A187776 -id:A187776 - cp's OEIS frontend

A003174 Positive integers D such that Q[sqrt(D)] is a quadratic field which is norm-Euclidean.

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

Offset: 1

These integers yield norm-Euclidean real quadratic fields. There are other positive integers, e.g., D=14 or D=69, for which Q[sqrt(D)] is Euclidean, but for a Euclidean function different from the field norm.

For further references see sequence A048981 which also lists negative D corresponding to (complex) norm-Euclidean fields. - M. F. Hasler, Jan 26 2014

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

H. Chatland and H. Davenport, Euclid's algorithm in real quadratic fields, Canadian J. Math. 2, (1950), 289-296.
S. R. Finch, Class number theory [Cached copy, with permission of the author]
Pierre Samuel, Unique factorization, Amer. Math. Monthly 75 (1968), 945-952.
Index entries for sequences related to quadratic fields

Cf. A003173, A003246, A048981, A187776, A263465.

is_A003174(n) = bittest(9444877083272958060780,n) \\ M. F. Hasler, Jan 26 2014

a(n) = A048981(n+5). - M. F. Hasler, Jan 26 2014

Definition corrected and comment rephrased by M. F. Hasler, Jan 26 2014

Definition corrected by Jonathan Sondow, Oct 19 2015

A236546 Discriminant of A048981(n) (= squarefree integers for which the quadratic field Q[sqrt(D)] is norm-Euclidean).

Original entry on oeis.org

-11, -7, -3, -8, -4, 8, 12, 5, 24, 28, 44, 13, 17, 76, 21, 29, 33, 37, 41, 57, 73

Offset: 1

M. F. Hasler, Jan 28 2014

sign

Note that here, values are not sorted by size, but in direct correspondence with terms of A048981. This is in contrast to A003246, where the same positive values are listed by size.

Cf. A003174, A003246, A037449, A048981, A204993, A187776.

for(D=-11,73,is_A048981(D)&&print1(quaddisc(D)","))

PARI
```
A236546 = n->quaddisc(A048981(n))
```

a(n) = A037449(A048981(n)) for n>5, a(n) = -A204993(-A048981(n)) for n <= 5.

A187777 3-smooth Euclidean fields that are not 2-smooth Euclidean.

Original entry on oeis.org

43, 46, 61, 69, 93, 109, 133, 157, 177, 181, 201, 217, 241, 409

Offset: 1

Arkadiusz Wesolowski, Jan 05 2013

Xavier Guitart and Marc Masdeu, Continued fractions in 2-stage euclidean quadratic fields, arXiv:1106.0856 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Quadratic Field
Index entries for sequences related to quadratic fields

Cf. A003174, A187776.

cp's OEIS Frontend

A003174 Positive integers D such that Q[sqrt(D)] is a quadratic field which is norm-Euclidean.

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A236546 Discriminant of A048981(n) (= squarefree integers for which the quadratic field Q[sqrt(D)] is norm-Euclidean).

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PARI

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A187777 3-smooth Euclidean fields that are not 2-smooth Euclidean.

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