

A003174


Positive integers D such that Q[sqrt(D)] is a quadratic field which is normEuclidean.
(Formerly M0619)


10



2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73
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OFFSET

1,1


COMMENTS

These integers yield normEuclidean 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) normEuclidean fields.  M. F. Hasler, Jan 26 2014


REFERENCES

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, 135, 1947. [Incorrectly gives 97 as a member of this sequence.]
W. J. LeVeque, Topics in Number Theory. AddisonWesley, 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.


LINKS

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), 289296.
S. R. Finch, Class number theory [Cached copy, with permission of the author]
Pierre Samuel, Unique factorization, Amer. Math. Monthly 75 (1968), 945952.
Index entries for sequences related to quadratic fields


FORMULA

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


PROG

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


CROSSREFS

Cf. A003173, A003246, A048981, A187776, A263465.
Sequence in context: A283599 A096530 A299157 * A238463 A166070 A053813
Adjacent sequences: A003171 A003172 A003173 * A003175 A003176 A003177


KEYWORD

fini,nonn,full,nice


AUTHOR

N. J. A. Sloane.


EXTENSIONS

Definition corrected and comment rephrased by M. F. Hasler, Jan 26 2014
Definition corrected by Jonathan Sondow, Oct 19 2015


STATUS

approved



