{"id":144,"identifier":"LOR-00144","url":"https://lifeofresearch.com/articles/144","title":"Shnoudi Research On the Almost Perfect Number Conjecture","abstract":"Summary\r\nA positive integer n is almost perfect if σ(n) = 2n − 1. Every power of 2 is almost perfect, and the\r\nconjecture that there are no others is open. This project does not prove it. It proves the conjecture for odd\r\nparts with at most two prime factors, with a machine-checked proof, and it reduces each remaining case\r\nto a precisely stated problem.\r\n Two prime factors are excluded completely, for every level and every exponent pattern; the\r\nproof is formally verified in Lean 4.\r\n Every odd almost perfect number has at least eight distinct prime factors; the previous bound,\r\nsix, dates from 1981.\r\n An odd almost perfect number with squarefree root would have a prime factor greater than\r\n1.6·1030 and more than 4.3·1027 prime factors; the corresponding bounds for quasiperfect\r\nnumbers improve the best known ones.\r\n For three prime factors there is a finite decision procedure for each k, run for all k ≤ 16 (k ≤\r\n20 when the odd part is squarefree), and a result valid for every k when the component\r\nquotient is at most 500,000.\r\n The squarefree case does not depend on primality: it is a question about integral points on K3\r\nsurfaces, some of whose genus-one slices violate the Hasse principle.\r\n For each number of prime factors and each k, the odd part of a counterexample is bounded\r\nexplicitly: b2\r\n < (5ωN)M with M = 4ω\r\n − 1.\r\n For odd almost perfect numbers with squarefree root the remaining step is identified exactly:\r\nan upper bound for certain primes that is equivalent to a level of distribution θ > 0.87 for the\r\nprimes in arithmetic progressions, which the Elliott–Halberstam conjecture would provide.\r\n There is no odd integer solution of ∏Φ(xi) = 2∏xi\r\n2\r\n − 1 with |xi| ≥ 3 and at most eight bases,\r\nprime or not, of either sign.\r\n All 212 formal statements, every computation and every figure are checked independently; the\r\nrecord of 94 checks is Appendix B of the report.\r\nEvery statement of this overview is proved, computed or formally verified in full in the four documents of\r\nthe Zenodo record https://doi.org/10.5281/zenodo.23674490, which a reader should consult for any detail.\r\nBold navy text marks what is new.","authors":[{"name":"Ibrahim Amin Shnoudi","affiliation":"Jordan University","orcid":""}],"type":"article","license":"CC-BY-4.0","keywords":["Mathematical"],"publication_date":"2026-10-27","version":1,"views":1,"downloads":0,"created_at":"2026-10-27 09:55:51","doi":"","source":"","source_url":"","files":[{"id":152,"name":"Shnoudi_prime_factors_odd_almost_perfect.pdf","size":342801,"mime":"application/pdf","downloads":0,"download_url":"https://lifeofresearch.com/articles/144/files/152/Shnoudi_prime_factors_odd_almost_perfect.pdf"}]}