OBJECTIVE AND SOURCE REVIEW
Erdős #265; independent attempt, 2026-09-23. How rapidly can increasing integers $a_n\ge2$ grow while both $\sum1/a_n$ and $\sum1/(a_n-1)$ are rational? I read the complete
265.tex, including its proposed exponent invariant and the imported quantitative Kovač--Tao lower range. Here I derive an explicit necessary denominator-growth constraint. I do not claim a better optimal exponent.
WORK: TWO DENOMINATOR CONSTRAINTS
Suppose $a_{n+1}\ge2a_n$ eventually and the two sums are $p/q$ and $r/s$, respectively, with positive integer denominators $q,s$. Put \[
Q_n=\prod_{i\le n}a_i,\qquad R_n=\prod_{i\le n}(a_i-1).
\] For all sufficiently large $n$, positivity and rationality of the tails give \[
\frac1{qQ_n}\le\sum_{i>n}\frac1{a_i}\le\frac2{a_{n+1}},
\qquad a_{n+1}\le2qQ_n.
\] Also $a_i-1\ge a_i/2$, so \[
\frac1{sR_n}\le\sum_{i>n}\frac1{a_i-1}\le\frac4{a_{n+1}},
\qquad a_{n+1}\le4sR_n.
\] Thus both constraints must hold simultaneously. In particular no such sequence can satisfy $a_{n+1}\ge a_n^{2+\delta}$ eventually for fixed $\delta>0$. Indeed, writing $b_n=\log a_n$ and summing the backward geometric bound gives \[
\log Q_n\le O(1)+\frac{2+\delta}{1+\delta}b_n.
\] But $\log a_{n+1}\ge(2+\delta)b_n$, contradicting $a_{n+1}\le2qQ_n$ as $n\to\infty$.
ADVERSARIAL VERIFICATION AND REMAINING GAP
All denominators are positive because $a_n\ge2$; allowing $a_1=1$ would make the shifted sum undefined. The doubling hypothesis is used for the tail estimate and is not automatic from an arbitrary average-growth specification. The pointwise power condition proved impossible is stronger than the global exponent condition in the source, so this argument is not a replacement proof of every claimed global barrier. It offers necessary inequalities for any proposed construction, but no construction near the optimal rate.
SOURCES CHECKED
Complete local source:
gpt_pro_5.4/265.tex. Primary paper checked: Kovač--Tao,
https://arxiv.org/html/2406.17593v3. Its simultaneous-rationality setting supplies context; the displayed obstruction is elementary and no new record is claimed.
FINAL
UNRESOLVED. Explicit tail constraints are verified, while the maximal admissible growth remains undetermined.
COMPLETION ESTIMATE
Conservative subjective progress toward the optimal growth problem.
COMPLETION: 10%