cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A003653 Discriminants of quadratic fields whose fundamental unit has norm -1.

Original entry on oeis.org

5, 8, 13, 17, 29, 37, 40, 41, 53, 61, 65, 73, 85, 89, 97, 101, 104, 109, 113, 137, 145, 149, 157, 173, 181, 185, 193, 197, 229, 232, 233, 241, 257, 265, 269, 277, 281, 293, 296, 313, 317, 328, 337, 349, 353, 365, 373, 389, 397, 401, 409, 421, 424, 433, 445, 449
Offset: 1

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Author

N. J. A. Sloane, Mira Bernstein

Keywords

References

  • D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Cf. A003658, A349649 (discriminants of real quadratic fields whose fundamental unit has norm 1).

A349419 Fundamental discriminants of real quadratic number fields with odd class number > 1 whose fundamental unit has norm 1.

Original entry on oeis.org

316, 321, 469, 473, 568, 817, 892, 993, 1016, 1101, 1257, 1304, 1393, 1436, 1509, 1641, 1756, 1761, 1772, 1897, 1929, 1957, 1996, 2021, 2101, 2177, 2429, 2589, 2636, 2908, 2913, 2981, 3173, 3261, 3356, 3569, 3736, 3873, 3928, 3941, 3957, 3981, 3997, 4009, 4193, 4281
Offset: 1

Views

Author

Jianing Song, Dec 29 2021

Keywords

Comments

Composite terms of A342368.
For a positive fundamental discriminant d, the class number of the real quadratic field of discriminant d is odd if and only if d = 8 or is of one of the three following forms: (i) p, where p is a prime congruent to 1 modulo 4; (ii) 4p or 8p, where p is a prime congruent to 3 modulo 4; (iii) pq, where p, q are distinct primes congruent to 3 modulo 4. See Theorem 1 and Theorem 2 of Ezra Brown's link. This sequence gives values for d in the cases (ii) and (iii) and that the real quadratic number field with discriminant d has odd class number > 1.

Examples

			316 is a term since the quadratic field with discriminant 316 (Q(sqrt(79))) has class number 3. The fundamental unit of that field (80+9*sqrt(79)) has norm 1.
321 is a term since the quadratic field with discriminant 321 (Q(sqrt(321))) has class number 3. The fundamental unit of that field (215+12*sqrt(321)) has norm 1.

Links

Crossrefs

Intersection of A342368 and A349649. Equals A342368 \ A350165.
Cf. A003658, A003656, A327297.

Programs

  • PARI
    isA349419(D) = if(!isprime(D) && (D>1) && isfundamental(D), my(h=quadclassunit(D)[1]); (h%2)&&(h>1), 0)
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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