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import FormalConjecturesUtilErdős Problem 445
[He00] Heath-Brown, D. R., Arithmetic applications of {K}loosterman sums. Nieuw Arch. Wiskd. (5) (2000), 380--384.
open Filter
namespace Erdos445
The property that there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$.
def Erdos445Prop (c : ℝ) (p n : ℕ) : Prop :=
∃ a b : ℕ,
n < a ∧ (a : ℝ) < (n : ℝ) + (p : ℝ) ^ c ∧
n < b ∧ (b : ℝ) < (n : ℝ) + (p : ℝ) ^ c ∧
a * b ≡ 1 [MOD p]
Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any $n\geq 0$, there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$?
This is discussed in this MathOverflow question [MathOverflow].
@[category research open, AMS 11]
theorem erdos_445 :
answer(sorry) ↔ ∀ c : ℝ, c > 1 / 2 →
∀ᶠ p : ℕ in atTop, p.Prime → ∀ n : ℕ, Erdos445Prop c p n := ⊢ True ↔ ∀ c > 1 / 2, ∀ᶠ (p : ℕ) in atTop, Nat.Prime p → ∀ (n : ℕ), Erdos445Prop c p n
All goals completed! 🐙
Heilbronn (unpublished) proved this for $c$ sufficiently close to $1$.
@[category research solved, AMS 11]
theorem erdos_445.variants.heilbronn :
∃ c₀ < 1, ∀ c : ℝ, c > c₀ →
∀ᶠ p : ℕ in atTop, p.Prime → ∀ n : ℕ, Erdos445Prop c p n := ⊢ ∃ c₀ < 1, ∀ c > c₀, ∀ᶠ (p : ℕ) in atTop, Nat.Prime p → ∀ (n : ℕ), Erdos445Prop c p n
All goals completed! 🐙
Heath-Brown [He00] used Kloosterman sums to prove this for all $c>3/4$.
@[category research solved, AMS 11]
theorem erdos_445.variants.heath_brown :
∀ c : ℝ, c > 3 / 4 →
∀ᶠ p : ℕ in atTop, p.Prime → ∀ n : ℕ, Erdos445Prop c p n := ⊢ ∀ c > 3 / 4, ∀ᶠ (p : ℕ) in atTop, Nat.Prime p → ∀ (n : ℕ), Erdos445Prop c p n
All goals completed! 🐙Small example: for $p=5$, $c=1$, $n=0$, the pair $(2,3) \in (0,5)$ satisfies $2 \cdot 3 = 6 \equiv 1 \pmod{5}$.
@[category test, AMS 11]
theorem erdos_445.test.small_example : Erdos445Prop 1 5 1 := ⊢ Erdos445Prop 1 5 1
refine ⟨2, 3, ⊢ 1 < 2 All goals completed! 🐙, ?_, ⊢ 1 < 3 All goals completed! 🐙, ?_, ⊢ 2 * 3 ≡ 1 [MOD 5] All goals completed! 🐙⟩
all_goals ⊢ ↑3 < ↑1 + ↑5; All goals completed! 🐙
end Erdos445