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Statement

The question (p. 10, Section 3). Properties (C) and (D) are those of the paper's theorem: a sequence stays complete after deleting any finite subsequence, and is not complete after deleting any infinite subsequence. The paper says that examples of sequences of positive integers with both properties "are rather elusive", and that it would be interesting to know whether there is such a sequence $T=(t_1,t_2,\ldots)$ "which is essentially different from S", for example one with

$$\lim_{n\to\infty}\frac{t_{n+1}}{t_n}\ne\frac{1+\sqrt5}{2}.$$

Here $S$ is the sequence with $n$th term $F_n-(-1)^n$. The paper does not define "essentially different" beyond this example.

Proof pointer

The paper proves nothing about the question; it is posed as a closing remark.

Read depth

Claims checked: Section 3 on p. 10 was read clause by clause on the page image of the print. Nothing here is independently reviewed.

Dependencies

None.

Source. R. L. Graham, A property of Fibonacci numbers, Fibonacci Quart. 2 (1964), no. 1, 1--10; the edition read is named on the source card.

Bears on

• Problem 346: the question concerns the same pair of deletion properties as the problem but asks a different thing. Graham asks whether some sequence with both properties is essentially different from $F_n-(-1)^n$, for example with a ratio limit other than $(1+\sqrt5)/2$; the problem asks whether ratios bounded below by $1+\epsilon$ force the ratio limit to be $(1+\sqrt5)/2$. The paper answers neither question.