Feit–Thompson conjecture

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 distinct primes $p,q$, the integers \[ A_{p,q}=\frac{p^q-1}{p-1}=\sum_{j=0}^{q-1}p^j,\qquad B_{p,q}=\frac{q^p-1}{q-1}=\sum_{j=0}^{p-1}q^j \] are geometric sums. The relation $a\mid b$ means $b=at$ for some integer $t$.

Statement recorded in the supplied source.

For every ordered pair of distinct primes $p,q$, is $A_{p,q}\nmid B_{p,q}$? [S2]

Short English statement

Can the stated prime-base geometric sum ever divide the analogous sum with the two primes interchanged?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400123 (AMR-093-0123). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Feit–Thompson conjecture. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 Wikipedia contributors, Feit–Thompson conjecture, revision 1353413994 (2026) Lead. https://en.wikipedia.org/w/index.php?title=Feit%E2%80%93Thompson+conjecture&oldid=1353413994.