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import FormalConjecturesUtilErdős Problem 434
References:
[Ki02] Kiss, G., On the extremal Frobenius problem in a new aspect. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (2002), 139–142.
namespace Erdos434open Erdos434 FinsetA natural $n$ is representable as a set $A$ if it can be written as the sum of finitely many elements of $A$ (with repetition allowed).
abbrev Nat.IsRepresentableAs (n : ℕ) (A : Set ℕ) :=
∃ (S : Multiset ℕ), (∀ a ∈ S, a ∈ A) ∧ S.sum = nThe number of naturals that cannot be written as the sum of finitely many elements of the set $A$, with repetition allowed.
noncomputable abbrev Nat.NcardUnrepresentable (A : Set ℕ) :=
{ n : ℕ | ¬n.IsRepresentableAs A }.ncardLet $k \le n$. What choice of $A\subseteq{1, \dots, n}$ (with $\text{gcd}(A) = 1$) of size $|A| = k$ maximises the number of integers not representable as the sum of finitely many elements from $A$ (with repetitions allowed)? Is it ${n, n - 1, \dots, n - k + 1}$?
The maximal choice is indeed ${n, \dots, n - k + 1}$, as proved by Kiss [Ki02].
The Lean theorem assumes $2 \le k$. When $k = 1 < n$, the proposed singleton ${n}$ does not have gcd $1$; the gcd condition instead forces the unique admissible choice $A = {1}$.
@[category research solved, AMS 11, formal_proof using lean4 at "https://www.erdosproblems.com/forum/thread/434#post-4437"]
theorem erdos_434.parts.i (n k : ℕ) (hn : 1 ≤ n) (hk : 2 ≤ k) (h : k ≤ n) :
IsGreatest
{ Nat.NcardUnrepresentable S | (S : Finset ℕ) (_ : S ⊆ Finset.Icc 1 n)
(_ : #S = k) (_ : S.gcd id = 1) }
(Nat.NcardUnrepresentable <| answer(Set.Icc (n - k + 1 : ℕ) n)) := n:ℕk:ℕhn:1 ≤ nhk:2 ≤ kh:k ≤ n⊢ IsGreatest {x | ∃ S, ∃ (_ : S ⊆ Icc 1 n) (_ : #S = k) (_ : S.gcd id = 1), Nat.NcardUnrepresentable ↑S = x}
(Nat.NcardUnrepresentable (Set.Icc (n - k + 1) n))
All goals completed! 🐙For $2 \le k \le n$, the interval $A = {n, n - 1, \dots, n - k + 1}$ maximises the number of integers not representable as the sum of finitely many elements from $A$ (with repetitions allowed), as proved by Kiss [Ki02].
@[category research solved, AMS 11, formal_proof using lean4 at "https://www.erdosproblems.com/forum/thread/434#post-4437"]
theorem erdos_434.parts.ii : answer(True) ↔ ∀ᵉ (n ≥ 1) (k ≥ 2), k ≤ n →
IsGreatest
{ Nat.NcardUnrepresentable S | (S : Finset ℕ) (_ : S ⊆ Finset.Icc 1 n)
(_ : #S = k) (_ : S.gcd id = 1)}
(Nat.NcardUnrepresentable <| Set.Icc (n - k + 1 : ℕ) n) := ⊢ True ↔
∀ n ≥ 1,
∀ k ≥ 2,
k ≤ n →
IsGreatest {x | ∃ S, ∃ (_ : S ⊆ Icc 1 n) (_ : #S = k) (_ : S.gcd id = 1), Nat.NcardUnrepresentable ↑S = x}
(Nat.NcardUnrepresentable (Set.Icc (n - k + 1) n))
All goals completed! 🐙end Erdos434