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import FormalConjecturesUtilErdős Problem 433
References:
P. Erdős and R. L. Graham, On a linear Diophantine problem of Frobenius, Acta Arithmetica 21 (1972), 399–408.
J. Dixmier, Proof of a conjecture by Erdős and Graham on Frobenius' coin problem, Proceedings of the AMS 109 (1990), 567–577.
namespace Erdos433open Filteropen scoped TopologyThe Frobenius number of a finite set $A$: the greatest natural number outside the additive submonoid generated by $A$.
noncomputable def frobeniusNumber (A : Finset ℕ) : ℕ :=
sSup {m | m ∉ AddSubmonoid.closure (A : Set ℕ)}The largest Frobenius number of a $k$-element subset of ${1,\ldots,n}$ whose elements have greatest common divisor one.
noncomputable def g (k n : ℕ) : ℕ :=
sSup {m | ∃ A : Finset ℕ, A ⊆ Finset.Icc 1 n ∧ A.card = k ∧ A.gcd id = 1 ∧
m = frobeniusNumber A}For every fixed $k \geq 2$, Erdős and Graham conjectured that $$g(k,n) \sim \frac{n^2}{k-1}.$$ The conjecture was proved by Dixmier [Di90].
@[category research solved, AMS 11]
theorem erdos_433 : answer(True) ↔ ∀ k, 2 ≤ k →
Tendsto (fun n : ℕ => (g k n : ℝ) / ((n : ℝ) ^ 2 / ((k : ℝ) - 1))) atTop (𝓝 1) := ⊢ True ↔ ∀ (k : ℕ), 2 ≤ k → Tendsto (fun n ↦ ↑(g k n) / (↑n ^ 2 / (↑k - 1))) atTop (𝓝 1)
All goals completed! 🐙end Erdos433