For any given integer a > 0, are there infinitely many primes p such that a^{p − 1} ≡ 1 (mod p^{2})

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 $a>0$. A prime $p$ satisfies the base-$a$ Wieferich condition when $a^{p-1}\equiv1\pmod{p^2}$. The congruence itself excludes primes dividing $a$. For $a=1$ it holds for every prime.

Statement recorded in the supplied source.

For every fixed integer $a>0$, are there infinitely many primes $p$ with $a^{p-1}\equiv1\pmod{p^2}$? [S2]

Short English statement

Does every positive integer base have infinitely many Wieferich primes? The base one case holds for all primes.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400172 (AMR-093-0172). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, For any given integer a > 0, are there infinitely many primes p such that a^{p − 1} $\equiv$ 1 (mod p^{2}). https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 Jonathan Sondow, Lerch quotients, Lerch primes, Fermat--Wilson quotients, and the Wieferich-non-Wilson primes $2,3,14771$, arXiv:1110.3113v5 (2012) Introduction and Section 3.2, pp. 1--2 and 9--10. https://arxiv.org/pdf/1110.3113.
• S4 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A258045: Table T(b, m) of largest exponents k such that for p = prime(m) and base b > 1 the congruence b^(p-1) == 1 (mod p^k) is satisfied, or 0 if no such k exists, read by antidiagonals (downwards)., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A258/A258045.seq.