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A048981
Squarefree values of n for which the quadratic field Q[ sqrt(n) ] is norm-Euclidean.
8
-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
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
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
KEYWORD
fini,sign,full,nice
EXTENSIONS
Name corrected by Marc A. A. van Leeuwen, Feb 15 2011
STATUS
approved

  NODES
orte 1
see 2
Story 1