Chojecki 2026 distinct consecutive products

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lemma_2_1: Chojecki's stability lemma: every finite prefix of the gap-greedy set, and the set itself, has distinct consecutive-block products, and a rejected prime gap has a witness whose later block is a full interval inside the gap and is shorter than the earlier block.

lemma_2_2: Chojecki's lemma that the chosen parent gap (p,q) of a rejected gap has both endpoints multiplied into the later interval [m,n] of the witness, and that either one multiplier j serves both endpoints or p^2 <= ng + ps.

lemma_2_3: Chojecki's lemma that, for all large Y, the prime gaps (p, p^+) with Y <= p < 2Y and length above p^{1/20} have total length O(Y (log Y)^{-D_0}), so the long gaps with left endpoint at most X have total length o(X).

proposition_4_3: Chojecki's forest bound: in the gap-greedy construction, the sum of the lengths of all short rejected prime gaps with right endpoint at most X is O(X^{9/10+o(1)}).

theorem_1_1: Chojecki's theorem that some set A of positive integers of natural density one has the products of its distinct consecutive blocks, in increasing order, pairwise distinct, answering a question of Erdős and Graham.

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Przemek Chojecki, Distinct Consecutive Products. preprint (ulam.ai) (2026).

Theorem 1.1 asserts a set A of natural density one in the naturals such that distinct consecutive blocks of its increasing enumeration have distinct products, answering the Erdos-Graham question of Old and New Problems, p. 84. The construction is greedy over prime gaps: starting at 2, each consecutive-prime gap interior is retained wholesale unless the tentative prefix has a product collision, in which case only the terminal prime is kept, so the complement of A is, apart from 1, exactly the union of the rejected gap interiors. Lemma 2.1 proves stability and gives every rejected gap a witness in a canonical separated form, the earlier block longer than the later interval, which is a full interval inside the gap; Lemma 2.2 splits parent-child edges of the induced forest of rejected gaps into equal and unequal types; Lemma 2.3 shows that the long gaps (p, p^+) with Y <= p < 2Y (length above p^{1/20}) contribute total length O(Y (log Y)^{-D_0}) for all large Y, via Li's theorem on primes in almost all short intervals. Uniform affine-curve point counts of Castryck-Cluckers-Dittmann-Nguyen control raw witnesses, and a scale-contracting forest argument (Proposition 4.3, exponents (6) and (7) decreasing in the path length, at most 9/10 and 115/156) bounds the total length of short rejected gaps with right endpoint at most X by O(X^{9/10+o(1)}), giving density one. The note is dated 13 July 2026 and states the proof was found by GPT-5.6 Sol while iterating on the author's earlier attempts. The paper presents Theorem 1.1 as an affirmative answer to the question of Erdos and Graham, which is problem 421; it is an unrefereed preprint.

Source: https://www.ulam.ai/research/chojecki-2026-erdos421-distinct-consecutive-products.pdf. The copy read for this card, retrieved from the hosting organization's site (https://www.ulam.ai/research/chojecki-2026-erdos421-distinct-consecutive-products.pdf), carries no arXiv stamp and prints no notice; the arXiv abstract page of the same paper, its only version with the same title and author, names arXiv's non-exclusive distribution license (https://arxiv.org/abs/2609.17543, read 2026-10-02), and the site's footer reads "© 2017-2026 ULAM" (read 2026-10-02), every other right reserved.

Read status: claims checked for Theorem 1.1, the construction, Lemmas 2.1, 2.2 and 2.3 and Proposition 4.3, read clause by clause on the page images of the print; the proofs of Theorem 1.1 and Lemmas 2.1 to 2.3 followed, and those of Proposition 3.1, Lemma 3.2 and Section 4 read for structure. The cited theorems of Castryck--Cluckers--Dittmann--Nguyen and of Li were not read. Nothing here is independently reviewed. Result pages: theorem_1_1, lemma_2_1, lemma_2_2, lemma_2_3 and proposition_4_3.

Bears on. #421: Theorem 1.1 (p. 1) asserts a set of natural density one whose distinct consecutive blocks, in increasing order, have distinct products, which is the affirmative answer to the problem's question; the paper presents it as answering the question of Erdos and Graham.

Results.

• Theorem 1.1 (p. 1): there is a set $A\subseteq\mathbb N$ of natural density one whose distinct consecutive blocks have distinct products.
• Lemma 2.1 (p. 2): every finite prefix $A_p$ and $A$ itself are collision-free, and a rejected gap $(P,Q)$ has a witness whose later block is an interval $[m,n]\subset(P,Q)$ shorter than the earlier block.
• Lemma 2.2 (p. 2): the chosen parent $(p,q)$ of a rejected gap has multiples $\alpha p,\beta q\in[m,n]$ with $\alpha,\beta\geq2$, and either one multiplier $j\geq2$ serves both endpoints (equal edge) or $p^2\leq ng+ps$ (unequal edge).
• Lemma 2.3 (p. 2): for all large $Y$ the prime gaps $(p,p^+)$ with $Y\leq p<2Y$ and length above $p^{1/20}$ have total length $O(Y(\log Y)^{-D_0})$.
• Proposition 4.3 (p. 4): the short rejected gaps with right endpoint at most $X$ have total length $O(X^{9/10+o(1)})$.

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