0,3

An indefinite quadratic form over the integers in two variables F(x,y) := a*x^2 + b*x*y + c*y^2 has discriminant D := b^2 - 4*a*c >0 not a square (a and c non-vanishing); that is D=D(n)= A079896(n) = [5,8,12,13,17,20,21,...], n>=0.

For a given discriminant D from A079896(n) a reduced form [a,b,c] is defined by b>0 and f(D)-min(|2*a|,|2*c|) =< b < f(D), with f(D) := ceiling(sqrt(D)).

For a given discriminant D from A079896(n) every primitive reduced form [a,b,c] defines a periodic chain of such forms by applying repeatedly the transformation R(t)*[a,b,c]=[a'(t),b'(t),c'(t)]=[c,-b+2*c*t,F(1,t)] with uniquely defined t= ceiling((f(D)+b)/(2*c))-1 if c>0 and t=-(ceiling((f(D)+b)/(2*|c|)-1)) if c<0. The number of such different) periodic chains of primitive reduced forms is called the class number for this (indefinite) discriminant D from A079896(n).

A primitive form [a,b,c] has gcd(a,b,c)=1.

A. Scholz and B. Schoeneberg, Einfuehrung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, ch. 31, pp. 112 ff.

S. R. Finch, Class number theory

W. Lang, Table for n=0,...,135.

n=2, D(2) = A079896(2) = 12, a(2) = 2 because there are the following two periodic chains of primitive reduced forms [a,b,c] (both with period length 2): [[ -2, 2, 1], [1, 2, -2]] and [[ -1, 2, 2], [2, 2, -1]].

n=13, D(13) = A079896(13) = 40, a(13) = 2 because there are the following two periodic chains of primitive reduced forms [a,b,c] (with period length 6 resp. 2): [[ -3, 2, 3], [3, 4, -2], [ -2, 4, 3], [3, 2, -3], [ -3, 4, 2], [2, 4, -3]] and [[ -1, 6, 1], [1, 6, -1]].

n=35, D(35) = A079896(35) = 89, a(35) = 1 because there is only one periodic chain of primitive reduced forms [a,b,c] (with period length 14): [[ -5, 3, 4], [4, 5, -4], [ -4, 3, 5], [5, 7, -2], [ -2, 9, 1], [1, 9, -2], [ -2, 7, 5], [5, 3, -4], [ -4, 5, 4], [4, 3, -5], [ -5, 7, 2], [2, 9, -1], [ -1, 9, 2], [2, 7, -5]]. See p. 116 of the Scholz/Schoeneberg reference which starts with the form [1, 9, -2].

n=62, D(62) = A079896(62) = 148, a(62) = 3 because there are three periodic chains of primitive reduced forms [a,b,c] (with period length 6 and 6 and 2, resp.): [[ -7, 6, 4], [4, 10, -3], [ -3, 8, 7], [7, 6, -4], [ -4, 10, 3], [3, 8, -7]] and [[ -4, 6, 7], [7, 8, -3], [ -3, 10, 4], [4, 6, -7], [ -7, 8, 3], [3, 10, -4]] and [[ -1, 12, 1], [1, 12, -1]]. See p. 116 of the Scholz/Schoeneberg reference which starts with the forms [4, 10, -3] and [3, 10, -4] and [1, 12, -1], resp.

See A006375 for another version. Cf. A079896.

Sequence in context: A122586 A079487 A069010 this_sequence A109700 A087742 A072530

Adjacent sequences: A087045 A087046 A087047 this_sequence A087049 A087050 A087051

nonn

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 07 2003