Claims

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1948_07_08_erdos: Erdős's 1948 theorem that the sum of 1/(t^n - 1) over all n is irrational for every integer t with |t| > 1 settles the problem for the support N and, through the base t = 2^d, for every set of multiples dN.

1965_12_13_erdos: Erdős's 1968 paper in the Mathematics Student proves that the sum of 1/(t^n - 1) over a pairwise coprime set with convergent reciprocal sum is irrational for every integer base t at least 2, a class of supports.

2019_08_14_duverney_tachiya: Duverney and Tachiya's 2019 paper proves the sum of 1/(q^n - 1) over the products of powers below s of a pairwise coprime, polynomially bounded sequence, the squarefree integers among them, irrational.

2025_12_01_tao_teravainen: Tao and Teräväinen's theorem that the sum over primes p of 1/(2^p - 1) is irrational settles the problem for the support consisting of the primes, the case that is Problem 69.

2026_09_11_cook: Cook's 2026 note, drafted with AI agents, claims that the sum of 1/(b^a - 1) over any infinite set with convergent reciprocal sum is irrational at every integer base b at least 2.

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