A Counterexample to a Prime-Counting Bound for 1-Separated Multiplicative Semigroups

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lemma_11: If reals 1 < a_1 < ... < a_n generate a multiplicative semigroup whose distinct elements are at least 1 apart, some a_(n+1) > a_n can be adjoined keeping that property; proved by bounding the measure of the excluded points of an interval [R, 2R], with no computation.

proposition_1: Five reals e^(beta_i), each beta_i an explicit integer combination of the powers of 2^(1/6), lie between 1 and 10 and generate a semigroup whose products for distinct exponent vectors differ by at least 1; certified by a norm bound and a validated interval-arithmetic enumeration, unreviewed.

theorem_1: A strictly increasing sequence of reals above 1 whose power products for distinct exponent vectors are at least 1 apart can have five terms below 10, so #{a_i <= x} <= pi(x) for all x >= 1 fails at x = 10; computer-assisted, from a note printing no author, unreviewed.

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A Counterexample to a Prime-Counting Bound for 1-Separated Multiplicative Semigroups, a note of 11 pages that prints no author, date, affiliation or reference list and names neither Problem 951 nor erdosproblems.com. Its attribution is the claim page's: the [[../wiki/problems/number_theory/E0951/claims/2026_01_27_leeham|claim page for the counterexample at x = 10]] credits it to Liam Price, the forum user Leeham. That page's sources are the erdosproblems.com thread for Problem 951, where Kevin Barreto posted the note on 27 January 2026 as the work of a chain of ChatGPT-5.2 Pro instances prompted by Leeham, and the problem's erdosproblems.com commentary, which credits the counterexample at $x=10$ to ChatGPT-5.2 Pro prompted by Leeham. Sothanaphan's note dated 2 February 2026 cites it as by "K. Barreto and L. Price". The note is not refereed, and the searches recorded on the problem page found no arXiv or journal version.

Version and provenance. The copy read for this card was fetched on 2026-10-14 from https://drive.google.com/uc?export=download&id=1CRBMOMCTdAuuVp8Wqace1AQ_dpK9BeTV, the direct-download address of the Google Drive file https://drive.google.com/file/d/1CRBMOMCTdAuuVp8Wqace1AQ_dpK9BeTV/view that the claim page records: 310,635 bytes, 11 pages, its metadata giving no title or author and a creation time of 2026-01-27T20:03:01Z. A Google Drive file has no version pin, so the file at that address can change; the card describes the copy of that date. No notice is printed in the note, and no terms came with the download; the term is unstated.

What it claims. For a strictly increasing sequence of reals $a_i>1$ the note calls the set of its power products $A(k)=\prod_ia_i^{k_i}$, over finitely supported vectors $k$ of nonnegative integers, 1-separated when $|A(k)-A(\ell)|\ge1$ for all distinct $k$ and $\ell$ (its condition (1), p. 1). Theorem 1 (p. 8) refutes the implication from (1) to $\#\{i:a_i\le x\}\le\pi(x)$ for all $x\ge1$ (its (2), p. 1), which the abstract calls a question of Erdős. Section 2 (p. 2) defines five generators $a_i=e^{\beta_i}$ with $\beta_i\in\mathbb Z[2^{1/6}]$, and Lemma 1 places them between 1 and 10, the largest about $9.84133$. Section 3 (pp. 2--5) proves their semigroup 1-separated (Proposition 1, p. 5): the $\beta_i$ are linearly independent over $\mathbb Q$, a norm bound in $\mathbb Q(2^{1/6})$ keeps every nonzero integer combination of them away from 0, so two products less than 1 apart would need exponents differing by at most 29 and logarithms below 43, and a validated interval-arithmetic enumeration of the 85,838 exponent vectors in that range finds every gap above 1, the least about $1.1067$. Section 4 (pp. 5--8) proves the extension step (Lemma 11, p. 6): a finitely generated 1-separated semigroup takes one more, larger generator and stays 1-separated, by a bound on the measure of the excluded points of an interval $[R,2R]$; iterating it gives the infinite sequence of Corollary 1 (p. 8). Remark 1 (p. 8) states that Proposition 1 is computer-assisted and Lemma 11 is not. Appendix A (pp. 8--11) prints a Python script using mpmath interval arithmetic for the numerical facts behind Lemmas 1, 6, 7 and 8, without its output.

AI disclosure. None is printed. The thread post and the erdosproblems.com commentary, as the claim page records them, attribute the note to ChatGPT-5.2 Pro prompted by Leeham.

Read status. Claims checked. The statements of Theorem 1, Proposition 1 and Lemma 11, with Lemma 1 and Corollary 1, were read clause by clause on the page images of the PDF; the proofs were read only to write the pointers on the result pages. No proof step, bound or computation was checked, and the script of Appendix A was not run. Nothing here awards verification, acceptance or a tier.

Relation to the other two notes. Sothanaphan's notes write $m(n)$ for the infimum of $x_n$ over reals $1<x_1<\cdots<x_n$ whose semigroup is free with elements at least 1 apart, so that the first $n$ primes give $m(n)\le p_n$. His note of 28 January 2026 ([[number_theory/sothanaphan_2026_one_separated_semigroups_prime_minimality/_index|filed here]]), posted a day after this one, does not cite it; it proves $m(3)<5$ with three rational generators and lists $m(n)<p_n$ as unknown for every $n\ge4$. His note dated 2 February 2026 ([[number_theory/sothanaphan_2026_probabilistic_box_certification_one_separated_semigroups/_index|filed here]]) cites this note for $m(5)<11=p_5$, from the five generators here, all below 10, and for linking $m(n)<p_n$ to a formulation of Problem 951; it proves $m(n)<p_n$ for $3\le n\le7$ by another method. Neither restates Lemma 11; the claim page for both applies it to their generators.

Bears on. #951: Theorem 1 answers the part every_x, the reading in which the inequality must hold for every $x$, in the negative at $x=10$; the problem page records it as a [[../wiki/problems/number_theory/E0951/claims/2026_01_27_leeham|pending partial claim]]. The note does not address the part large_x, the reading for all large $x$; the curator's reply recorded on that claim page notes that the construction may satisfy the inequality for all large $x$.

Results.

• Theorem 1 (p. 8): the separation condition (1) does not imply $\#\{i:a_i\le x\}\le\pi(x)$ for all $x\ge1$, since an infinite sequence satisfying (1) can have five terms below 10.
• Proposition 1 (p. 5): the five explicit generators $e^{\beta_i}$ with $\beta_i\in\mathbb Z[2^{1/6}]$, all between 1 and 10 by Lemma 1, generate a 1-separated semigroup.
• Lemma 11 (p. 6): if $1<a_1<\cdots<a_n$ generate a 1-separated semigroup, some $a_{n+1}>a_n$ can be adjoined keeping it 1-separated.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.