A003246 Discriminants of real quadratic norm-Euclidean fields (a finite sequence). (Formerly M3778)

5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 41, 44, 57, 73, 76

Offset

1,1

Comments

Euclidean fields that are not norm-Euclidean, such as Q(sqrt(14)) and Q(sqrt(69)), are not included. Actually, assuming GCH, a real quadratic field is Euclidean if and only if it is a PID (equivalently, if and only if it is a UFD). - Jianing Song, Jun 09 2022

References

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

Table of n, a(n) for n=1..16.

S. R. Finch, Class number theory [Cached copy, with permission of the author]

Erich Kaltofen and Heinrich Rolletschek, Computing greatest common divisors and factorizations in quadratic number fields, Mathematics of Computation 53.188 (1989): 697-720. See page 698.

A. M. Odlyzko, Letters to N. J. A. Sloane Feb 1974

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

Peter J. Weinberger, On Euclidean rings of algebraic integers, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 321-332.

Index entries for sequences related to quadratic fields

Formula

A003246 = A037449(A003174) as a set, not composition of functions (values are sorted by size; it turns out that a(n) is different from A037449(A003174(n)) for all n=1,...,16. - M. F. Hasler, Jan 26 2014

Mathematica

A003174 = {2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73}; Sort[ NumberFieldDiscriminant /@ Sqrt[A003174]] (* Jean-François Alcover, Jul 18 2012 *)

Prog

(PARI) for(n=1, 99, is_A003174(n) && print1(quaddisc(n)", ")) \\ M. F. Hasler, Jan 26 2014

Crossrefs

Cf. A003174, A003656.

Sequence in context: A133315 A003658 A003656 * A143748 A124378 A066299

Adjacent sequences: A003243 A003244 A003245 * A003247 A003248 A003249

Keyword

fini,full,nonn,nice

Author

N. J. A. Sloane

Status

approved