Paper (*cross-listing*): hep-th/0401052 From: Simon Davis Dr <davis@math.uni-potsdam.de> Date: Thu, 8 Jan 2004 23:56:42 GMT (9kb) Title: A Proof of the Odd Perfect Number Conjecture Authors: Simon Davis Comments: TeX, 13 pages Report-no: RFSC 04-01 Subj-class: High Energy Physics - Theory; Number Theory The finiteness of the number of solutions to the rationality condition for the existence of odd perfect numbers is deduced from the prime decomposition of the product of repunits defined by the sum of the divisors, $\sigma(N)$. All of the solutions to this rationality condition satisfy the inequality ${{\sigma(N)}\over N}\ne 2$. This technique is then used to demonstrate that there are no odd perfect numbers $N=(4k+1)^{4m+1} \prod_{i=1}^\ell q_i^{2\alpha_i}$. http://arXiv.org/abs/hep-th/0401052 13 pages ---------------------- I looked at this, but haven't puzzled through it yet. A crank? Putting the paper in hep-th is curious, and the author affiliation is unusual. The surface markers look mostly reasonable: real references, real notation. The logic is abbreviated, and it's hard to tell if it's merely explanatory. Page 1 leaps immediately into complicated algebra, eschewing background. There are well known required shapes for hypothetical odd perfect numbers. (These arise from equating 2N with the formula for the sum of divisors of N, and analyzing the even/odd properties of base P repunits. "Elementary" number theory.) One requirement is that all but one prime divisor have even exponent, making a square of a prime or prime power. One part of Davis's argument is that the divisor sum term for these primes (a factor of P^2 leads to a divisor-sum factor of P^2 + P + 1, a base P repunit 111) leads to new primes > P, which must then appear as new divisors of the hypothetical perfect number. For example, 7^2 -> 7^2 + 7 + 1 = 57 = 3 * 19 giving the new prime 19. Continuing the argument, 19^2 -> 381 = 3 * 127, and so on. If this always led to new larger primes, it would prove NoOPN. But it doesn't always give new primes (67^2 -> 4557 = 3 * 7 * 7 * 31 and 79^2 -> 6321 = 3 * 7 * 7 * 43), and I think his argument is concerned with handling the exceptional cases. Rich rcs@cs.arizona.edu