{"id":"palomar/PALOMAR-2026-10-21-000020","url":"https://github.com/wcook04/plectis-erdos-lean/tree/52f29ad173b04e3bac941b3663f2b9aebe5de0bb","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-21-000020-v1.json","repo":"https://github.com/wcook04/plectis-erdos-lean","tags":["math.NT","math.CA","math.DS","msc:11J72","msc:11A25","msc:11A07","msc:11B37","msc:11B83","msc:28A80","msc:11K16"],"score":0,"title":"Erdős problem #257: all base irrationality of the reciprocal Mersenne subseries for every infinite reciprocal summable support, with weighted mass and positive cover extensions, achievement set geometry, and exact greedy membership criteria","trust":"high","domain":"math.NT","source":"palomar","authors":["Will Cook"],"reviews":0,"subject":"math.NT","summary":"Erdős problem 257 asks whether the sum over a in A of 1/(2^a - 1) is irrational for every infinite set A of positive integers. The principal theorem compared here settles the whole reciprocal summable regime at every integer base: if A is infinite and the sum over a in A of 1/a converges, then the sum over a in A of 1/(b^a - 1) is irrational for every integer b at least 2 (ReciprocalSupport.irrational_supportPowerSeries_of_summable_reciprocal). No pairwise coprimality, periodicity, density, or powerful support hypothesis appears. Paul Erdős printed the pairwise coprime case of exactly this…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-21","reactions":{},"formalized":true}