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Rational D(q)-quintuples

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

For a nonzero rational number q, a rational D(q)-n-tuple is a set of n distinct nonzero rationals \(\{a_1, a_2, \dots , a_n\}\) such that \(a_ia_j+q\) is a square for all \(1 \leqslant i < j \leqslant n\). We investigate for which q there exist infinitely many rational D(q)-quintuples. We show that assuming the Parity Conjecture for the twists of several explicitly given elliptic curves, the density of such q is at least \(295026/296010\approx 99.5\%\).

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Notes

  1. https://web.math.pmf.unizg.hr/~duje/dtuples.html.

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Acknowledgements

The author was supported by the Croatian Science Foundation under the project no. 1313. The author is grateful to Andrej Dujella for sharing his unpublished results, as well as motivation for this paper and to Julie Desjardins for helpful comments. The author is very grateful to his mentor Matija Kazalicki for guidance through this paper.

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  1. Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 10000, Zagreb, Croatia

    Goran Dražić

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  1. Goran Dražić
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Correspondence to Goran Dražić.

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Dražić, G. Rational D(q)-quintuples. RACSAM 116, 9 (2022). https://doi.org/10.1007/s13398-021-01146-9

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