A simple proof for Ji-Kim-Oh's theorem
Main Article Content
Abstract
In 1911, Dubouis determined all positive integers represented by sums of $k$ nonvanishing squares for all $k \geq 4$. 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 $k \geq 5$. In this article, we give a simple proof for Ji-Kim-Oh's theorem for all $k \geq 10$.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.
Supporting Agencies
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.) Google Scholar
[2] E. Grosswald, Representations of integers as sums of squares, Springer-Verlag, New York, 1985. Google Scholar
[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. Google Scholar
[4] B. M. Kim, On nonvanishing sum of integral squares of Q(sqrt{5}), Kangweon-Kyungki Math. Jour. 6 (1998), 299–302. Google Scholar
[5] B. M. Kim, On nonvanishing sum of integral squares of Q(sqrt{6}), preprint. Google Scholar
[6] B. M. Kim, J. Y. Kim, Sums of nonvanishing integral squares in real quadratic fields, J. Number Theory 177 (2017), 497–515. Google Scholar
[7] O. T. O’Meara, Introduction to Quadratic Forms, Springer-Verlag, Berlin, 1973. Google Scholar