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Source. Printed p. 95, the italicized statement following "Remarquons que nous avons incidemment démontré la proposition générale suivante", and the paragraph after it. Read on the page image (physical PDF p. 3).

Statement

Let $c_n$ be a sequence of integers $>0$ such that

$$n^{-2}c_n\to0,\qquad n\to\infty .$$

Whenever the sequence

$$\frac{c_n}{n}-\left[\frac{c_n}{n}\right]\qquad(3)$$

does not tend to $1$, the sum of the series $\sum_{n\ge1}c_n/n!$ is irrational.

Proof

The paper gives none beyond "incidemment démontré": Steps 1--3 of the main theorem use only $c_n>0$ and $c_n=o(n^2)$. In detail: if $\sum c_n/n!=a/b$, then for every $k>b$ the number $\{c_k/k\}+R_k$ is a positive integer, where $R_k=\sum_{j\ge1}c_{k+j}/(k(k+1)\cdots(k+j))$ satisfies $0<R_k\le13\sup_{m\ge k}c_m/m^2\to0$; hence $\{c_k/k\}\ge1-R_k$ for all $k>b$, and since $\{c_k/k\}<1$ this forces $\{c_k/k\}\to1$. So a rational sum makes (3) tend to $1$, which is the proposition in contrapositive form. $\blacksquare$ (Complete by reference to the steps on the main-theorem page; author-recorded like them.)

Remarks

• The paper continues (p. 95): it suffices to establish that (3) does not tend to $1$; for $c_n=p_n$ one can even show that the sequence (2) is dense in $(0,1)$, but this needs the prime number theorem with the remainder $o(x/\log^2x)$ [3], and it would be of interest to find a more elementary proof.
• For $c_n=p_n^k$ with $k\ge2$ the hypothesis $c_n=o(n^2)$ fails, so the proposition gives nothing for the higher powers; this is consistent with the paper proving only $k=1$.
• Later exact criteria replace the growth hypothesis by an increment hypothesis: Erdős–Straus 1974, Corollary 2.10 and the remark following it, Tijdeman–Yuan 2002, Theorem 3.1 (integers $b_n$ with $b_{n+1}-b_n=o(n)$: $\sum b_n/n!$ is rational if and only if $b_n/(n-1)$ is eventually constant) and Hančl–Tijdeman 2004, Corollary 4.2.

Bears on. #251 (context only; the proposition concerns factorial denominators).