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The cubic congruence \(x^3+ax^2+bx+c\equiv 0\pmod p\) and binary quadratic forms \(F(x,y)=ax^2+bxy+cy^2\). II. (English) Zbl 1212.11064

Summary: Let \(F(x,y)=ax^2+bxy+cy^2\) be an integral binary quadratic form of discriminant \(\Delta=b^2-4ac\), let \(p\geq 5\) be a prime number and let \(\mathbb F_p\) be a finite field. In the first section, we give some preliminaries from cubic congruence and binary quadratic forms. In the second section, we consider the number of integer solutions of quadratic congruence \(x^2\equiv \pm k\pmod p\), where \(k\) is an integer such that \(1\leq k\leq 10\). In the third section, we consider the number of integer solutions of cubic congruences \(x^3+ax^2+bx+c\equiv 0\pmod p\) over \(\mathbb F_p\) for two specific binary quadratic forms \(F^k_1(x,y)=2kx^2+kxy+2k^2y^2\) and \(F^k_2(x,y)=-3kx^2-kxy+3k^2y^2\). In the last section, we consider representation of primes by \(F^k_1\) and \(F^k_2\).
For Part I, see Ars Comb. 85, 257–269 (2007; Zbl 1204.11068).

MSC:

11D79 Congruences in many variables
11E16 General binary quadratic forms
11E25 Sums of squares and representations by other particular quadratic forms
11D25 Cubic and quartic Diophantine equations

Citations:

Zbl 1204.11068
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