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Fundamental Discriminant

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An integer d is a fundamental discriminant if it is not equal to 1, not divisible by any square of any odd prime, and satisfies d=1 (mod 4) or d=8,12 (mod 16). The function FundamentalDiscriminantQ[d] in the Wolfram Language version 5.2 add-on package NumberTheory`NumberTheoryFunctions` tests if an integer d is a fundamental discriminant.

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 -3, -4, -7, -8, -11, -15, -19, -20, -23, -24, -31, ... (OEIS A003657).

See also

Class Number, Dirichlet L-Series, Discriminant, Regulator

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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."

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Fundamental Discriminant

Cite this as:

Weisstein, Eric W. "Fundamental Discriminant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FundamentalDiscriminant.html

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