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Erdős Problem #124 - Discussion thread

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In [BEGL96], the problem is formulated in a way that only allows powers of d i 𝑑 𝑖 greater than d 0 i =1 𝑑 𝑖 0 = 1 to be added. However, in [Er97] and [Er97e], it's formulated so that 1 1 s are allowed. Incidentally, this means that all the proofs in [BEGL96] actually prove slightly stronger statements. [Note: this comment was written before 2025/12/01, when the problem text was updated.] Aristotle from Harmonic has solved this problem all by itself, working only from the formal statement! Type-check it online! A formal statement of the conjecture was available in the Formal Conjectures project. Unfortunately, there is a typo in that statement, wherein the comment says ≥1 ≥ 1 in the display-style equation while the corresponding Lean says "= 1". (That makes the statement weaker.) Accordingly, I have also corrected that issue and included a proof of the corrected statement. Finally, I removed a lot of what I believed were unnecessary aspects of the statement, and Aristotle

124 Discussion Thread | Erdős Problems Forum Inbox Favourites Tags More FAQ Prizes Problem Lists Definitions Links Forum Menu Inbox Favourites Tags FAQ Prizes Problem Lists Definitions Links Go Go Dual View Random Solved Random Open OPEN This is open, and cannot be resolved with a finite computation. For any $d\geq 1$ and $k\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\geq k$. Let $3\leq d_1<d_2<\cdots <d_r$ be integers such that\[\sum_{1\leq i\leq r}\frac{1}{d_r-1}\geq 1.\]Can all sufficiently large integers be written as a sum of t

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