2004/01/08 by Simon Davis, Davis, Simon
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/0401052
TeX, 27 pages. The introduction has been expanded and the factor deleted by other sources from Eq.(4.3) is restored
openalex publication_date 2004/01/08 · arxiv created 2008/05/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∏i=1^ℓ ~ qi2αi 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 U2αi+1(qi+1,qi) and U2αj+1(qj+1,qj) with qi,qj,2αi+1,2αj+1 being odd primes. The occurrence of new prime divisors in each quotient (4k+1)4m+2-1\over 4k, qi2αi+1-1\over qi-1, i=1,...,ℓ 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≠ 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.