Infinitely Many Amicable Pairs

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

An amicable pair is an unordered pair $\{a,b\}$ of distinct positive integers satisfying $\sigma(a)=\sigma(b)=a+b$. Infinitude means that there are infinitely many different such pairs, or equivalently examples with unbounded larger member. No parity or ratio restriction is added. [S1] [S2]

Statement recorded in the supplied source.

Are there infinitely many pairs of amicable numbers? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Prove an infinite family of distinct amicable pairs or prove that only finitely many exist. [S1]

Short English statement

Do amicable pairs continue to occur at arbitrarily large sizes?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1254 (NT-047), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 Mariano Garcia, A Million New Amicable Pairs, Journal of Integer Sequences 4 (2001). https://cs.uwaterloo.ca/journals/JIS/VOL4/GARCIA/millionc.html. Bibliographic information from the supplied record.

• S3 Erdos Problems, Problem 830: Are there infinitely many amicable pairs?. https://www.erdosproblems.com/830. Bibliographic information from the supplied record.