PreprintCC BY 4.0arXiv
Amicable numbers
Abstract
Let s(n):=∑d| 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)≤ xexp{−(1/2+o(1))log x log3 x/log2 x} as x→∞. Here logk denotes the kth iterate of the natural logarithm.
Related
BibTeX
@misc{pollack2026amicable,
title = {{Amicable numbers}},
author = {Paul Pollack},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05661},
eprint = {2610.05661},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05661},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Amicable numbers
AU - Paul Pollack
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05661
UR - https://arxiv.org/abs/2610.05661
AB - Let s(n):=∑_d| 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)≤ xexp{−(1/2+o(1))log x log_3 x/log_2 x} as x→∞. Here log_k denotes the kth iterate of the natural logarithm.
ER -