Positive definite quadratic forms representing integers of the form an 2+b
Article
First Online:
Received:
Accepted:
- 118 Downloads
Abstract
For any subset S of positive integers, a positive definite integral quadratic form is said to be S-universal if it represents every integer in the set S. In this article, we classify all binary S-universal positive definite integral quadratic forms in the case when S=S a ={an 2∣n≥2} or S=S a,b ={an 2+b∣n∈ℤ}, where a is a positive integer and ab is a square-free positive integer in the latter case. We also prove that there are only finitely many S a -universal ternary quadratic forms not representing a. Finally, we show that there are exactly 15 ternary diagonal S 1-universal quadratic forms not representing 1.
Keywords
S-universal quadratic formsMathematics Subject Classification (2000)
11E12 11E20Preview
Unable to display preview. Download preview PDF.
References
- 1.Bhargava, M.: On the Conway–Schneeberger fifteen theorem. In: Quadratic Forms and Their Applications Dublin. Contemporary Math., vol. 272, pp. 27–37. Am. Math. Soc, Providence (2000) CrossRefGoogle Scholar
- 2.Bhargava, M., Hanke, J.: Universal quadratic forms and the 290 theorem. Invent. Math. (to appear) Google Scholar
- 3.Delone, B.N.: Geometry of positive quadratic forms. Usp. Math. Nauk 4 (1938) Google Scholar
- 4.Dujella, A., Jadrijević, B.: A family of quartic Thue inequalities. Acta Arith. 111, 61–76 (2004) MathSciNetMATHCrossRefGoogle Scholar
- 5.Jagy, W.C.: Five regular or nearly-regular ternary quadratic forms. Acta Arith. 77, 361–367 (1996) MathSciNetMATHGoogle Scholar
- 6.Kaplansky, I.: Ternary positive forms that represent all odd positive integers. Acta Arith. 120, 209–214 (1995) MathSciNetGoogle Scholar
- 7.Kim, B.M., Kim, M.-H., Oh, B.-K.: A finiteness theorem for representability of quadratic forms by forms. J. Reine Angew. Math. 581, 23–30 (2005) MathSciNetMATHCrossRefGoogle Scholar
- 8.Kitaoka, Y.: Arithmetic of Quadratic Forms. Cambridge University Press, Cambridge (1993) MATHCrossRefGoogle Scholar
- 9.Oh, B.-K.: Representations of arithmetic progressions by positive definite quadratic forms. Int. J. Number Theory doi: 10.1142/S1793042111004915
- 10.O’Meara, O.T.: Introduction to Quadratic Forms. Springer, New York (1963) MATHGoogle Scholar
- 11.Watson, G.L.: Determination of a binary quadratic form by its values at integer points. Mathematika 26, 72–75 (1979) MathSciNetMATHCrossRefGoogle Scholar
Copyright information
© Springer Science+Business Media, LLC 2011