Answers to some recent conjectures and questions in classical analysis, with computer assistance

We answer, or make progress on, several questions and conjectures posed in recent arXiv papers in classical analysis. (1) We prove two conjectures of Q. Fan on the Riesz capacity of three-point sets on a circle and of the vertex sets of regular odd polygons. (2) We show that a conjecture of C. Xu and J. Zhao on the sign thresholds of higher-order Gregory coefficients is false, and compute these thresholds rigorously (computer-assisted, interval arithmetic) for orders up to 10. (3) We prove, by computer-assisted ball arithmetic, that the reciprocal of the diagonal Padé denominator of the exponential has positive Taylor coefficients for every odd degree 3 ≤ m ≤ 361 — a case left open in the literature on a conjecture of Fasi and Higham — and describe numerically where positivity first fails for fixed m. (4) We give an exact algorithm and explicit values for series of products of consecutive harmonic numbers, short answers to questions of S. Finch on Abel's equation and on iterates of exponentials, two of the three open problems of H. Wiśniewska on Gurland's ratio, and numerical values of two completely monotonic degrees of Stirling remainders. Theorems, computer-assisted proofs and numerical results are clearly separated in the text; open problems are listed at the end. AI disclosure. This work was carried out with extensive use of an AI system, Claude (Anthropic), via the Claude Code software, under the direction of Jacob Goodchild. The AI system chose the problems from the recent literature, carried out the derivations, proofs and computations, searched and checked the literature, and wrote the manuscript. Proofs were checked by independent AI referee passes and by computer algebra and interval arithmetic, but have not been checked by a human expert or peer reviewed. Novelty assessments are based on literature searches and may be incomplete. Jacob Goodchild directed the work and is responsible for its publication. Preprint; not peer reviewed. All proofs and computations were produced and checked with AI assistance (see the AI disclosure). Computer-assisted statements are listed with exactly what was computed; numerical results are not theorems.