Open Research Index

Erdős #257: irrationality for every reciprocal-summable support and every integer base

Will Cook

Palomar Registrypalomar/PALOMAR-2026-10-21-000018Oct 21, 2026math.NTscore 0

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…

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