Claims

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1959_10_01_birch: Proves that for coprime a and b every large integer is a sum of distinct numbers of the form a to the k times b to the l; refereed in 1959, credited by the site's curator, and later formalized in Lean by others.

1960_01_01_cassels: Cassels's 1960 Theorem I, a circle-method criterion under which every large integer is a sum of distinct elements of a set, of which the completeness of the numbers a to the k times b to the l is an immediate consequence.

2000_01_01_hegyvari: Hegyvári (Acta Math. Hungar. 86 (2000)) proves that the numbers p^a q^b with b at most an explicit K(p,q) already form a complete set, the first effective form of Birch's theorem; refereed.

2015_07_08_bergelson_simmons: Bergelson and Simmons (Acta Arith. 177 (2017)) prove that the numbers a^n b^k with at most 4a-4 distinct exponents k, including 0, form a complete set, so K(a,b) is at most 4a-5; refereed.

2017_05_10_fang_chen: Fang and Chen (Acta Arith. 178 (2017)) give explicit towers K and B such that every integer at least B is a sum of distinct numbers p^a q^b with b at most K; refereed.

2024_01_01_yu: Yu (Publ. Math. Debrecen 104 (2024)) proves that every large n is a sum of distinct numbers p^a q^b, all greater than c n/(log n)^(1+eps); refereed.

2026_09_03_song_yue: Claims that for coprime p and q above one the numbers p to the a times q to the b with b at most 2p minus 3 already form a complete set, a quantitative strengthening of the Erdős–Birch theorem; posted in 2026 and unreviewed.

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