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An Elementary Proof of a Theorem About the Representation of Primes by Quadratic Forms

Published online by Cambridge University Press:  20 November 2018

W. E. Briggs
W. E. Briggs*
Affiliation:
University of Colorado
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The theorem that every properly primitive binary quadratic form is capable of representing infinitely many prime numbers was first proved completely by H. Weber (5). The purpose of this paper is to give an elementary proof of the case where the form is ax2 + 2bxy + cy2, with a > 0, (a, 2b, c) = 1, and D = b2 — ac not a square. The cases where the form is ax2 + bxy + cy2 with b odd, and the case where the form is ax2+ 2bxy + cy2 with D a square, can be settled very simply once the first case is taken care of, and this is done in a page and a half in the Weber paper.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 6 , 1954 , pp. 353 - 363
Copyright
Copyright © Canadian Mathematical Society 1954

References

1. Dirichlet, P. G. L., Vorlesungen über Zahlentheorie (Brauschweig, 1879).Google Scholar
2. Landau, E., Vorlesungen über Zahlentheorie, vol. 1 (New York, 1950).Google Scholar
3. Selberg, A., An elementary proof of Dirichlet's theorem about primes in an arithmetic progression, Annals of Math., 50 (1949), 297–304.Google Scholar
4. Selberg, A., An elementary proof of the prime-number theorem for arithmetic progressions, Can. J. Math., 2 (1950), 66–78.Google Scholar
5. Weber, H., Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähigist, Math. Annalen, 20 (1882), 301–329.Google Scholar