The cubic congruence x3 + ax2 + bx + c ≡ = 0(mod p) and binary quadratic forms F(x, y) = ax2 + bxy + cy 2

TEKCAN A.

Ars Combinatoria, vol.85, pp.257-269, 2007 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 85
  • Publication Date: 2007
  • Journal Name: Ars Combinatoria
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.257-269
  • Keywords: Binary quadratic form, Cubic congruence, Cubic residue, Representation of primes by binary quadratic forms
  • Bursa Uludag University Affiliated: Yes

Abstract

Let F(x, y) = ax2; + bxy + cy2 be a binary quadratic form of discriminant Δ = b2 - 4ac for a, b, c ∈ Z, let p be a prime number and let Fp be a finite field. In this paper we formulate the number of integer solutions of cubic congruence x3 + ax2 + bx + c ≡ 0 (mod p) over Fp for two specific binary quadratic forms Flk (x, y) = x2 + kxy + ky2 and F2k(x, y) = kxy + kxy + k 2y2 for integer k such that 1 < k < 9. Later we consider representation of primes by F1k and F 2k.