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A003174
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Positive integers D such that Q[sqrt(D)] is a quadratic field which is norm-Euclidean.
(Formerly M0619)
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12
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2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73
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OFFSET
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1,1
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COMMENTS
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These integers yield norm-Euclidean real quadratic fields. There are other positive integers, e.g., D=14 or D=69, for which Q[sqrt(D)] is Euclidean, but for a Euclidean function different from the field norm.
For further references see sequence A048981 which also lists negative D corresponding to (complex) norm-Euclidean fields. - M. F. Hasler, Jan 26 2014
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REFERENCES
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H. Cohn, A Second Course in Number Theory, Wiley, NY, 1962, p. 109.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 213.
K. Inkeri, Über den Euklidischen Algorithmus in quadratischen Zahlkörpern. Ann. Acad. Sci. Fennicae Ser. A. 1. Math.-Phys., No. 41, 1-35, 1947. [Incorrectly gives 97 as a member of this sequence.]
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 2, p. 57.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 294.
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LINKS
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FORMULA
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PROG
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CROSSREFS
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KEYWORD
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fini,nonn,full,nice
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AUTHOR
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EXTENSIONS
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Definition corrected and comment rephrased by M. F. Hasler, Jan 26 2014
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STATUS
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approved
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