A003174 Positive integers D such that Q[sqrt(D)] is a quadratic field which is norm-Euclidean.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• fini
• nonn
• full
• nice

Comments

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

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, 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.

Links

• H. Chatland and 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

Crossrefs

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

Programs

PARI

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

Formula

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

Extensions

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

Definition corrected by Jonathan Sondow, Oct 19 2015