{"id":"hexagon/2610.00454","pdf":"https://hexagonmath.org/pdf/2610.00454v1","url":"https://hexagonmath.org/abs/2610.00454v1","tags":["msc:11A25","msc:11A41","msc:11A51","msc:11D68","msc:11N36","Carmichael numbers","Euler's totient function","Fermat primes","Lehmer's totient problem","large sieve"],"score":0,"title":"Lehmer's totient problem with fewer than sixteen prime factors","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Deep Bhattacharjee"],"reviews":0,"subject":"math.NT","summary":"A composite number $n$ whose totient divides $n-1$ was known to have at least fourteen prime factors. We prove that it has at least sixteen, at least $16001$ if $(n-1)/\\varphi(n)\\ge3$, and more than $10^8$ if $3\\mid n$, and that $n<2^{2^{k-7}}$ if $n$ has $k$ prime factors. For the companion equation $\\varphi(n)\\mid n+1$ we show that the nine known solutions are the only ones with at most eight prime factors, and that a further solution prime to $3$ has at least sixteen. Each statement is reduced by hand to a finite computation, which independent programs carry out. Lehmer's conjecture…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code"],"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}