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Source. P. Erdős, Sur l'irrationalité d'une certaine série, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 17, 765--768. The unnumbered paragraph following the Théorème on p. 765. Bibliographic details are on the source card.

Statement

For an increasing sequence of integers $a_1<a_2<\cdots$, Erdős writes that it seems very likely that

$$\lim_{n\to\infty}\frac{a_n}{n}=+\infty$$

is a sufficient condition for

$$\sum_{n\ge1}\frac{a_n}{2^{a_n}}$$

to be irrational (p. 765).

In the same paragraph he adds that he could not find a sequence with $\limsup(a_{n+1}-a_n)=+\infty$ and a rational sum, but he thinks such an example exists and could be easy to find (p. 765).

After the proof (p. 767) he adds that the hypothesis $a_{n+1}-a_n\to\infty$ of the Théorème can be weakened a little, but that for the moment he cannot replace it by a substantially better condition.

No proof is offered; this is a conjecture.

Bears on

• Problem 260: the problem's question is this conjecture, the irrationality of the sum under $a_n/n\to\infty$. The paper offers it as a conjecture and proves nothing about it beyond the case of the Théorème.