Price 2026 counterexample erdos problem 346

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lemma_2: If a sequence of positive integers eventually satisfies x_{n+2} = x_{n+1} + x_n - (-1)^n, then every tail of it is complete, and for every fixed k the ratios x_{n+1}/x_n and (x_k + ... + x_n)/x_{n+1} both tend to the golden ratio.

theorem_1: A strictly increasing sequence of positive integers stays complete after deleting any finite subsequence, becomes incomplete after deleting any infinite one, has consecutive ratios at least 6/5, and has subsequential ratio limits phi and phi + 1/4, so its ratios do not converge.

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GPT Pro, A counterexample to Erdős Problem 346, preprint (2026), 5 pp., posted by Liam Price. The author line prints GPT PRO alone. No notice is printed in the paper; the hosting service's terms (https://www.overleaf.com/legal, read 2026-10-02) say "We don't claim any ownership of your stuff" and grant readers of a shared project no license; the term is unstated.

Theorem 1 (p. 1) constructs a strictly increasing sequence A = (a_n) of positive integers such that A minus any finite subsequence is complete, A minus any infinite subsequence is incomplete, a_{n+1}/a_n >= 6/5 for every n, and, along some increasing index sequence (N_j), a_{N_j}/a_{N_j-1} tends to phi while a_{N_j+1}/a_{N_j} tends to phi + 1/4, where phi = (1+sqrt 5)/2; hence a_{n+1}/a_n does not converge. The paper states the question of Erdős and Graham (its reference [1], p. 57) as whether the two deletion properties and a_{n+1}/a_n >= 1+eps for some eps > 0 force a_{n+1}/a_n -> phi, and its abstract says the theorem gives a negative answer to Erdős Problem 346. Section 2 (pp. 1--3) proves Lemma 2: a sequence of positive integers eventually satisfying Graham's recurrence x_{n+2} = x_{n+1} + x_n - (-1)^n has every tail complete, and x_{n+1}/x_n -> phi and (x_k+...+x_n)/x_{n+1} -> phi for every fixed k. Section 3 (pp. 3--5) builds admissible sequences a_{n+1} = S_{n-1} + b_n with 0 <= b_n <= a_n/4, proves that deleting any infinite subsequence from one leaves an incomplete sequence (Lemma 3, p. 3), gives a finite interval certificate that keeps a tail complete under all later admissible choices (Lemma 4, p. 4) and shows that unperturbed stretches produce such certificates (Lemma 5, p. 4), and then sets b_{N_j} = floor(a_{N_j}/4) at sparse indices N_j (p. 5).

Source: https://www.overleaf.com/read/tgrrgqpbjpht.

Read status: claims checked. Theorem 1 and Lemma 2 were read clause by clause on the page images of the print; the proofs were read for structure only. Result pages: theorem_1 and lemma_2.

Bears on.

• #346: Theorem 1 (p. 1) gives a sequence with both deletion properties and a_{n+1}/a_n >= 6/5 whose consecutive ratios do not converge, so these hypotheses do not force a_{n+1}/a_n -> phi; it does not address sequences whose ratios are assumed to converge.

Results to transcribe.

• Theorem 1 (p. 1): There is a strictly increasing sequence A of positive integers with: A minus any finite subsequence complete, A minus any infinite subsequence incomplete, a_{n+1}/a_n >= 6/5 for every n, and subsequential ratio limits phi and phi + 1/4 along a_{N_j}/a_{N_j-1} and a_{N_j+1}/a_{N_j}, so the ratios do not converge.
• Lemma 2 (p. 1): If (x_n) eventually satisfies x_{n+2} = x_{n+1} + x_n - (-1)^n then every tail of (x_n) is complete, and x_{n+1}/x_n -> phi and (x_k+...+x_n)/x_{n+1} -> phi for each fixed k.

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