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.
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Y.-S. Ji and M.-H. Kim were supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (2010-0015670).
B.-K. Oh was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2010-0019516).
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Ji, YS., Kim, MH. & Oh, BK. Positive definite quadratic forms representing integers of the form an 2+b . Ramanujan J 27, 329–342 (2012). https://doi.org/10.1007/s11139-011-9323-0
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DOI: https://doi.org/10.1007/s11139-011-9323-0