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On Euclid's algorithm in real quadratic fields

Published online by Cambridge University Press:  24 October 2008

H. Heilbronn
H. Heilbronn
Affiliation:
Trinity College
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Extract

The object of this paper is to complete the proof of the

Theorem. Let P(√d) be the quadratic field of discriminant d > 0. Then Euclid's algorithm does not hold in P(√d) if d is sufficiently large.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 34 , Issue 4 , October 1938 , pp. 521 - 526
Copyright
Copyright © Cambridge Philosophical Society 1938

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References

† Kungl. Fysiogr. Sällskapeti Lund Förhandlingar, 5 (1935), 5.Google Scholar

‡ Journal für Math. 174 (1936), 192–205.Google Scholar

§ Journal London Math. Soc. 13 (1938), 3–8.Google Scholar

∥ We say that u is a quadratic residue mod v in the naive sense, if the congruence y 2 ≡ u (mod v) has a solution.

† Vinogradov, , Trans. Amer. Math. Soc. 29 (1927), 218–26Google Scholar proved His proof is easily generalized to obtain Lemma 3. See also Erdös and Ko, loc. cit. Lemma 3.

† It may, of course, happen that a 1 and a 2 are both residues.