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There are 913 regular ternary forms. (English) Zbl 0923.11060
A positive definite integral quadratic form is said to be regular if it represents every integer represented by its genus. The systematic study of such forms was initiated by L. E. Dickson [Ann. Math. (2) 28, 333-341 (1927; JFM 53.0133.03)]. In an unpublished thesis [Some problems in the theory of numbers, Ph.D. thesis, University College, London (1953)], G. L. Watson proved that there are only finitely many inequivalent primitive regular forms in three variables, and determined all such forms with discriminants not divisible by 4. (Watson’s much later paper on the subject [J. Lond. Math. Soc. (2) 13, 97-102 (1976; Zbl 0319.10024)] treats only the case of square-free discriminants.) The finiteness also follows from the more general considerations of Watson in [Mathematika 1, 104-110 (1954; Zbl 0056.27201)] regarding the size of the exceptional set of integers represented by the genus of a primitive positive definite ternary quadratic form but not by the form itself.
In the present paper, the authors announce the results of a continuation of the work of Watson on the problem of determining all primitive ternary regular forms. The paper consists primarily of a list containing representatives of all equivalence classes of forms of this type. Of the 913 forms on the list, the regularity of 22 remains to be proven, although these forms have been checked by the authors to represent all eligible integers up to two million. A future paper, containing detailed proofs and descriptions of the computations used, is promised.

MSC:
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
11E12 Quadratic forms over global rings and fields
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[1] DOI: 10.1112/jlms/s2-13.1.97 · Zbl 0319.10024 · doi:10.1112/jlms/s2-13.1.97
[2] Watson, Mathematika 22 pp 1– (1975)
[3] Kaplansky, Mathematika 42 pp 444– (1995)
[4] Brandt, Abh. Sächs. Akacl. Wiss. Math.-Nat. Kl 45 (1958)
[5] Hsia, Mathematika 28 pp 231– (1981)
[6] DOI: 10.2307/1968378 · JFM 53.0133.03 · doi:10.2307/1968378
[7] Jones, Ada Mathematics 70 pp 165– (1939)
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