Cubic congruences and binary quadratic forms

2025/03/17 by Sun, Zhi-Hong
#11B37 #11B39 #11E16 #FOS: Mathematics #Number Theory (math.NT) #Primary 11A07 #Secondary 11A15

paper · doi:10.48550/arxiv.2503.12861

Abstract

Let p>3 be a prime, a₁,a₂,a₃∈\Bbb Z and let N_p(x³+a₁x²+a₂x+a₃) denote the number of solutions to the congruence x³+a₁x²+a₂x+a₃≡ 0\pmod p. In this paper, we give an explicit criterion for N_p(x³+a₁x²+a₂x+a₃)=3 via binary quadratic forms.

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