there are any. Indeed, any odd perfect number $N$ must satisfy:

1.) $N> 10^{1500}$,

2.) $N$ is not divisible by 105.

3.) $N$ is of the form $N ≡ 1 (mod 12), N ≡ 117 (mod 468)$, or $N ≡ 81 (mod 324)$.

and so on. For references see For relations the abc conjecture, see

Edit: And here is more discussion at MO: Algebraic Attacks on the Odd Perfect Number Problem.

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Why is "N is not divisible by 105" so important? – Zsbán Ambrus Sep 26 '13 at 18:28
Is it important ? It is one of many restrictions. – Dietrich Burde Sep 26 '13 at 18:32
related, someone asked something on MSE that turned out to be about these: For Guy's book, section B2, I think we should define a class of pluperfect numbers...No, already been done, – Will Jagy Sep 26 '13 at 18:45
@WillJagy, is the MSE question that you are referring to this one? =) – user11235813 Oct 11 '13 at 19:07
@JoseArnaldoDris, yes, that's it. – Will Jagy Oct 11 '13 at 19:51

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