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.
closed as too broad by Eric Wofsey, Andy Putman, Andrey Rekalo, Ramiro de la Vega, Kevin P. Costello Sep 26 '13 at 20:29There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs. If this question can be reworded to fit the rules in the help center, please edit the question. 


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. 

