Goormaghtigh 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 integers $z>1$ and $r>2$, write $R_r(z)=1+z+\cdots+z^{r-1}=(z^r-1)/(z-1)$. The known equalities are $R_3(5)=R_5(2)=31$ and $R_3(90)=R_{13}(2)=8191$.

Statement recorded in the supplied source.

Are the only integer solutions of $R_m(x)=R_n(y)$ with $x>y>1$ and $m,n>2$ the two quadruples $(x,y,m,n)=(5,2,3,5)$ and $(90,2,3,13)$? [S2]

Short English statement

Are thirty-one and eight thousand one hundred ninety-one the only integers which have these two distinct-base repunit descriptions with at least three digits?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400108 (AMR-093-0108). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Tomohiro Yamada, On the Ratat--Goormaghtigh equation and integer points close to the graph of a smooth function, arXiv:2510.11252v1 (2025) Introduction, equation (1.1), p. 1. https://arxiv.org/pdf/2510.11252.
• S3 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Goormaghtigh conjecture. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.