A Proof of the Odd Perfect Number Conjecture
| dc.creator | Davis, Simon | |
| dc.date | 2004-01-08 | |
| dc.date | 2008-05-31 | |
| dc.date.accessioned | 2026-07-07T09:41:56Z | |
| dc.date.available | 2026-07-07T09:41:56Z | |
| dc.description | It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer $N=(4k+1)^{4m+1}\prod_{i=1}^\ell ~ q_i^{2α_i}$ to establish that there do not exist any odd integers with equality between $σ(N)$ and 2N. The existence of distinct prime divisors in the repunits in $σ(N)$ follows from a theorem on the primitive divisors of the Lucas sequences $U_{2α_i+1}(q_i+1,q_i)$ and $U_{2α_j+1}(q_j+1,q_j)$ with $q_i,q_j,2α_i+1,2α_j+1$ being odd primes. The occurrence of new prime divisors in each quotient ${{(4k+1)^{4m+2}-1}\over {4k}}$, ${{q_i^{2α_i+1}-1}\over {q_i-1}}, i=1,...,\ell$ also implies that the square root of the product of $2(4k+1)$ and the sequence of repunits will not be rational unless the primes are matched. Although a finite set of solutions to the rationality condition for the existence of odd perfect numbers is obtained, it is verified that they all satisfy ${{σ(N)}\over N}\ne 2$ because the repunits in the product representing $σ(N)$ introduce new prime divisors. Minimization of the number of prime divisors in $σ(N)$ leads to an infinite set of repunits of increasing mangitude or prime equations with no integer solutions. It is proven then that there exist no odd perfect numbers. | |
| dc.description | TeX, 27 pages. The introduction has been expanded and the factor deleted by other sources from Eq.(4.3) is restored | |
| dc.identifier | https://arxiv.org/abs/hep-th/0401052 | |
| dc.identifier | http://arxiv.org/abs/hep-th/0401052 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/162005 | |
| dc.subject | High Energy Physics - Theory | |
| dc.title | A Proof of the Odd Perfect Number Conjecture | |
| dc.type | text |