Open Research Index

Erdős 252: irrationality of the factorial divisor-sum series

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Palomar Registrypalomar/PALOMAR-2026-10-12-000001Oct 12, 2026math.NTscore 0

For every natural number k, the series whose nth term is the sum of the kth powers of the positive divisors of n, divided by n factorial, is irrational. This gives the positive-exponent statement of Erdős problem 252 and also includes exponent zero. The proof combines factorial-tail integrality, a finite coprime dilation grid with exact finite-difference cancellation, and incompatible progression means obtained by a fresh-prime refinement. It has no prime-pattern conjecture or unproved number-theoretic hypothesis. The Lean development also proves convergence and the equivalence of its…

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