Claims

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1963_03_17_graham: Graham's 1963 theorem that every large integer is a sum of distinct integers above any bound whose reciprocals sum to one gives the strong completeness of n + 1/n, the case p(x) = x of the problem; refereed.

2025_09_15_van_doorn: Wouter van Doorn's note that the set of n squared plus 1/n is strongly complete, the case p(x) = x^2 of the problem, adapting Graham's 1963 proof with Alekseyev's theorem; an unrefereed note credited in the site's remarks.

2026_05_03_price_barreto: The GPT 5.5 Pro argument for Problem 283, posted by Liam Price and edited by Kevin Barreto, yields that p(n) + 1/n is strongly complete for every rational polynomial with positive leading coefficient; reviewed, accepted by the site.

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