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Claim. For $f(n)=an+b$ with integers $a\ge1$ and $b\ge1-a$ the series of Problem 270 is

$$C_{a,b}=\sum_{n\ge1}\frac{n!}{((a+1)n+b)!},$$

and the manuscript Algebraic Independence in the Affine Case of Erdős Problem 270 by Costa Lambrinoudis states that every such $C_{a,b}$ is irrational, and more: that for each $a\ge1$ the $a+1$ numbers $C_{a,0},\ldots,C_{a,a-1},C_{a,a+1}$ are algebraically independent over $\mathbb{Q}$, that every other $C_{a,b}$ with $b\ge1-a$ is a rational affine combination of them with nonzero linear part, and hence that every $C_{a,b}$ in that range is transcendental. The manuscript's elementary irrationality argument covers only $0\le b\le a$: there, with $D_n=((a+1)n+b)!/n!$, the numerator of $D_{n+1}/D_n=\prod_{j=1}^{a+1}((a+1)n+b+j)/(n+1)$ has the factor $(a+1)(n+1)$ at $j=a+1-b$, so the ratios are integers tending to infinity, $D_N$ times the $N$-th partial sum is an integer, and $D_N$ times the tail lies strictly between $0$ and a quantity tending to $0$, which no rational value allows. For $b>a$ or $b<0$ the ratios need not be integers (for $a=1$, $b=2$, $D_3/D_2=56/3$), and irrationality there rests only on the claimed transcendence. The transcendence of $C_{1,0}$ follows, by the manuscript, from the Gaussian representation $C_{1,0}=e^{1/4}\int_0^{1/2}e^{-t^2}\,dt$ and the Siegel–Shidlovsky theorem; the algebraic independence from the criteria of Salikhov and of Viskina and Salikhov for hypergeometric $E$-functions, a logarithmic obstruction at infinity and a trace-descent argument for the extra function, and Beukers's refinement of the Siegel–Shidlovsky theorem. The manuscript records that Kovač's thread post of 2026-07-12 gives the case $a=1$, $b=0$, found independently. The manuscript's disclosure says that generative AI did nearly all of the mathematical work, the literature search, the writing and the Lean, under the author's direction, and names no system; the thread post that announces it (2026-08-24) calls the result a claim by GPT. The repository was created on 2026-08-22, and the links are pinned to its commit of 2026-08-24, whose version strengthens an earlier transcendence statement to algebraic independence.

Covers. The instances $f(n)=an+b$ with $a\ge1$ and $b\ge1-a$: for them the answer to the question is yes, the sum is irrational (by the elementary argument for $0\le b\le a$, and for the other intercepts only through the claimed transcendence). The general question, asked of every $f(n)\to\infty$, is answered no on the accepted page Crmarić and Kovač 2025; this page settles only the affine instances.

Formalization. The repository's Lean 4 project proves the irrationality of $C_{a,b}$ unconditionally for $a\ge1$ and $0\le b\le a$ (its theorem constant_irrational), by the author's report with no sorry, admit or axiom and only the standard axioms. The transcendence results are conditional on hypotheses stated in its ExternalTheorems.lean. For $C_{1,0}$ they are the Gaussian identity and a special-value consequence of the Siegel–Shidlovsky theorem, and Lean checks the deduction from them. For the other values the hypothesis is the algebraic independence of $C_{a,0},\ldots,C_{a,a-1},C_{a,a+1}$ itself, the manuscript's main claim, which the file says packages the Salikhov and Viskina–Salikhov results, the manuscript's own logarithmic-obstruction and trace-descent arguments and Beukers's theorem. Lean proves without this hypothesis that every $C_{a,b}$ lies in the rational affine span of that basis, and derives from it only the transcendence of every $C_{a,b}$ with $b\ge1-a$ and the transcendence degree. Nothing is built or audited in this corpus, so no formalized evidence is listed.

Standing. Claimed. The result is a dated manuscript posted to a public repository and announced on the site's discussion thread; the site's page does not mention it, no named mathematician has reviewed it and there is no refereed publication.

Depends on. No page of this wiki.