Are there any pairs of betrothed numbers which have same parity

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 $\sigma(n)=\sum_{d\mid n,\ d>0}d$ and $s(n)=\sigma(n)-n$. Distinct positive integers $m,n$ form a betrothed pair when $s(m)=n+1$ and $s(n)=m+1$.

Statement recorded in the supplied source.

Does there exist a betrothed pair $m,n$ with $m\equiv n\pmod2$? [S2]

Short English statement

Can a betrothed pair consist of two integers of the same parity?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400020 (AMR-093-0020). 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 betrothed numbers which have same parity. 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.