Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

A SIMPLE PROOF FOR JI-KIM-OH'S THEOREM

  • Byeong Moon Kim (Department of Mathematics, Gangneung-Wonju National University) ;
  • Ji Young Kim (Department of Mathematical Sciences, Seoul National University)
  • Received : 2022.10.30
  • Accepted : 2023.03.29
  • Published : 2023.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

In 1911, Dubouis determined all positive integers represented by sums of k nonvanishing squares for all k ≥ 4. As a generalization, Y.-S. Ji, M.-H. Kim and B.-K. Oh determined all positive definite binary quadratic forms represented by sums of k nonvanishing squares for all k ≥ 5. In this article, we give a simple proof for Ji-Kim-Oh's theorem for all k ≥ 10.

Keywords

Acknowledgement

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A3B07048195, NRF-2020R1I1A1A01053318).

References

  1. R. Descartes, Oeuvres(Adam and Tannery, editors), II volumes. Paris, 1898. (See especially letters to Mersenne from July 27 and Aug. 23, 1638 in vol. II, pp. 256 and 337-338, respectively.)
  2. E. Grosswald, Representations of integers as sums of squares, Springer-Verlag, New York, 1985.
  3. Y.-S. Ji, M.-H. Kim, B.-K. Oh, Binary quadratic forms represented by a sum of nonzero squares, J. Number Theory 148 (2015), 257-271. https://doi.org/10.1016/j.jnt.2014.09.009
  4. B. M. Kim, On nonvanishing sum of integral squares of ${\mathbb{Q}}({\sqrt{5}})$, Kangweon-Kyungki Math. Jour. 6 (1998), 299-302.
  5. B. M. Kim, On nonvanishing sum of integral squares of ${\mathbb{Q}}({\sqrt{6}})$, preprint.
  6. B. M. Kim, J. Y. Kim, Sums of nonvanishing integral squares in real quadratic fields, J. Number Theory 177 (2017), 497-515. https://doi.org/10.1016/j.jnt.2017.01.006
  7. O. T. O'Meara, Introduction to Quadratic Forms, Springer-Verlag, Berlin, 1973.