Are there any Wieferich primes in base 47

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 an integer base $a\ge1$, a prime $p$ is Wieferich to base $a$ if $p\nmid a$ and $a^{p-1}\equiv1\pmod{p^2}$. Equivalently $p$ divides the integer Fermat quotient $(a^{p-1}-1)/p$.

Statement recorded in the supplied source.

Does there exist a prime $p\ne47$ such that $47^{p-1}\equiv1\pmod{p^2}$? [S2]

Short English statement

Does any prime satisfy the square-modulus Fermat congruence for base forty-seven?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400143 (AMR-093-0143). 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 any Wieferich primes in base 47. 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, equation (1), and Section 3.2, pp. 9--10. https://arxiv.org/pdf/1110.3113.