Agrawal's conjecture

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 positive coprime integers $n,r$, polynomial congruence modulo $n$ and $X^r-1$ means equality in the quotient ring $(\mathbb Z/n\mathbb Z)[X]/(X^r-1)$; equivalently the difference belongs to the ideal $(n,X^r-1)$ in $\mathbb Z[X]$. Primality always means an integer greater than one with no positive divisors other than one and itself.

Statement recorded in the supplied source.

Whenever $\gcd(n,r)=1$ and $(X-1)^n\equiv X^n-1\pmod{n,\ X^r-1}$, must either $n$ be prime or $n^2\equiv1\pmod r$? [S2]

Short English statement

Does this single polynomial identity force either primality or the stated square congruence for every coprime pair of positive integers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400114 (AMR-093-0114). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Agrawal's conjecture. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 Wikipedia contributors, Agrawal's conjecture, revision 1295268932 (2026) Formal statement. https://en.wikipedia.org/w/index.php?title=Agrawal%27s+conjecture&oldid=1295268932.