A048981 Squarefree values of n for which the quadratic field Q[ sqrt(n) ] is norm-Euclidean.

-11, -7, -3, -2, -1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73

Offset

1,1

Comments

These are norm-Euclidean fields, excluding for instance Q[sqrt(69)] which is Euclidean but not for norm. - Marc A. A. van Leeuwen, Feb 15 2011

References

H. Cohn, A Second Course in Number Theory, Wiley, NY, 1962, pp. 107, 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.

H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 294.

Links

Table of n, a(n) for n=1..21.

Alexander Bogomolny, Strange Integers

Kyle Bradford and Eugen J. Ionascu, Unit Fractions in Norm-Euclidean Rings of Integers, arXiv:1405.4025 [math.NT], May 2014 (see p. 3).

Eugen J. Ionascu and Kyle Bradford, Unit Fractions in Norm-Euclidean Rings of Integers, Acta Mathematica Universitatis Comenianae, 86(1), 127-141.

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

Eric Weisstein's World of Mathematics, Quadratic Field

Wikipedia, Norm-Euclidean field.

Index entries for sequences related to quadratic fields

Formula

a(n) = -A003173(6-n) = -A263465(6-n) for n = 1, 2, 3, 4, 5. - Jonathan Sondow, Dec 09 2015

Maple

select(t -> traperror(numtheory:-factorEQ(-1, t)) <> lasterror, [$-11..77]); # Robert Israel, Jul 20 2016

Crossrefs

Cf. A003173, A003174, A263465.

Sequence in context: A341195 A005603 A109828 * A132361 A305273 A236546

Adjacent sequences: A048978 A048979 A048980 * A048982 A048983 A048984

Keyword

fini,sign,full,nice

Author

N. J. A. Sloane, Jud McCranie

Extensions

Name corrected by Marc A. A. van Leeuwen, Feb 15 2011

Status

approved