Agrawal's conjecture — open problem in number theory

Agrawal's conjecture

Agrawal's Primality Conjecture. Does the congruence (X-1)^n ≡ X^n - 1 pmodn, X^r-1 imply n is prime (with a specific exception for n^2 ≡ 1 pmodr)? While the "if" direction is a known theorem, the "only if" direction remains a conjecture.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-agrawal,
  title        = {Agrawal's conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/agrawal}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Agrawal's Primality Conjecture.

Does the congruence $(X - 1)^{n} \equiv X^{n} - 1 (mod n, X^{r} - 1)$ imply $n$ is prime (with a specific exception for $n^{2} \equiv 1 (mod r)$ )?

While the "if" direction is a known theorem, the "only if" direction remains a conjecture.

Agrawal's conjecture is a stronger version of the theorem that forms the basis of the AKS primality test. If true, it would significantly improve the efficiency of primality testing.

The conjecture states that for coprime $n$ and $r$ , if the polynomial congruence $(X - 1)^{n} \equiv X^{n} - 1 (mod n, X^{r} - 1)$ holds, then $n$ is either prime or $n^{2} \equiv 1 (mod r)$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Agrawal. answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem agrawal_conjecture :
  answer(sorry) ↔
    ∀ (n r : ℕ), n > 1 → r > 0 → n.gcd r = 1 →
      let R := Polynomial (ZMod n)
      let X : R := Polynomial.X
      let I : Ideal R := Ideal.span ({X^r - 1} : Set R)
      Ideal.Quotient.mk I ((X - 1)^n) = Ideal.Quotient.mk I (X^n - 1) →
      (n.Prime ∨ (n^2 : ZMod r) = 1)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

agrawal_conjecture.variants.popovych — Roman B. Popovych Conjecture. A stronger version of Agrawal's conjecture, which also considers the congruence (X+2)^n ≡ X^n + 2 pmodn, X^r-1.

References

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.