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import FormalConjecturesUtilAgrawal's 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 $(X-1)^n \equiv X^n-1 \pmod{n, X^r-1}$ holds, then $n$ is either prime or $n^2 \equiv 1 \pmod{r}$.
open Polynomial
namespace AgrawalConjecture
Agrawal's Primality Conjecture.
Does the congruence $(X-1)^n \equiv X^n - 1 \pmod{n, X^r-1}$ imply $n$ is prime (with a specific exception for $n^2 \equiv 1 \pmod{r}$)?
While the "if" direction is a known theorem, the "only if" direction remains a conjecture.
@[category research open, AMS 11]
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) := ⊢ True ↔
∀ (n r : ℕ),
n > 1 →
r > 0 →
n.gcd r = 1 →
let R := (ZMod n)[X];
let X := X;
let I := Ideal.span {X ^ r - 1};
(Ideal.Quotient.mk I) ((X - 1) ^ n) = (Ideal.Quotient.mk I) (X ^ n - 1) → Nat.Prime n ∨ ↑n ^ 2 = 1
All goals completed! 🐙
Roman B. Popovych Conjecture. A stronger version of Agrawal's conjecture, which also considers the congruence $(X+2)^n \equiv X^n + 2 \pmod{n, X^r-1}$. If both congruences hold, then $n$ is either prime or $n^2 \equiv 1 \pmod{r}$. This variant was proposed by Roman B. Popovych in 2018.
@[category research open, AMS 11]
theorem agrawal_conjecture.variants.popovych :
∀ (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) →
Ideal.Quotient.mk I ((X + 2)^n) = Ideal.Quotient.mk I (X^n + 2) →
(n.Prime ∨ (n^2 : ZMod r) = 1) := ⊢ ∀ (n r : ℕ),
n > 1 →
r > 0 →
n.gcd r = 1 →
let R := (ZMod n)[X];
let X := X;
let I := Ideal.span {X ^ r - 1};
(Ideal.Quotient.mk I) ((X - 1) ^ n) = (Ideal.Quotient.mk I) (X ^ n - 1) →
(Ideal.Quotient.mk I) ((X + 2) ^ n) = (Ideal.Quotient.mk I) (X ^ n + 2) → Nat.Prime n ∨ ↑n ^ 2 = 1
All goals completed! 🐙
end AgrawalConjecture