Consecutive Powerful Numbers and Pell Equations: Triples, Pairs, and Further Observations

Abstract

Note: the data deposit ZIP (deposit.zip) is temporarily withdrawn and will be provided again soon in a cleaned and usable English version.A positive integer is powerful if every prime that divides it divides it at least twice. Pairs of consecutive powerful numbers exist in abundance, but no three consecutive powerful numbers are known, and Erdős asked whether they exist (Erdős Problems #364 and #365). This volume collects three parts. Part I turns a hypothetical triple into an address on a Pell equation, proves a lower bound for its middle in terms of its kernel, and reduces the existence of a triple to primes that divide a Pell number exactly once. Using Reinhart's complete list of the kernels m up to 1.5·10^12 that divide U_1(m), it shows by computation that no triple has a middle below 1.83·10^18. Independently of this bound, it asks whether the 961 building blocks attached to the lattice points it examines can be powerful: 954 are shown not to be, by witnesses proved prime, and 7 remain open. Part II establishes the explicit lower bound liminf N_1(x)/log x ≥ 1.6822 for the number N_1(x) of pairs n, n+1 with n+1 ≤ x, a logarithmic lower bound at distance two, and shows that the number of pairs at distance two up to x differs by O(x^(2/9)) from the number of kernels m with m | U_1(m), T_1(m) even and T_1(m) ≤ x. Part III collects the gap spectrum of powerful numbers, powerful values of cyclotomic polynomials, and tests of the independence assumptions behind the heuristic, each with the power of the test. Every computed number is generated from deposited data or from a cited constant, every numbered statement is labeled by the kind of evidence behind it (proved, taken from the literature, computed, measured, heuristic, or open), and the open questions are collected as Problems A to I. This record contains the volume and the complete data deposit (core and archive: scripts, result files, certificates, a manifest with hashes). The text and the data are licensed under CC BY 4.0, the code under Apache-2.0. Part of a series of four linked records. Part I: 10.5281/zenodo.23127703 · 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.