Problems & Puzzles: Conjectures

Conjecture 11. Is there any odd Perfect Number ?

A perfect number is a number equal to the sum of the aliquot part of its proper divisors.

The smaller two smaller ones are : 6=1+2+3  and 28=1+2+4+7+14.

Euclid knew that 2p-1(2p-1) is a perfect number if 2p-1 is prime. If 2p-1 is prime then p is prime, but not the converse.

Euler showed that all the even perfect numbers are of the form given by Euclid.

O.K…., but, are there only even perfect numbers ? exist one, some or infinite odd perfect numbers ?

Up today nobody knows the answer to that question. Neither we have devised one of these named odd perfect numbers, nor nobody has proved the nonexistence of those odd perfect numbers.

Up today the only sure thing we know about this question is that if N is such odd perfect number then N is greater than 10300 (Brent, Cohen & Te Riele, 1991)

(Ref. 2, p. 44)

***

Samuli Larvala sent today (11/08/98) the following interesting information about where to find the work of Brent 'et alia' mentioned above:

" This paper can also be found on the Interent at: <ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/rpb116.ps.gz> or rpb116.dvi.gz. The proof three mentioned in the paper can also be found at the same address. The file is rpb116p.ps.gz or rpb116p.dvi.gz."

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