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Statement

Example 1 (p. 132). Let $a$ and $b_k$ be integers with $a>1$ and $\sum|b_k|a^{-2^k}<\infty$, and put $n_k=a^{2^k}+b_k$. Then $\sum1/n_k$ is irrational.

The paper marks the example with its reference [1], printed as J. W. Golomb, A Special Case, On the sum of the reciprocals of the Fermat numbers and related irrationalities, Canad. J. Math. 15 (1963), 475--478; the article is by S. W. Golomb. The case $a=2$, $b_k=1$ is the sum of the reciprocals of the Fermat numbers $2^{2^k}+1$.

Proof pointer

Pp. 132--133. The convergence hypothesis gives $n_k^2/n_{k+1}\to1$, which is condition (i) of Theorem 1, and $N_k^*/n_{k+1}=a^{-1}\prod_{l=1}^k(1+b_la^{-2^l})/(1+b_{k+1}a^{-2^{k+1}})$ is bounded, which gives condition (ii) since $N_k\le N_k^*$. A rational sum would then force the recurrence, which reads $b_{k+1}=2a^{2^k}b_k+b_k^2-a^{2^k}-b_k+1$. Then $b_k\ne0$ forces $|b_{k+1}|>a^{2^k}$ for large $k$ (11), and $b_k=0$ forces $b_{k+1}=1-a^{2^k}\ne0$; iterating (11) keeps $|b_{k+l}|a^{-2^{k+l}}$ above $a^{-2^k}$, so it does not tend to $0$, against the hypothesis.

Dependencies

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. 132 and the proof on pp. 132--133 for its structure. Nothing here is independently reviewed.

Bears on

• Problem 243: an instance. The sequences of Example 1 satisfy the problem's hypothesis $a_n/a_{n-1}^2\to1$, and the proof shows they do not satisfy the recurrence for all large $n$; their reciprocal sums are irrational, so the problem's statement holds for this family.