Transcendence in the affine case of Erdős Problem 270

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Crmarić and Kovač, On the irrationality of certain super-polynomially decaying series - answers Problem 270 as posed evidence for this event

The manuscript claims $C_{a,b}$ is transcendental for every $a\ge1$ and $b\ge1-a$, settling the positive integer-valued affine subclass of Erdős Problem 270. Two pieces of context matter. Problem 270 as Erdős and Graham posed it, for every $f(n)\to\infty$, was already answered no by Crmarić and Kovač in 2025: for any $\alpha>0$ some such $f$ makes the series sum to $\alpha$. What survives is the non-decreasing case, and the affine family sits inside it. Separately, the checkable parts here were already known - the short irrationality proof for $C_{1,0}$ is Crmarić and Kovač's, posted by Kovač on the Erdős Problems forum in July 2026 and credited in the repository, and base-case transcendence follows a 2023 MathOverflow argument. The new content is the extension to the whole affine family, which is the part with neither formalization nor review. parent of this event

Kovač's earlier elementary irrationality proof evidence for this event

Erdős Problem #270 evidence for this event

Transcendence in the affine case of Erdős Problem 270 evidence for this event

VibeMathed record: Transcendence in the affine case of Erdős Problem 270 evidence for this event

Transcendence in the affine case of Erdős Problem 270 parent of this event