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A003174 Positive integers D such that Q[sqrt(D)] is a quadratic field which is norm-Euclidean.
(Formerly M0619)
2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 (list; graph; refs; listen; history; text; internal format)



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 an 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


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.


Table of n, a(n) for n=1..16.

H. Chatland, H. Davenport, Euclid’s algorithm in real quadratic fields, Canadian J. Math. 2, (1950), 289-296.

S. R. Finch, Class number theory [Cached copy, with permission of the author]

Pierre Samuel, Unique factorization, Amer. Math. Monthly 75 (1968), 945-952.

Index entries for sequences related to quadratic fields


a(n) = A048981(n+5). - M. F. Hasler, Jan 26 2014


(PARI) is_A003174(n) = bittest(9444877083272958060780, n) \\ M. F. Hasler, Jan 26 2014


Cf. A003173, A003246, A048981, A187776, A263465.

Sequence in context: A283599 A096530 A299157 * A238463 A166070 A053813

Adjacent sequences:  A003171 A003172 A003173 * A003175 A003176 A003177




N. J. A. Sloane.


Definition corrected and comment rephrased by M. F. Hasler, Jan 26 2014

Definition corrected by Jonathan Sondow, Oct 19 2015



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Last modified January 17 16:53 EST 2019. Contains 319235 sequences. (Running on oeis4.)