Claimed AI-led

Lambert series values, including the Erdős-Borwein constant, are not quadratic irrationals

A self-published paper with a Lean formalization claims that $F(q) = \sum_{n \ge 1} 1/(q^n - 1)$ is neither rational nor quadratic irrational for every integer $q \ge 2$ , the Erdős-Borwein constant $F(2)$ included, and widens the region where $F(a/b)$ is known to be irrational from $\log b/\log a < 0.405683$ to $\log b/\log a < 0.5728$ , with generative AI credited with nearly all of the mathematics.

Lab: Independent
Model: Generative AI, not named in the repository
Field: Mathematics
Date: 2026-10-08
Human collaborators: Costa Lambrinoudis

Sources

Original work

GitHubclambro/lambert-nonquadraticity, paper, Lean formalization and AI disclosure↗

What was found

Erdős proved in 1948 that $F(q)$ is irrational for integer $q \ge 2$ , and Erdős Problem 1049, a conjecture of Chowla, asks for irrationality at every rational base $t > 1$ . The paper proves a degree-exclusion theorem for every real algebraic base: if $e\lambda \log M(q) < \log q$ , with $\lambda \approx 0.4272$ and $M(q)$ the Mahler measure, then $F(q)$ satisfies no polynomial relation of degree at most $e$ over $\mathbb{Q}(q)$ . The argument builds Gram determinants that are strictly positive at the true value and decay exponentially in $N^3$ , clears their cyclotomic denominators without losing the decay, and closes with the product formula. For rational bases this widens the irrationality region from Cook's 2026 threshold of 0.405683 to 0.5728, or 0.573290 with a refinement that is formalized but only remarked on in the paper. Problem 1049 itself stays open, since bases such as $3/2$ fall outside the region.

Novelty check

erdosproblems.com listed Problem 1049 as open with no proof claims on 2026-10-20. The paper's literature account runs from Erdős (1948) through Bundschuh and Väänänen's threshold of $1/2 - 1/\pi^2$ to W. Cook's self-published 2026 refinement of Zudilin's method, which gave 0.405683, and vibemathed's record cites the same history. A web search on 2026-10-20 for quadratic irrationality or irrationality measures of the Erdős-Borwein constant found Erdős's and Borwein's irrationality proofs and a conditional proof by Pratt, and no earlier result excluding a quadratic relation.

Caveats and known objections

Self-published and unrefereed, and not registered at Palomar. vibemathed records the Lean as checked but its statement as unaudited, and notes that there is no CI and that the build was not reproduced; the repository's check script, which builds the development and audits its axioms, was not run here. The statements rest on the repository's own definition of the Lambert value rather than a Mathlib or Formal Conjectures one, which is why the grade is claimed rather than formal. Cook's 0.405683 threshold, the baseline the headline improvement is measured against, is itself self-published. The AI disclosure names no model; vibemathed's submitter metadata names GPT-6 Astra through Codex, but that is the aggregator's label, not the author's. Autonomy is ai-led on the disclosure that generative AI developed the determinant construction and the degree arguments, searched the literature, wrote the manuscript and produced the Lean, while the author directed and supervised.

Also recorded at

vibemathednonquadraticity-of-lambert-series-values-and-improved-bounds-for-erdos-problem-1

Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.

Nobody outside the lab has checked this yet.

Reading the primary source closely enough to say whether it supports the claim counts as a check, and you are credited on the entry.

Submit a check Challenge the grade

Or on GitHub: submit a check challenge the grade send a correction or send a pull request

Entry history (1 event)

2026-10-20AddedEntered the registry graded Claimed and AI-led.

Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.

Graded claimed for verification and ai-led for autonomy. What these mean.

Cite this entry

Plain text

whataifound.org. (2026). Lambert series values, including the Erdős-Borwein constant, are not quadratic irrationals. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-10-08-lambert-series-nonquadraticity

BibTeX

@misc{whataifound-independent-2026-nonquadraticity,
  title        = {Lambert series values, including the Erdős-Borwein constant, are not quadratic irrationals},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by Independent. Verification: Claimed. Autonomy: AI-led.},
  url          = {https://whataifound.org/finding/2026-10-08-lambert-series-nonquadraticity}
}

Related findings

Formally verifiedErdős Problem 408: an unconditional normal order for the number of totient iterationsPlasma AI · 2026-10-19 Formally verifiedErdős Problem 173: a two-coloring of the plane misses at most one triangle, which is equilateralPlasma AI · 2026-10-19 ClaimedErdős Problem 111: graphs that are not countably colorable need superlinear deletions to become bipartitePlasma AI · 2026-10-19

number-theory irrationality erdos-problems lean argument

← All mathematics findings in the registry