Erdős problem 123 · problems.science

Let $a,b,c>1$ be pairwise coprime integers. Is every large integer a sum of distinct numbers of the form $a^k b^l c^m$ ( $k,l,m\ge 0$ ), none dividing another?

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5 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

123.lean

Retained formal statement3 of 4

A simpler case: the set of numbers of the form $2^k 3^l$ ( $k, l ≥ 0$ ) is d-complete.

This was initially conjectured by Erdős in 1992, who called it a "nice and difficult" problem, but it was quickly proven by Jansen and others using a simple inductive argument: - If $n = 2m$ is even, apply the inductive hypothesis to $m$ and double all summands. - If $n$ is odd, let $3^k$ be the largest power of $3$ with $3^k ≤ n$ , and apply the inductive hypothesis to $n - 3^k$ (which is even).

FormalConjectures/ErdosProblems/123.leanErdos123.erdos_123.variants.powers_2_31 lineExact file

1Erdos123.IsDComplete (↑(Submonoid.powers 2) * ↑(Submonoid.powers 3))

SolvedStatement only, no proof

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