Claims

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1996_01_01_burr_erdos_graham_li: Section 3 of the 1996 Acta Arithmetica paper gives the largest integer not a sum of distinct positive-exponent powers from {3,4,7}, {3,5,7,13}, {3,6,7,13,21} and {3,4,5}, settling the second question at k = 1; refereed.

2015_07_08_bergelson_simmons: Bergelson and Simmons (Acta Arith. 2017) prove that the powers of four disjoint sets of bases, three with reciprocal sums at least 1 and one with gcd 1, are strongly complete, answering the second question there; refereed.

2025_11_29_alexeev: A Lean proof found by Aristotle from Harmonic and posted by Boris Alexeev: when the reciprocals of d_i - 1 sum to at least 1, every integer is a sum of one number with base-d_i digits 0 and 1 for each i; claimed.

2026_09_09_fan: Theorem 1.5 of Fan's arXiv preprint (version of 9 September 2026) proves the powers of bases split into two parts with reciprocal sums at least 1 and one part with gcd 1 are strongly complete, answering the second question there.

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