FormalConjectures.ErdosProblems.«246»

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import FormalConjecturesUtil

Erdős Problem 246

References:

erdosproblems.com/246

[Bi59] Birch, B. J., Note on a problem of Erd\H{o}s. Proc. Cambridge Philos. Soc. (1959), 370-373.

[Ca60] Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math. (Szeged) (1960), 111-124.

[FaCh17] Fang, Jin-Hui and Chen, Yong-Gao, A quantitative form of the {E}rd\H{o}s-{B}irch theorem. Acta Arith. (2017), 301--311.

[He00b] Hegyv'{a}ri, N., On the completeness of an exponential type sequence. Acta Math. Hungar. (2000), 127--135.

[Yu24] Yu, Wang-Xing, On the representation of an exponential type sequence. Publ. Math. Debrecen (2024), 253--261.


namespace Erdos246

The set ${a^k b^l : k,l \geq 0}$.

def Gamma (a b : ℕ) : Set ℕ :=
  {x | ∃ k l : ℕ, x = a ^ k * b ^ l}

Let $(a,b)=1$. The set ${a^kb^l: k,l\geq 0}$ is complete - that is, every large integer is the sum of distinct integers of the form $a^kb^l$ with $k,l\geq 0$.

We state the nontrivial case $a,b\geq 2$, proved by Birch [Bi59].

@[category research solved, AMS 11,
  formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos246.lean"]
theorem declaration uses 'sorry'erdos_246 (a b : ℕ) (ha : 2 ≤ a) (hb : 2 ≤ b) (hab : Nat.Coprime a b) :
    IsAddComplete (Gamma a b) := bya:ℕb:ℕha:2 ≤ ahb:2 ≤ bhab:a.Coprime b⊢ IsAddComplete (Gamma a b)
  sorryAll goals completed! 🐙


end Erdos246