Infinitely Many Perfect 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 $\sigma(n)=\sum_{d\mid n,\ d>0}d$ be the sum of its positive divisors. A perfect number satisfies $\sigma(n)=2n$, equivalently $n$ is the sum of its positive proper divisors. Both even and odd positive integers are included.

Statement recorded in the supplied source.

Is the set $\{n\in\mathbb N:\sigma(n)=2n\}$ infinite? Equivalently, for every $B\in\mathbb N$, must there exist $n>B$ with $\sigma(n)=2n$? [S2] The Euclid--Euler description identifies the even perfect numbers with $2^{p-1}(2^p-1)$ for prime Mersenne numbers $2^p-1$. Infinitely many Mersenne primes would imply the target, but the recorded question does not exclude possible odd perfect numbers.

Short English statement

Are there infinitely many positive integers equal to the sum of their proper positive divisors, including both parity classes?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1248 (NT-041). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 John Cowles and Ruben Gamboa, Perfect Numbers in ACL2, EPTCS 192 (2015), pp. 53--59, Sections 1 and 2; doi:10.4204/EPTCS.192.5. https://arxiv.org/pdf/1509.06081.