A225953 Table for period length of periods of primitive reduced binary quadratic forms with discriminants D(n) = A079896(n).

2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 4, 4, 6, 6, 2, 10, 2, 2, 2, 2, 2, 2, 10, 2, 4, 4, 6, 6, 2, 2, 2, 2, 6, 6, 6, 2, 4, 4, 2, 2, 18, 6, 6, 2, 2, 2, 2, 6, 6, 6, 2, 6, 6, 14, 4, 4, 2, 2, 4, 4, 2, 2, 18, 6, 6, 2, 6, 6, 4, 4, 2, 2, 14, 4, 4, 14, 10, 2, 2, 4, 4, 2, 2, 8, 8, 2

Offset: 1

Author

Wolfdieter Lang, May 27 2013

Keywords

• nonn
• tabf

Comments

The indefinite binary quadratic forms [a,b,c] have discriminant D := b^2 - 4*a*c > 0, not a square, given in A079896.

Primitive forms satisfy gcd(a,b,c) = 1. For the definition of reduced binary quadratic forms see a comment under A087048.

The number of periods of equivalent primitive reduced forms is given in A087048 (the class number).

Here the lengths of these periods is recorded. The computation is based on the book by Scholz and Schoeneberg. The row sums give A082174(n), the number of primitive reduced forms for D(n).

Two forms [a,b,c] and [a',b',c'] are properly equivalent if the 2 x 2 coefficient matrices A := [[a,b/2],[b/2,a]] and A' := [[a',b'/2],[b'/2,a']] satisfy A' = S^{-1,T} A S^{-1} with some matrix S, det S = +1 (T stands for transposed). The indeterminates (x,y) and (x',y') which represent the same number k = (x,y) A (x,y)^T = (x',y') A' (x',y')^T are related then by (x',y')^T = S (x,y)^T.

For the periods of primitive reduced forms for D(n), n = 1, ..., 101, see the link. See also the Buell reference, with the examples on p. 30, giving the periods for n = 1, ..., 20. They coincide with the ones given in the link up to the cyclic order in the periods.

All period lengths are even. See Buell, Proposition 3.6 on p. 24.

Examples

			The irregular table a(n,k) begins:
n/k  1   2  ...   D(n)    A087048(n)   A082174(n)
1:   2              5        1             2
2:   2              8        1             2
3:   2   2         12        2             4
4:   2             13        1             2
5:   6             17        1             6
6:   2             20        1             2
7:   2   2         21        2             4
8:   2   2         24        2             4
9:   4   4         28        2             8
10:  2             29        1             2
11:  2   2         32        2             4
12:  4   4         33        2             8
13:  6             37        1             6
14:  6   2         40        2             8
15: 10             41        1            10
16:  2   2         44        2             4
17:  2   2         45        2             4
18:  2   2         48        2             4
19: 10             52        1            10
20:  2             53        1             2
21:  4   4         56        2             8
... for rows up to n = 101 see the link.
a(1) = 2 because there are 2 = A082174(n) primitive reduced forms with discriminant D = 5, namely  [-1, 1, 1] and  [1, 1, -1], which are equivalent, and they give 1 = A087048(1) period [[[-1, 1, 1],[1, 1, -1]]] of length 2 = a(1,1).
Row n=3 is  2  2, because there are 4 primitive reduced forms for D = 12, appearing in 2 periods, each of length 2: [[[-1, 2, 2], [2, 2, -1]], [[1, 2, -2], [-2, 2, 1]]].
		

References

• D. A. Buell, Binary quadratic forms, 1989, Springer, especially Ch. 3.
• A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, ch. 31, pp. 112 ff.

Links

• Robin Visser, Table of n, a(n) for n = 1..10000
• Wolfdieter Lang, Table for n = 0..100 and more. [Uses the old offset of 0].
• Wolfdieter Lang, Periods of Indefinite Binary Quadratic Forms, Continued Fractions and the Pell +/-4 Equations. [Uses the old offset of 0].

Crossrefs

Cf. A079896, A082174, A087048.

Programs

SageMath

def a(n, k):
    i, D, S = 1, Integer(5), []
    while(i < n):
        D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
    for b in range(1, isqrt(D)+1):
        if ((D-b^2)%4 != 0): continue
        for a in Integer((D-b^2)/4).divisors():
            if gcd([a, b, (D-b^2)/(4*a)]) > 1: continue
            Q = BinaryQF(a, b, -(D-b^2)/(4*a)).reduced_form()
            if all([(not Q.is_equivalent(t)) for t in S]): S.append(Q)
    P = [len(Q.cycle(proper=True)) for Q in S]; P.sort(reverse=True);
    return P[k-1]  # Robin Visser, Jun 06 2025

Formula

a(n,k), n >= 1, k = 1, 2, ..., A087048(n), is the length of the period of the k-th primitive reduced forms for discriminant D(n) = A079896. The order in row n is nonincreasing.

Extensions

Offset corrected by Robin Visser, Jun 02 2025