Are there infinitely many Wieferich primes

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.

A Wieferich prime (to base $2$) is a prime $p$ with $2^{p-1}\equiv1\pmod{p^2}$, meaning $p^2$ divides $2^{p-1}-1$. This forces $p$ to be odd.

Statement recorded in the supplied source.

Are there infinitely many such primes? [S2]

Short English statement

Are there infinitely many primes satisfying the base-two Fermat congruence modulo their squares?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400164 (AMR-093-0164). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Are there infinitely many Wieferich primes. 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, A001220: Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A001/A001220.seq.