A simple proof for Ji-Kim-Oh's theorem
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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 $k \geq 10$.
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 $k \geq 10$.
참고문헌 (7)
- R. Descartes, "Oeuvres II volumes", ~ (1898)
- E. Grosswald, "Representations of integers as sums of squares" (1985), Springer-Verlag
- Y. -S. Ji, "Binary quadratic forms represented by a sum of nonzero squares", J. Number Theory, 148, pp.257~271 (2015)
- B. M. Kim, "On nonvanishing sum of integral squares of Q(√5)", Kangweon-Kyungki Math. Jour, 6, pp.299~302 (1998)
- B. M. Kim, "On nonvanishing sum of integral squares of Q(√6)", ~
- B. M. Kim, "Sums of nonvanishing integral squares in real quadratic fields", J. Number Theory, 177, pp.497~515 (2017)
- O. T. O’Meara, "Introduction to Quadratic Forms" (1973), Springer-Verlag