Equal radicals of consecutive products
Ruiyi Zhang · Preprint, 2026-10-29 · 12 pages
PDFHTMLDOI 10.5281/zenodo.23758120Source and checks
Abstract
For every fixed k ≥ 2 and ε > 0, F_{k,k}(X) ≪_{k,ε} X^(2/3+ε). For k ≥ 3 this improves the X^(1+o(1)) bound of Lebowitz–Lockard. For k = 2, Kane's localized estimates give the large-quotient endpoint bound. Our independent proof combines a Pell bound for partner multiplicities, a quotient-tail estimate, and Farey spacing for small quotients. Reuss's determinant estimate also gives N(H, Y) ≪_ε H^ε (Y/H + H^(7/12)), uniformly for H ≥ 2 and 1 ≤ Y < H^2, where N(H, Y) counts integers t ∈ (H, 2H] with rad(t(t + 1)) ≤ Y. A cubic family gives F(X) ≥ (4^(1/3)/3) X^(1/3) − O(1).
Cite
Ruiyi Zhang, Equal radicals of consecutive products, preprint, Zenodo, 2026. https://doi.org/10.5281/zenodo.23758120