The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.



(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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: A096530 A299157 A325327 * A238463 A340076 A340150

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



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 23 06:03 EDT 2021. Contains 343199 sequences. (Running on oeis4.)