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Statement

Setting (p. 129). An Ahmes series is a series $\sum1/n_k$ of reciprocals of positive integers. For a sequence $\{n_k\}$ write $N_k=\operatorname{lcm}(n_1,\ldots,n_k)$. The model is Sylvester's series (1), $1=\frac12+\frac13+\frac17+\frac1{43}+\cdots$, where $n_{k+1}=N_k+1=n_k^2-n_k+1$.

Theorem 1 (p. 129). Let $\{n_k\}$ be an increasing sequence of positive integers such that

• (i) $\limsup n_k^2/n_{k+1}\le1$;
• (ii) $\{N_k/n_{k+1}\}$ is bounded.

Then $\sum1/n_k$ is rational if and only if $n_{k+1}=n_k^2-n_k+1$ for all $k\ge k_0$, and in that case (2)

$$\sum\frac1{n_k}=\frac1{n_1}+\cdots+\frac1{n_{k_0-1}}+\frac1{n_{k_0}-1}.$$

Proof pointer

Pp. 129--130. If $\sum1/n_k=a/b$, write $bN_k=c_kn_{k+1}-d_k$ with integers $c_k,d_k$ and $0\le d_k<n_{k+1}$; by (ii) the $c_k$ are positive and bounded. Multiplying the tail by $bN_k$ and reading the result modulo 1 gives the congruence (4) for $d_k$ modulo $n_{k+1}$, whence $d_k\le c_k$ for large $k$ (5). Comparing $bN_{k+1}$ with $n_{k+1}\cdot bN_k$ gives $c_{k+1}\le c_kn_{k+1}^2/n_{k+2}+o(1)\le c_k+o(1)$ (6), so the integers $c_k$ are eventually constant, which forces $\lim n_k^2/n_{k+1}=1$ (7), then $d_k=c$ and finally the recurrence (9). The closed form follows from the telescoping identity $\frac1{n_{k_0}-1}=\frac1{n_{k_0}}+\cdots+\frac1{n_{k-1}}+\frac1{n_k-1}$ under the recurrence; the identity displayed on p. 130 omits its final term $\frac1{n_k-1}$. The converse direction is this identity in the limit.

Dependencies

None. On p. 131 the paper shows that finiteness of $\limsup n_k^2/n_{k+1}$ alone, with (ii), does not suffice (examples on p. 131); Theorem 3 replaces (ii) by a weaker condition, and Example 1 applies 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 was read clause by clause on the page image of p. 129 and the proof on pp. 129--130 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 1 then gives the problem's conclusion for every such sequence that also satisfies (ii). The problem asks for the conclusion without (ii); the authors say on p. 132 that Theorem 1 may well remain valid without it. Theorem 3 is the paper's stronger form.