International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 5, Pages 289-292
doi:10.1155/S0161171201005294
Kaplansky's ternary quadratic form
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.
Abstract
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.