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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• fini
• full
• nonn
• nice

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

• 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

Crossrefs

Cf. A003174, A003656.

Programs

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 *)

PARI

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

Formula

Equals 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