On several irrationality problems for Ahmes series

TL;DR

This paper investigates the irrationality of series of distinct unit fractions, constructs sequences with specific rational sum properties, and addresses longstanding questions in number theory related to the growth of such sequences.

Contribution

It introduces new constructions of sequences with prescribed rational sums and irrationality properties, extending previous results and answering open questions by Erdős and Stolarsky.

Findings

Constructed double exponentially growing sequences with rational sum properties.

Proved the existence of sequences with all sums rational, countering Stolarsky's conjecture.

Extended results to multiple series and addressed growth rate questions.

Abstract

Using basic tools of mathematical analysis and elementary probability theory we address several problems on the irrationality of series of distinct unit fractions, $\sum_k 1/a_k$. In particular, we study subseries of the Lambert series $\sum_k 1/(t^k-1)$ and two types of irrationality sequences $(a_k)$ introduced by Paul Erd\H{o}s and Ronald Graham. Next, we address a question of Erd\H{o}s, who asked how rapidly a sequence of positive integers $(a_k)$ can grow if both series $\sum_k 1/a_k$ and $\sum_k 1/(a_k+1)$ have rational sums. Our construction of double exponentially growing sequences $(a_k)$ with this property generalizes to any number $d$ of series $\sum_k 1/(a_k+j)$, $j=0,1,2,\ldots,d-1$, and, in particular, also gives a positive answer to a question of Erd\H{o}s and Ernst Straus on the interior of the set of $d$-tuples of their sums. Finally, we prove the existence of a sequence $(a_k)$ such that all well-defined sums $\sum_k 1/(a_k+t)$, $t\in\mathbb{Z}$, are rational numbers, giving a negative answer to a conjecture by Kenneth Stolarsky.