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Integers with a sum of co-divisors yielding a square

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Abstract

Finding elliptic curves with high ranks has been the focus of much research. Recently, with the goal of generating elliptic curves with a large rank, some authors used large integers n which have many divisors, amongst which one can find divisors d such that \(d+n/d\) is a perfect square. This strategy is in itself a motivation for studying the function \(\tau _\Box (n)\) which counts the number of divisors d of an integer n for which \(d+n/d\) is a perfect square. We show that \(\sum _{n\le x} \tau _\Box (n) = c_\Box x^{3/4} +O(\sqrt{x})\) for some explicit constant \(c_\Box \). Moreover, letting \(\rho _1(n):=\max \{d\mid n: d\le \sqrt{n}\}\) and \(\rho _2(n):=\min \{d\mid n: d\ge \sqrt{n}\}\) stand for the middle divisors of n, we show that the order of magnitude of the number of positive integers \(n\le x\) for which \(\rho _1(n)+\rho _2(n)\) is a perfect square is \(x^{3/4}/\log x\).

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Data Availability

Section 6 of the paper includes the data supporting the results proved in the paper. The data in Tables 1 and 2 was generated using the software system Mathematica.

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Acknowledgements

The authors are grateful to the referee for the careful reading of the paper and for helpful suggestions. The work of the first author was funded partially by a grant from the Natural Sciences and Engineering Research Council of Canada (Grant No. 02485).

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Author notes

  1. A. Arthur Bonkli Razafindrasoanaivolala and Hans Schmidt Ramiliarimanana have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Université Laval, 1045 Avenue de la médecine, Québec, QC, G1V0A6, Canada

    Jean-Marie De Koninck & A. Arthur Bonkli Razafindrasoanaivolala

  2. AIMS Rwanda Centre, Sector Remera, KN3, Kigali, Rwanda

    Hans Schmidt Ramiliarimanana

Authors
  1. Jean-Marie De Koninck
  2. A. Arthur Bonkli Razafindrasoanaivolala
  3. Hans Schmidt Ramiliarimanana

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Correspondence to Jean-Marie De Koninck.

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De Koninck, JM., Razafindrasoanaivolala, A.A.B. & Ramiliarimanana, H.S. Integers with a sum of co-divisors yielding a square. Res. number theory 10, 30 (2024). https://doi.org/10.1007/s40993-024-00520-x

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