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Representation of primes by binary quadratic forms of discriminant –256q and –128q

Published online by Cambridge University Press:  18 May 2009

Franz Halter-Koch
Franz Halter-Koch
Affiliation:
Institut Für MathematikKarl-Franzens-UniversitatHeinrichstrasse 36/IVA–8010 Graz, Osterreich.
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Abstract

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Recently, P. Kaplan and K. S. Williams [10] considered (as an example) the representation of primes by binary quadratic forms of discriminant –768. These forms fall into 4 genera, each consisting of two classes. In particular, they considered the forms

F=3X2+642 and G = 12X2+12XY+19Y2.

It follows from genus theory (as explained in [10]) that every prime p ≡ 19 mod 24 is represented by exactly one of the forms F and G. Based on numerical data, they conjectured that a prime p ≡ 19 mod 24 is represented by

where

Vo = 2, V1 = -4, Vn+2=-4Vn+1 -Vn (n∨0).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 35 , Issue 2 , May 1993 , pp. 261 - 268
Copyright
Copyright © Glasgow Mathematical Journal Trust 1993

References

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