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Statement

A sequence $X$ of positive integers is complete when every sufficiently large positive integer is a sum of distinct terms of $X$ (p. 1). Write $\varphi=(1+\sqrt5)/2$.

Lemma 2 (p. 1). Let $(x_n)_{n\ge1}$ be a sequence of positive integers which eventually satisfies

$$x_{n+2}=x_{n+1}+x_n-(-1)^n .$$

Then every tail of $(x_n)$ is complete, and for every fixed $k$

$$\frac{x_{n+1}}{x_n}\longrightarrow\varphi, \qquad \frac{x_k+\cdots+x_n}{x_{n+1}}\longrightarrow\varphi .$$

The recurrence is the paper's (2.1), the limits its (2.2). The paper notes (p. 1) that Graham had earlier established the two deletion properties of the Erdős-Graham question for the sequence $F_n-(-1)^n$, and that the lemma's bounded-gap descent follows the broad strategy of Graham's paper; the lemma's proof is stated to be self-contained.

Source. GPT Pro, A counterexample to Erdős Problem 346, preprint (2026), 5 pp.; Lemma 2 on p. 1, its proof on pp. 1--3. The edition read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page. The proof was read for structure only.

Proof pointer

pp. 1--3. With $y_n=x_n+(-1)^n$ the recurrence becomes the Fibonacci recurrence, so $x_n=c\varphi^n+O(1)$ with $c>0$, which gives (2.2) and eventual monotonicity. For a tail starting at $x_k$, the finite subset-sum sets $Q_r=\Sigma(x_k,\dots,x_r)$ satisfy $Q_{r+1}=Q_r\cup(x_{r+1}+Q_r)$ and have gaps bounded independently of $r$, so the subset sums of the tail have some finite gap bound. Explicit identities, the paper's (2.3) to (2.6), show that every integer within a fixed distance $R$ of $x_n$ is a subset sum of the tail for large $n$ (the paper's (2.7)). A descent step then lowers any gap bound $w\ge1$ to $w-1$ beyond some threshold, and iterating reaches gap bound $0$, that is, completeness.

Bears on

• Problem 346: the lemma makes every tail of a sequence eventually obeying Graham's recurrence complete, with ratios tending to $\varphi$; the paper uses it for the unperturbed stretches of the sequence of Theorem 1. On its own it does not answer the problem.