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
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…
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