On universal zero-free ternary quadratic form representations of primes in arithmetic progressions

JOURNAL OF NUMBER THEORY

卷:171

Article

Hurlimann, Werner¹ 

[1] Swiss Math Soc, CH-1700 Fribourg, Switzerland

关键词: Ternary quadratic form;    Prime;    Theorem of Gauss;    Class number;    Fundamental discriminant;   

DOI  :  10.1016/j.jnt.2016.07.014

来源: Elsevier

【 摘 要 】

For the specific set of fifteen ternary quadratic forms x(2) + by(2) + cz(2), b, c is an element of {1, 2,4, 8}, (b, c) is an element of {(2,16), (8,16), (1,3), (2, 3), (1, 5)}, it is shown that the distinct zero-free representations of an odd prime by these forms depend upon the class numbers h(-kp), k is an element of {1, 3, 4, 5,8, 12, 20, 24}. We determine when such a form is universal zero-free for an arithmetic progression of primes, i.e., when a prime from such a progression can be represented without zero components. The exceptional primes, which cannot be represented in this way, fall into two distinct classes. They are either infinite in number and belong to arithmetic progressions of primes, so-called infinite exceptional sets, or they are finite in number and build so-called finite exceptional sets. These exceptional sets are determined. Moreover, we show how to derive the finite number of primes expressible by a form x(2) + by(2) + cz(2) in essentially m ways, and illustrate the method. Reinterpreting results by Dirich-let [12], Dickson [10] and Kaplansky [23], we show that the forms (b, c) is an element of {(1, 2), (1, 3), (2, 3), (2, 4)1 are the only strictly universal zero-free forms of type x(2) + by(2) + cz(2), i.e., they can be represented without zero components for all primes up to a known finite number of primes. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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