{"id":"openai-math/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers","pdf":"https://github.com/openai/math/blob/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026","tags":["number-theory","totient-function","prime-numbers"],"score":0,"title":"Weighted dilation graphs, smooth shifted primes and totient fibers","domain":"Number theory","family":"011","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"We prove Erdős's conjecture on the largest fibers of Euler's totient function: for every ε > 0, infinitely many positive integers n have more than $`n^{1-\\varepsilon }`$ preimages. We also show that, for every fixed δ > 0, there are at least $`x^{1-o(1)}`$ primes p in $`2x\\lt p\\le5x`$ whose predecessors have no prime factor exceeding x δ .","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":false,"family_title":"Prime-factor statistics of $`p-1`$"}