Bennett 2024 computing four term arithmetic progressions powerful numbers

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evidence/: Exact integer check of the published 111-digit four-term progression of powerful numbers.

section_4_record_example: Records and directly verifies the published Bennett-Walsh record example of signature [1,3,5,7].

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Michael A. Bennett and P. G. Walsh, "Computing four-term arithmetic progressions of powerful numbers," INTEGERS 24A (2024), article A3, 5 pp. DOI.

The paper searches for coprime four-term progressions with a fixed signature $[\alpha,\beta,\gamma,\delta]$, meaning that the terms are respectively $\alpha^3x^2,\beta^3y^2,\gamma^3z^2,\delta^3w^2$ for positive integers $\alpha,\beta,\gamma,\delta$; the search takes $\alpha,\beta,\gamma$ odd and squarefree. It parametrizes the first three terms by a ternary quadratic form, obtains a genus-one quartic from the fourth, maps that curve to an elliptic curve, and searches a computed subgroup of its rational points.

Section 4 publishes a 111-digit example of signature $[1,3,5,7]$, improving the 190-digit example of Bajpai, Bennett and Chan (Int. J. Number Theory 20 (2024), 19–45). The abstract calls it the current "record" example, and Section 4 reports that "the smallest quadruple with signature $[1, 3, 5, 7]$ has integers with only 111 digits" (p. 4). It is not a proof that the example is least possible: the authors say only that it may be minimal and that a proof would need a careful analysis of the quadratic forms and of generators for the elliptic curves.

Canonical PDF. Published paper, published May 27, 2024; publisher-hosted source. No notice is printed; the journal's home page states "All works of this journal are licensed under a Creative Commons Attribution 4.0 International License so that all content is freely available without charge to the users or their institutions." (https://math.colgate.edu/~integers/, read 2026-10-02): the Creative Commons Attribution 4.0 license.

Bears on. #937.