Do odd perfect numbers exist?

Still open as of 2026.

Around 300 BC Euclid showed that $2^{p-1}(2^p-1)$ is perfect whenever $2^p-1$ is prime, and Euler proved two thousand years later that every even perfect number has this form. Odd perfect numbers are another matter, and Euler himself called their existence a most difficult question. He showed that an odd perfect number must have the shape $q^{\alpha }{m}^2$ with $q$ prime and $q\equiv \alpha \equiv 1(\mathrm{mod}4)$. Modern work shows that one would have to exceed $10^{1500}$ and have at least 10 distinct prime factors. Carl Pomerance has given a heuristic argument that none exist, but there is no proof.

References

• Perfect number (Wikipedia)