Further Observations on Powerful Numbers: Gaps, Cyclotomic Values, and Independence Tests

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Note: the data deposit ZIP (deposit.zip) is temporarily withdrawn and will be provided again soon in a cleaned and usable English version.We collect observations on powerful numbers that arose in the study of triples and pairs of consecutive powerful numbers. Up to 10^14, only 0.10 percent of the gap values between consecutive powerful numbers are powerful, but the most frequent gap values are, and a lemma of Bajpai, Bennett and Chan gives a mechanism for this. We give two explicit infinite chains, each giving powerful values of both x^2 + x + 1 and x^2 − x + 1, list the powerful values of the cyclotomic polynomials Φ_d with 3 ≤ d ≤ 12 up to 10^12, and show that x^4 + 1 = q^3 a^2 has no solution for primes q < 2·10^11. We test the independence of rare Wieferich-type events along the divisors of an index, with the power of the test, and measure the Wieferich fractions of Pell units against the equidistribution predicted by a conjecture of Katz. Every computed number is generated from the deposited data, 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 I 10.5281/zenodo.23127703 · Part II 10.5281/zenodo.23127705. Further observations on the smallest witnesses and the factor searches on three open building blocks will be published as new versions of this record.