Wikipedia number-theory item 173: For any given integer b which is not a perfect power and not of the form −4k^{4} for integer k, are…

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.

Fix an integer base $b$ with $|b|\ge2$. For $n\in\mathbb N$ define \[R_n(b)=1+b+\cdots+b^{n-1}=\frac{b^n-1}{b-1}.\] A repunit prime to base $b$ is a value $R_n(b)$ which is a positive prime. Here a perfect power is an integer $m^r$ with $m\in\mathbb Z$ and integer $r>1$; negative bases are included in this convention.

Statement recorded in the supplied source.

For every integer $b$ with $|b|\ge2$ which is neither a perfect power nor of the form $-4k^4$ for an integer $k$, is the set \[\{R_n(b):n\in\mathbb N,\ R_n(b)\text{ is a positive prime}\}\] infinite? [S2]

Short English statement

For every admissible positive or negative base, do its repunits take infinitely many positive prime values?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400173 (AMR-093-0173). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Wikipedia number-theory item 173: For any given integer b which is not a perfect power and not of the form −4k^{4} for integer k, are…. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 Wikipedia contributors, Repunit, revision 1363388346 (2026) Definition; Algebra factorization of generalized repunit numbers, infinitude conjecture paragraph. https://en.wikipedia.org/w/index.php?title=Repunit&oldid=1363388346.