A003656 Discriminants of real quadratic fields with unique factorization.

5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 41, 44, 53, 56, 57, 61, 69, 73, 76, 77, 88, 89, 92, 93, 97, 101, 109, 113, 124, 129, 133, 137, 141, 149, 152, 157, 161, 172, 173, 177, 181, 184, 188, 193, 197, 201, 209, 213, 217, 233, 236, 237, 241, 248, 249, 253, 268, 269

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• nice

Comments

Discriminants of real quadratic fields with class number 1.

Other than the term 8, every term 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. In fact, 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 the form (i), (ii) or (iii). See Theorem 1 and Theorem 2 of Ezra Brown's link. - Jianing Song, Feb 24 2021

References

• D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
• H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, p. 534.
• H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 576.
• Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.
• Henri Cohen and X.-F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Math. Comp. 69 (2000), 1229-1244.
• Eric Weisstein's World of Mathematics, Class Number
• Index entries for sequences related to quadratic fields

Crossrefs

Cf. A003652, A003658, A014602 (imaginary case).

For discriminants of real quadratic number fields with class number 2, 3, ..., 10, see A094619, A094612-A094614, A218156-A218160; see also A035120.

Programs

Mathematica

maxDisc = 269; t = Table[ {NumberFieldDiscriminant[ Sqrt[n] ], NumberFieldClassNumber[ Sqrt[n] ]}, {n, Select[ Range[2, maxDisc], SquareFreeQ] } ]; Union[ Select[ t, #[[2]] == 1 && #[[1]] <= maxDisc & ][[All, 1]]] (* Jean-François Alcover, Jan 24 2012 *)

Sage

is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 1
A003656 = lambda n: filter(is_fund_and_qfbcn_1, (1,2,..,n))
A003656(270) # Peter Luschny, Aug 10 2014

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 15 2002