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Brown, Ezra

Representations of discriminantal divisors by binary quadratic forms. (English) Zbl 0216.04203

J. Number Theory 3, 213-225 (1971).
It is implicit in the work of Gauss on binary quadratic forms that the principal class of discriminant \(d\) represents, besides 1, exactly one divisor of \(d\) from a certain set of those divisors (the so-called discriminantal divisors). It is natural to ask, which divisor is represented? (The case of the divisor being \(-1\) has been studied since the time of Fermat in connection, with the equation, \(t^2 - du^2 = -1.\) In this paper, discriminants \(d = 4pq\), where \(p\equiv q \pmod 4\) are distinct primes, are studied with reference to this problem. The main tool is the arithmetic theory of generalized quaternions, which is used to obtain a complete parametric solution of \(p = x_1^2 + x_2^2 = -x_3^2 + qx_4^2 =x_5^2 + qx_6^2\). The solutions of this system are used to obtain complete answers to the above question in certain cases.
Other relevant articles are by G. Pall [J. Number Theory 1, 525–533 (1969; Zbl 0186.08602)], A. Scholz [Math. Z. 39, 95–111 (1934; Zbl 0009.29402; JFM 60.0126.03)], L. Rédei [J. Reine Angew. Math. 173, 193–221 (1935; Zbl 0012.24602; JFM 61.0138.02)] and G. L. Dirichlet [“Werke”, Vol. I. Berlin: Reimer (1889; JFM 21.0016.01), p. 221–236].
The author has written more articles on this subject, to appear in [J. Number Theory and J. Reine Angew. Math.
Reviewer: Ezra Brown (Blacksburg, VA)
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Cited in 2 Documents

MSC:

11E16 General binary quadratic forms

Keywords:

binary quadratic forms; discriminantal divisors

Citations:

Zbl 0186.08602; Zbl 0009.29402; JFM 60.0126.03; Zbl 0012.24602; JFM 61.0138.02; JFM 21.0016.01
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Full Text: DOI

References:

[1] Cantor, G., Zwei Sätze aus der Theorie der binären quadratischen Formen, Z. Math. Physik, 13, 259-261 (1868) · JFM 01.0054.01
[2] Dirichlet, G. L., (Werke, Vol. I (1889), G. Reimer: G. Reimer Berlin) · JFM 21.0016.01
[3] Leveque, W. J., (Topics In Number Theory, Vol. II (1956), Addison-Wesley: Addison-Wesley Reading, Mass) · Zbl 0070.03803
[4] Pall, G., On Generalized Quaternions, Trans. Amer. Math. Soc., 59, 280-332 (1946) · Zbl 0060.11004
[5] Pall, G., Discriminantal Divisors of Binary Quadratic Forms, J. Number Theory, 1, 525-533 (1969) · Zbl 0186.08602
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