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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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Did you look at en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers –  Lucia Sep 26 '13 at 18:23
    
You can likewise try looking at arxiv.org/find/all/1/all:+AND+theory+AND+number+AND+odd+perfect/… –  Jose Arnaldo Dris Oct 1 '13 at 16:50
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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 http://en.wikipedia.org/wiki/Perfect_numbers#Odd_perfect_numbers. For relations the abc conjecture, see http://www.math.dartmouth.edu/~carlp/LucaPomeranceNYJMstyle.pdf.

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
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Is it important ? It is one of many restrictions. –  Dietrich Burde Sep 26 '13 at 18:32
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related, someone asked something on MSE that turned out to be about these: en.wikipedia.org/wiki/Almost_perfect_number For Guy's book, section B2, I think we should define a class of pluperfect numbers...No, already been done, en.wikipedia.org/wiki/Pluperfect_number –  Will Jagy Sep 26 '13 at 18:45
    
@WillJagy, is the MSE question that you are referring to this one? =) –  Jose Arnaldo Dris Oct 11 '13 at 19:07
    
@JoseArnaldoDris, yes, that's it. –  Will Jagy Oct 11 '13 at 19:51
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