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Statement

Notation as in Theorem 1: $\{n_k\}$ is an increasing sequence of positive integers and $N_k=\operatorname{lcm}(n_1,\ldots,n_k)$.

Theorem 3 (p. 132). Let $\{n_k\}$ satisfy (i), $\limsup n_k^2/n_{k+1}\le1$, and

$$\text{(ii}''\text{)}\qquad \limsup\frac{N_k}{n_{k+1}}\Bigl(\frac{n_{k+1}^2}{n_{k+2}}-1\Bigr)\le0.$$

Then $\sum1/n_k$ is rational if and only if $n_{k+1}=n_k^2-n_k+1$ for all $k\ge k_0$.

The paper notes (p. 132) that (i) and (ii) of Theorem 1 imply (ii$''$), but (i) and (ii$''$) do not imply (ii); so Theorem 3 contains the equivalence of Theorem 1. The closed form (2) of the sum is not restated, but it follows from the recurrence exactly as in Theorem 1.

Proof pointer

P. 132. Condition (i) gives $N_k/n_{k+1}\le N_k^*/n_{k+1}<C(1+\delta)^k$ for every $\delta>0$, so $c_k=o(e^{\delta k})=o(n_k)$ and the congruence (4) of Theorem 1's proof still holds. Then (ii$''$) gives $c_kn_{k+1}^2/n_{k+2}\le c_k+o(1)$ (10), so (5) and (6) hold and the rest of Theorem 1's proof applies unchanged. The authors say that bound (ii) is used in Theorem 1's proof mainly to make $c_k$ eventually constant from (6), and that this derivation can be made under weaker hypotheses.

Dependencies

The proof of Theorem 1.

Source. P. Erdős and E. G. Straus, On the irrationality of certain Ahmes series, J. Indian Math. Soc. (N.S.) 27 (1964), 129--133; the edition read is named on the source card.

Read depth. Claims checked: the statement and the note after it were read clause by clause on the page image of p. 132, and the proof for its structure. Nothing here is independently reviewed.

Bears on

• Problem 243: the problem's hypothesis $a_n/a_{n-1}^2\to1$ gives (i), and Theorem 3 then gives the problem's conclusion for every such sequence that also satisfies (ii$''$). Sequences for which the limit superior in (ii$''$) is positive are not covered. The problem's claim page for this paper records the theorem as a partial result.