2025/01/25 by Li, Jiamin, Liu, Jianya
#11D09 #11N36 #11P32 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2501.15033
Let f(x1, x2, x3) be an indefinite anisotropic integral quadratic form with determinant d(f), and t a non-zero integer such that d(f)t is square-free. It is proved in this paper that, as long as there is one integral solution to f(x1, x2, x3) = t, there are infinitely many such solutions for which (i) x1 has at most 6 prime factors, and (ii) the product x1 x2 has at most 16 prime factors. Various methods, such as algebraic theory of quadratic forms, harmonic analysis, Jacquet-Langlands theory, as well as combinatorics, interact here, and the above results come from applying the sharpest known bounds towards Selberg's eigenvalue conjecture. Assuming the latter the number 6 or 16 may be reduced to 5 or 14, respectively.