Fundamental Discriminant

An integer  is a fundamental discriminant if it is not equal to 1, not divisible by any square of any odd prime, and satisfies  or . The function FundamentalDiscriminantQ[d] in the Wolfram Language version 5.2 add-on package NumberTheory`NumberTheoryFunctions` tests if an integer  is a fundamental discriminant.

It can be implemented as:

  FundamentalDiscriminantQ[n_Integer] := n != 1&&
    (Mod[n, 4] == 1 [Or]
      ! Unequal[Mod[n, 16], 8, 12])&&
        SquareFreeQ[n/2^IntegerExponent[n, 2]]

The first few positive fundamental discriminants are 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, ... (OEIS A003658). Similarly, the first few negative fundamental discriminants are , , , , , , , , , , , ... (OEIS A003657).

REFERENCES:

Atkin, A. O. L. and Morain, F. "Elliptic Curves and Primality Proving." Math. Comput. 61, 29-68, 1993.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 294, 1987.

Cohn, H. Advanced Number Theory. New York: Dover, 1980.

Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, 2005a.

Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 2005b.

Dickson, L. E. History of the Theory of Numbers, Vol. 3: Quadratic and Higher Forms. New York: Dover, 2005c.

Sloane, N. J. A. Sequences A003657/M2332 and A003658/M3776 in "The On-Line Encyclopedia of Integer Sequences."