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import FormalConjecturesUtilErdős Problem 1214
[CoSc97] Corrales-Rodrigáñez, Capi and Schoof, René, The support problem and its elliptic analogue. J. Number Theory (1997) [Volume 64, Issue 2], 276--290.
namespace Erdos1214
Let $x,y\geq 1$ be integers such that, for all $n\geq 1$, the set of primes dividing $x^{n}-1$ is equal to the set of primes dividing $y^n-1$. Must $x=y$?
Erdős asked this at a 1988 number theory conference in Banff.
A positive answer was given by Corrales-Rodrigáñez and Schoof [CoSc97].
@[category research solved, AMS 11]
theorem erdos_1214 :
answer(True) ↔ ∀ x y : ℕ, x ≥ 1 → y ≥ 1 →
(∀ n : ℕ, n ≥ 1 → { p : ℕ | p.Prime ∧ p ∣ x ^ n - 1 } = { p : ℕ | p.Prime ∧ p ∣ y ^ n - 1 }) →
x = y := ⊢ True ↔
∀ (x y : ℕ), x ≥ 1 → y ≥ 1 → (∀ n ≥ 1, {p | Nat.Prime p ∧ p ∣ x ^ n - 1} = {p | Nat.Prime p ∧ p ∣ y ^ n - 1}) → x = y
All goals completed! 🐙
end Erdos1214