{"id":"palomar/PALOMAR-2026-10-21-000018","url":"https://github.com/wcook04/plectis-erdos-lean/tree/c39cea6a976747d132a848d26f110b9c608d5a45","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-21-000018-v1.json","repo":"https://github.com/wcook04/plectis-erdos-lean","tags":["math.NT","msc:11J72","number-theory","irrationality","erdos-problems","diophantine-approximation"],"score":0,"title":"Erdős #257: irrationality for every reciprocal-summable support and every integer base","trust":"high","domain":"math.NT","source":"palomar","authors":["Will Cook"],"reviews":0,"subject":"math.NT","summary":"For every infinite set A of exponents whose reciprocal sum converges and every integer base b at least two, the series with terms 1/(b^a - 1) supported on A is irrational. The theorem settles the whole reciprocal-summable support regime at once and uniformly in the base, with no pairwise-coprimality, periodicity, density or powerful-support hypothesis, and with the zero exponent normalised to zero by real division. Any counterexample to the universal Erdős Problem 257 must therefore be supported on a set whose reciprocal sum diverges. The mechanism produces close returns of shifted binary…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-21","reactions":{},"formalized":true}