Can a prime p satisfy $2^{p-1}\equiv 1\pmod{p^2}$ and $3^{p-1}\equiv 1\pmod{p^2}$ simultaneously

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 a prime $p\nmid a$, being Wieferich to base $a$ means $a^{p-1}\equiv1\pmod{p^2}$. A prime satisfying this for both bases $2$ and $3$ must be different from $2$ and $3$.

Statement recorded in the supplied source.

Does there exist a prime $p$ satisfying both $2^{p-1}\equiv1\pmod{p^2}$ and $3^{p-1}\equiv1\pmod{p^2}$ simultaneously? [S2]

Short English statement

Can one prime be Wieferich for both bases two and three?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400168 (AMR-093-0168). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Can a prime p satisfy $2^{p-1}\equiv 1\pmod{p^2}$ and $3^{p-1}\equiv 1\pmod{p^2}$ simultaneously. 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.