Are there any pairs of relatively prime amicable numbers

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $n\in\mathbb N$, let $s(n)=\sum_{d\mid n,\ 1\le d<n}d$. Distinct positive integers $m,n$ are amicable if $s(m)=n$ and $s(n)=m$. They are relatively prime if $\gcd(m,n)=1$.

Statement recorded in the supplied source.

Does there exist an amicable pair $m,n$ with $\gcd(m,n)=1$? [S2]

Short English statement

Can two relatively prime positive integers form an amicable pair?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400021 (AMR-093-0021). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Are there any pairs of relatively prime amicable numbers. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 Zhipeng Chen, Haolun Tang and Jingyi Zhan, Formalization of Amicable Numbers Theory, arXiv:2601.07444 (2026) Definitions 2.2 and 2.7, and Section 2.3. https://arxiv.org/pdf/2601.07444.