---

Claim. P. Erdős and E. G. Straus, On the irrationality of certain Ahmes series, J. Indian Math. Soc. (N.S.) 27 (1964), 129--133, received 22 January 1964. Theorem 3 (printed p. 132) states: let $\{n_k\}$ be an increasing sequence of positive integers with

(i) $\limsup n_k^2/n_{k+1}\le1$ and

(ii$''$) $\limsup(N_k/n_{k+1})\bigl(n_{k+1}^2/n_{k+2}-1\bigr)\le0$,

where $N_k=\operatorname{lcm}(n_1,\ldots,n_k)$; then $\sum1/n_k$ is rational if and only if $n_{k+1}=n_k^2-n_k+1$ for all $k\ge k_0$. Theorem 1 (printed p. 129) is the case where $\{N_k/n_{k+1}\}$ is bounded, condition (ii), which together with (i) implies (ii$''$), as the paper notes; it adds that the rational sum then equals $1/n_1+\cdots+1/n_{k_0-1}+1/(n_{k_0}-1)$. The proof writes $bN_k=c_kn_{k+1}-d_k$ for a rational sum $a/b$ and shows the integers $c_k$ eventually constant, which forces the recurrence; the source card erdos_1964_irrationality_certain_ahmes_series digests the paper. On p. 132 the authors say that Theorem 1 may well remain valid without condition (ii), the question of Problem 243. The site's remark on the problem writes the factor of (ii$''$) as $(a_n^2/a_{n+1}-1)$, one index earlier than the paper's $(n_{k+1}^2/n_{k+2}-1)$.

Covers. The sequences of Problem 243 that also satisfy (ii$''$): in the problem's indexing, $a_n/a_{n-1}^2\to1$ gives (i), and the claim applies when $\limsup([a_1,\ldots,a_n]/a_{n+1})(a_{n+1}^2/a_{n+2}-1)\le0$, in particular whenever $[a_1,\ldots,a_n]/a_{n+1}$ stays bounded (Theorem 1). Not covered: the sequences for which the lcm-weighted error has a positive limit superior, which is the gap the problem concerns.

Acceptance. Refereed: J. Indian Math. Soc. (N.S.) 27 (1964), 129--133. The site labels the problem OPEN and records this theorem in its remark, so no reviewed evidence is listed. The proof is not checked here.

Depends on. Nothing in this wiki.