Counting powerful products of consecutive integers

A number is powerful if every prime that divides it divides it at least twice. This paper shows that the number of n ≤ x for which n and n+1 are both powerful is O(x^(1/4+ε)), improving the exponent 29/100 of Reuss. For each fixed k ≥ 3, the number of n ≤ x for which (n+1)(n+2)···(n+k) is powerful is O(x^(77/320+ε)), and the same exponent holds uniformly over all widths k ≥ 3. These are bounds on how often such products can be powerful; the paper does not settle Erdős’s question whether they ever are.

The new ingredient is a per-interval count in Reuss’s form of Heath-Brown’s approximate determinant method. In lattice-reduced coordinates, the solutions on an auxiliary curve of bidegree (α, β) become integral points of bounded height on an affine curve of degree 2α + 3β, to which the uniform bound of Castryck, Cluckers, Dittmann and Nguyen applies.

All theorems are formalized in Lean 4 with Mathlib. They are checked by Lean’s kernel, replayed by the independent kernel Nanoda, and matched against independently written statement files. Six theorems from the literature (Stewart; Castryck–Cluckers–Dittmann–Nguyen; Gelfond; Reuss; Heath-Brown; Tao) enter the formal statements as explicit, named hypotheses, each compared with its source text. The Lean sources, replay receipts and the record of assumptions are in the accompanying repository, release v1.0.0: https://github.com/dannyply/math-frontier (archived at https://doi.org/10.5281/zenodo.23614472).

The results, proofs and text were produced with AI systems.