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Statement

Setting (p. 1). $P(A)$ is the set of sums of finitely many distinct terms of a sequence of integers $A$, and $A$ is complete when every sufficiently large integer lies in $P(A)$. The Fibonacci sequence is $F=(F_1,F_2,\ldots)$ with $F_0=0$, $F_1=1$ and $F_{n+2}=F_{n+1}+F_n$ for $n\ge0$; it is complete.

Properties (A) and (B) (p. 1). The paper states that $F$ satisfies

• (A) if any one term is removed from $F$, the resulting sequence is complete;
• (B) if any two terms are removed from $F$, the resulting sequence is not complete.

The paper proves (B) (pp. 1--2) and refers to Brown for a simple proof of (A).

Proof pointer

Pp. 1--2. Remove $F_r$ and $F_s$ with $r<s$ to form $F^*$. The paper shows by induction on $k\ge0$ that $F_{s+2k+1}-1\notin P(F^*)$, comparing with the sum of all terms of $F^*$ below the target, computed from $\sum_{k=1}^{n}F_k=F_{n+2}-1$. So infinitely many integers lie outside $P(F^*)$.

Read depth

Claims checked: the definitions, (A), (B) and the proof of (B) were read clause by clause on the page images of the print. Property (A) is cited, not proved, in the paper and was not read. Nothing here is independently reviewed.

Dependencies

None in the corpus. External input named by the paper: J. L. Brown, On complete sequences of integers, Amer. Math. Monthly 68 (1961), 557--560, for the completeness of $F$ and for (A).

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

None directly. The paper uses (B) as the contrast for its theorem on the sequence $F_n-(-1)^n$.