Five regular or nearly-regular ternary quadratic forms
1. Introduction. In a recent article , 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 . A more recent article with a short history and bibliography of work on regular ternary forms is .
-  H. Brandt und O. Intrau, Tabelle reduzierten positiver ternärer quadratischer Formen, Abh. Sächs. Akad. Wiss. Math.-Natur. Kl. 45 (1958).
-  J. S. Hsia, Two theorems on integral matrices, Linear and Multilinear Algebra 5 (1978), 257-264.
-  J. S. Hsia, Regular positive ternary quadratic forms, Mathematika 28 (1981), 231-238.
-  B. W. Jones and G. Pall, Regular and semi-regular positive ternary quadratic forms, Acta Math. 70 (1939), 165-191.
-  I. Kaplansky, The first nontrivial genus of positive definite ternary forms, Math. Comp. 64 (1995), 341-345.
-  I. Kaplansky, Ternary positive quadratic forms that represent all odd positive integers, Acta Arith. 70 (1995), 209-214.