Triples of Consecutive Powerful Numbers: Pell Addresses, a Height Inequality, and Witnesses

Abstract

Note: the data deposit ZIP (deposit.zip) is temporarily withdrawn and will be provided again soon in a cleaned and usable English version.We study the question of Erdős whether three consecutive integers can all be powerful. The middle n of such a triple is a Pell number T_k(m) at a lattice point (m, k) with a squarefree kernel m ≡ 7 (mod 8). We prove a height inequality that splits the kernels into two wings by whether m divides U_1(m), and we recast the reduction of Mollin and Walsh as a statement about witnesses, primes that divide a Pell number exactly once, organized by rank. We compute: all 445 lattice points with kernel up to 10^8 and T_k < 10^2000 are excluded by parity or by a certified witness, and with the height inequality and Reinhart's list of the kernels dividing U_1 up to 1.5·10^12 this shows that no triple has a middle below 1.83·10^18; further regions are excluded whatever the size of the middle; and of the 961 building blocks M_d of these lattice points, 954 are shown not to be powerful by a certified witness, a prime proved prime that divides the block exactly once, and 7 remain open. A profile theorem collects the conditions on a counterexample that we prove or compute, and the weak point of the bound, the kernels dividing U_1, is stated as an open problem. Every computed number is generated from the deposited data or from a cited constant, and every numbered statement is labeled as proved, taken from the literature, computed, measured, heuristic, or open. The data deposit is part of the record of the collected volume, 10.5281/zenodo.23127547; the text is licensed under CC BY 4.0. Part of a series of four linked records: collected volume 10.5281/zenodo.23127547 · Part II 10.5281/zenodo.23127705 · Part III 10.5281/zenodo.23127707. Further observations on the smallest witnesses and the factor searches on three open building blocks will be published as new versions of this record.