International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 5, Pages 289-292

Kaplansky's ternary quadratic form

James Kelley

Department of Mathematics, University of California at Berkeley, Berkeley 94709, CA, USA

Received 26 February 1998; Revised 17 May 2000

Copyright © 2001 James Kelley. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper proves that if N is a nonnegative eligible integer, coprime to 7, which is not of the form x2+y2+7z2, then N is square-free. The proof is modelled on that of a similar theorem by Ono and Soundararajan, in which relations between the number of representations of an integer np2 by two quadratic forms in the same genus, the pth coefficient of an L-function of a suitable elliptic curve, and the class number formula prove the theorem for large primes, leaving 3 cases which are easily numerically verified.