2026-10-27T11:45:19Z https://lifeofresearch.com/oai2

oai:lifeofresearch.com:144 2026-10-27T09:55:51Z type:article openaire

https://lifeofresearch.com/articles/144 Ibrahim Amin Shnoudi Jordan University Life of Research 2026 Mathematical Journal article Creative Commons Attribution 4.0 Summary A positive integer n is almost perfect if σ(n) = 2n − 1. Every power of 2 is almost perfect, and the conjecture that there are no others is open. This project does not prove it. It proves the conjecture for odd parts with at most two prime factors, with a machine-checked proof, and it reduces each remaining case to a precisely stated problem.  Two prime factors are excluded completely, for every level and every exponent pattern; the proof is formally verified in Lean 4.  Every odd almost perfect number has at least eight distinct prime factors; the previous bound, six, dates from 1981.  An odd almost perfect number with squarefree root would have a prime factor greater than 1.6·1030 and more than 4.3·1027 prime factors; the corresponding bounds for quasiperfect numbers improve the best known ones.  For three prime factors there is a finite decision procedure for each k, run for all k ≤ 16 (k ≤ 20 when the odd part is squarefree), and a result valid for every k when the component quotient is at most 500,000.  The squarefree case does not depend on primality: it is a question about integral points on K3 surfaces, some of whose genus-one slices violate the Hasse principle.  For each number of prime factors and each k, the odd part of a counterexample is bounded explicitly: b2 < (5ωN)M with M = 4ω − 1.  For odd almost perfect numbers with squarefree root the remaining step is identified exactly: an upper bound for certain primes that is equivalent to a level of distribution θ > 0.87 for the primes in arithmetic progressions, which the Elliott–Halberstam conjecture would provide.  There is no odd integer solution of ∏Φ(xi) = 2∏xi 2 − 1 with |xi| ≥ 3 and at most eight bases, prime or not, of either sign.  All 212 formal statements, every computation and every figure are checked independently; the record of 94 checks is Appendix B of the report. Every statement of this overview is proved, computed or formally verified in full in the four documents of the Zenodo record https://doi.org/10.5281/zenodo.23674490, which a reader should consult for any detail. Bold navy text marks what is new.