Math-4200 · mathematics · Number theory

Are there any composite c satisfying 2^{c − 1} ≡ 1 (mod c^{2})

Status Open · unsolvedmath.comLLM status Open unverified

Statement

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 composite integer is an integer greater than one which is not prime. Congruence $u\equiv v\pmod m$ means $m$ divides $u-v$.

Statement recorded in the supplied source.

Does there exist a composite integer $c>1$ such that $c^2\mid2^{c-1}-1$? [S2]

Short English statement

Can a composite integer satisfy the base-two Wieferich congruence modulo its square?

Sources

S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400141 (AMR-093-0141). 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 composite c satisfying 2^{c − 1} $\equiv$ 1 (mod c^{2}). https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
S3 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A240719: Numbers k such that 2^k == 1 (mod (k+1)^2)., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A240/A240719.seq.
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.

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

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Record details

Subjects: Number theory
Collection: Top Open Problems
Mathematics rank: 4200
UnsolvedMath: Catalogue entry