Claims

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1963_03_17_graham: Graham's 1963 theorem that every integer above 77 is a sum of distinct positive integers whose reciprocals sum to one, the case p(x) = x of the problem with the exact threshold; refereed.

2018_01_18_alekseyev: Alekseyev's theorem that 8542 is the largest integer not a sum of squares of distinct positive integers whose reciprocals sum to one, the case p(x) = x^2 of the problem; a published book chapter, refereeing not recorded.

2025_08_31_van_doorn: Van Doorn's manuscript reduces the binomial case p(x) = ax^d + b to a finite computation and carries it out for x + b (b up to 5000), 5x + b (b below 375, prime to 5) and x^2 + b (b up to 800); unrefereed, with Lean for part.

2026_05_03_price: The general case of the problem, with 1 replaced by any positive rational, by an argument that GPT 5.5 Pro generated at Price's prompting; accepted by the site's curator, with third-party Lean formalizations not built locally.

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