There's often a version 2, or larger. Note that v2 is twice as large as v1, while v3 is 4 years later. Submission history From: Simon Davis Dr [view email] [v1] Thu, 8 Jan 2004 23:56:42 GMT (9kb) [v2] Mon, 14 Jun 2004 19:34:03 GMT (17kb) [v3] Sat, 31 May 2008 00:59:08 GMT (20kb) The sum-of-divisors of a number p^j * q^k * ... is the product of (1+p+...+p^j)(1+q+...+q^k)... . The (symbolic) factors are repunits base p,q, etc. For N to be an odd perfect number, that product (aka sigma(N)) must Have exactly one factor of 2. This implies that one special prime, say P, Must have an odd exponent in the factorization of N, while the other Primes have even exponents. The special prime P must be a 4K+1 prime And its exponent must be 4L+1. So one approach to the OPN problem is try matching up divisors on the LHS & RHS of the equation sigma(N) = 2N. This suggests the idea of Trying a prime on RHS, perhaps 7, and guessing that its exponent is 2. Then sigma(N) will have a factor of (1 + 7 + 7^2) (or 111 heptal) = 57. 57 = 3.19, so there must be a divisor 19^2M dividing N. Again guessing That the exponent is 2, we find a factor of (1+19+19^2) =381 dividing the LHS. This has a divisor of 127, leading to 1+127+127^2 = 16257 = 3 * 5419. Unless this series of primes terminates, our potential OPN is infinite. So one tries to analyze ways out of the problem: looking for the primes & powers that don't increase the sequence, like 67^2: 4557 = 3 * 7^2 * 31. So far, unless Davis has done it, noone's gotten very far in the proof Direction. It's a useful numerical ingredient for establishing the various Known restrictions and lower bounds on a hypothetical OPN. Rich PS: Running a comment facility isn't easy. Expect moby human labor & costs To keep it civil. -----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of James Propp Sent: Wednesday, July 22, 2015 10:14 AM To: math-fun Subject: [EXTERNAL] Re: [math-fun] Purported proof that there are no odd perfect numbers Yes, authors can modify their papers. That includes retracting claims. Jim Propp On Wednesday, July 22, 2015, Allan Wechsler <acwacw@gmail.com> wrote:
Do they even permit *authors* to modify their papers (or retract their claims) after posting? For all I know Davis accepted a correction long ago.
On Wed, Jul 22, 2015 at 11:42 AM, Warren D Smith <warren.wds@gmail.com <javascript:;>> wrote:
link to http://arxiv.org/abs/hep-th/0401052, a paper by Simon Davis with the intriguing title, *A Proof of the Odd Perfect Number Conjecture*.
--Simon Davis' only other paper on arXiv was http://arxiv.org/abs/hep-th/0312101 "Effectively Closed Infinite-Genus Surfaces and the String Coupling"
If he proved the odd perfect number conjecture in 2004 it is somewhat,
er,
odd, he got no attention. Although given that he published it on hep-th (high energy physics) topic rather than "number theory" it would be rather easy for number theorists to miss it.
This is yet another excellent example of why the arrogant irresponsible damned idiots in charge of arxiv.org have been massively remiss by refusing to include a commenting facility.
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