2024/08/27 by Stephan Baier, Baier, Stephan, Aishik Chattopadhyay +1 · 3 citations
Computer Science · Mathematics · #11D79 #11E04 #11E25 #11L40 #11T24 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2408.15360
openalex publication_date 2024/08/27 · openalex created_date 2024/09/29 · openalex updated_date 2026/07/28
We study small non-trivial solutions of quadratic congruences of the form x12+α2x22+α3x32≡ 0 \bmodq, with q being an odd natural number, in an average sense. This extends previous work of the authors in which they considered the case of prime power moduli q. Above, α2 is arbitrary but fixed and α3 is variable, and we assume that (α2α3,q)=1. We show that for all α3 modulo q which are coprime to q except for a small number of α3's, an asymptotic formula for the number of solutions (x1,x2,x3) to the congruence x12+α2x22+α3x32≡ 0 \bmodq with max\|x1|,|x2|,|x3|\≤ N and (x3,q)=1 holds if N≥ q11/24+ε and q is large enough. It is of significance that we break the barrier 1/2 in the above exponent. Key tools in our work are Burgess's estimate for character sums over short intervals and Heath-Brown's estimate for character sums with binary quadratic forms over small regions whose proofs depend on the Riemann hypothesis for curves over finite fields. We also formulate a refined conjecture about the size of the smallest solution of a ternary quadratic congruence, using information about the Diophantine properties of its coefficients.