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

Author

N. J. A. Sloane, Jud McCranie

Keywords

• fini
• sign
• full
• nice

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

• 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

Crossrefs

Cf. A003173, A003174, A263465.

Programs

Maple

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

Formula

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

Extensions

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