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Cubic congruences and binary quadratic forms
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DOI：10.4064/aa250326-1-10.png)
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Abstract
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Let p > 3 be a prime, a(1), a(2), a(3) is an element of Z, and let Np(x(3) + a(1)x(2 )+ a(2)x + a3) denote the number of solutions to the congruence x(3) + a(1)x(2) + a(2)x + a(3) equivalent to 0 (mod p). We give an explicit criterion for N-p(x(3) + a(1)x(2 )+ a(2)x + a(3)) = 3 via binary quadratic forms.
Keywords:
cubic congruence
cubic Jacobi symbol
binary quadratic form
third-order recurrence sequence