Transcendence in the affine case of Erdős Problem 270
The series in Erdős Problem 270 is claimed transcendental for every positive integer-valued affine choice of the defining function.
- Lab
- Independent
- Model
- GPT-5.6 Sol (Codex)
- Field
- Mathematics
- Date
- 2026-08-22
- Problem posed
- 1980 · open 46 yrs
Sources
Original work
Independent commentary
What was found
Erdős Problem 270 concerns the series whose n-th term is 1 over the product of the f(n) consecutive integers starting at n+1. For the affine choice f(n) = an + b this is the sum of n! over ((a+1)n + b)!, and the manuscript claims that constant is transcendental for every a at least 1 and b at least 1 - a, settling the positive integer-valued affine subclass. It also determines the algebraic structure of the family at each fixed slope, and ships a Lean development covering part of the argument.
Novelty check
Two pieces of prior work bound what is left to prove, and both are recorded in the repository rather than found later. Problem 270 as Erdős and Graham posed it, for every f tending to infinity, was already answered in the negative by Crmarić and Kovac in 2025, who showed some such f makes the series sum to any prescribed value; what survives is the non-decreasing case, inside which the affine family sits. Separately, the short irrationality proof for the base constant is Crmarić and Kovac's, posted by Kovac on the Erdős Problems forum in July 2026 and credited in the repository, and base-case transcendence follows from known results. So the checkable parts were already known and the new content is the affine extension.
Caveats and known objections
Announced through a repository rather than a paper, unrefereed, with no independent check on record, and vibemathed lists it as a candidate pending review. The Lean development is explicitly partial: the repository separates what is fully verified from what is not yet verified, so the transcendence claim as a whole is not machine-checked and the grade stays at claimed. The novelty is narrow for the reason given above, since the pieces a reader can most easily check are the ones already in the literature. Autonomy is graded ai-led on the repository's own AI disclosure, which is unusually sweeping: generative AI performed nearly all of the mathematical work, independently rediscovering the elementary denominator argument, developing the affine extensions and the transcendence-theoretic arguments, conducting the literature search, writing the manuscript and producing the Lean formalization, with the repository owner directing and supervising. No human author is named on the work.
Also recorded at
vibemathedtranscendence-in-the-affine-case-of-erdos-problem-270
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.
Or on GitHub: submit a check challenge the grade send a correction or send a pull request
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Entry history (1 event)
- AddedEntered 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
whataifound.org. (2026). Transcendence in the affine case of Erdős Problem 270. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-22-erdos-270-affine-transcendenceBibTeX
@misc{whataifound-independent-2026-transcendence,
title = {Transcendence in the affine case of Erdős Problem 270},
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/2026-08-22-erdos-270-affine-transcendence}
}