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import FormalConjecturesUtilErdős Problem 246
References:
[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 Erdos246The 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 erdos_246 (a b : ℕ) (ha : 2 ≤ a) (hb : 2 ≤ b) (hab : Nat.Coprime a b) :
IsAddComplete (Gamma a b) := a:ℕb:ℕha:2 ≤ ahb:2 ≤ bhab:a.Coprime b⊢ IsAddComplete (Gamma a b)
All goals completed! 🐙end Erdos246