Year: 2026
Type: preprint
Abstract: Let $s(n):=\sum_{d\mid n,\,d < n} d$ denote the sum of the proper divisors of $n$. Distinct positive integers $n,m$ form an amicable pair if $s(n)=m$ and $s(m)=n$. Let $A(x)$ count the positive integers not exceeding $x$ that belong to an amicable pair. We report on a proof, found by ChatGPT Astra, that $$ A(x)\le x\exp\{-(1/2+o(1))\log x\,\log_3 x/\log_2 x\}$$ as $x\to\infty$. Here $\log_k$ denotes the $k$th iterate of the natural logarithm.
Source: arXiv (Cornell University)
Author: Pollack, Paul
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220722940