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Source. Theorem 4.2, its one-line proof and the consequence for $e^m$, preprint p. 14. Read on the rendered page. The edition read is identified on the source card.

Statement

Let $m$ be a positive integer and $(b_n)_{n\ge1}$ a sequence of positive integers such that $b_N,b_{N+1},\ldots,b_{4N}$ form an arithmetic progression for infinitely many positive integers $N$. Suppose that $b_n=o(n\log n)$. Then $\sum_{n=1}^{\infty}m^{b_n}/n!$ is irrational.

The case $b_n=n$ (p. 14). It gives $e^m=\sum_{n=0}^{\infty}m^n/n!\notin\mathbb{Q}$ for every positive integer $m$.

Proof pointer

The printed proof is "Apply Theorem 3.2." With $a_n=m^{b_n}$, an arithmetic run of $b_n$ is a geometric run of $a_n$ with positive terms, and $b_n=o(n\log n)$ gives $a_n=o(n^{n/7})$.

Dependencies

Theorem 3.2 of the same paper.

Bears on

No catalog problem directly.