Five regular or nearly-regular ternary quadratic forms

Volume 77 / 1996

William Jagy Acta Arithmetica 77 (1996), 361-367 DOI: 10.4064/aa-77-4-361-367

Abstract

1. Introduction. In a recent article [6], the positive definite ternary quadratic forms that can possibly represent all odd positive integers were found. There are only twenty-three such forms (up to equivalence). Of these, the first nineteen were proven to represent all odd numbers. The next four are listed as "candidates". The aim of the present paper is to show that one of the candidate forms h = x² + 3y² + 11z² + xy + 7yz does represent all odd (positive) integers, and that it is regular in the sense of Dickson. We will consider a few other forms, including one in the same genus as h that is a "near miss", i.e. it fails to represent only a single number which it is eligible to represent. Our methods are similar to those in [4]. A more recent article with a short history and bibliography of work on regular ternary forms is [3].

Authors

  • William Jagy

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