Prime Numbers

5.6 Binary quadratic forms 239
f(jB)modn for j =1, 2,...,B, then we would be in business to check each
gcd and find the first that exceeds 1.
Algorithm 9.6.7 provides the computation of f(x) as a polynomial in Zn[x]
(that is, the coefficients are reduced modulo n) and the evaluation of each
f(jB) modulo n for j =1, 2,...,B in O B ln 2 B = O n 1/4 ln 2 n arithmetic
operations with integers the size of n. This latter big-O expression then stands
as the complexity of the Pollard–Strassen polynomial evaluation method for
factoring n.
5.6 Binary quadratic forms
There is a rich theory of binary quadratic forms, as developed by Lagrange,
Legendre, and Gauss in the late 1700s, a theory that played, and still plays,
an important role in computational number theory.
5.6.1 Quadratic form fundamentals
For integers a, b, c we may consider the quadratic form ax 2 + bxy + cy 2 .Itisa
polynomial in the variables x, y, but often we suppress the variables, and just
refer to a quadratic form as an ordered triple (a, b, c) of integers.
We say that a quadratic form (a, b, c) represents an integer n if there are
integers x, y with ax 2 + bxy + cy 2 = n. So attached to a quadratic form
(a, b, c) is a certain subset of the integers, namely those numbers that (a, b, c)
represents. We note that certain changes of variables can change the quadratic
form (a, b, c) to another form (a ′ ,b ′ ,c ′ ), but keep fixed the set of numbers that
are represented. In particular, suppose
x = αX + βY, y = γX + δY,
where α, β, γ, δ are integers. Making this substitution, we have
ax 2 + bxy + cy 2 = a(αX + βY ) 2 + b(αX + βY )(γX + δY )+c(γX + δY ) 2
= a ′ X 2 + b ′ XY + c ′ Y 2 , (5.1)
say. Thus every number represented by the quadratic form (a ′ ,b ′ ,c ′ )isalso
represented by the quadratic form (a, b, c). We may assert the converse
statement if there are integers α ′ ,β ′ ,γ ′ ,δ ′ with
That is, the matrices
α β
,
γ δ
X = α ′ x + β ′ y, Y = γ ′ x + δ ′ y.
α ′ β ′
γ ′ δ ′
are inverses of each other. A square matrix with integer entries has an inverse
with integer entries if and only if its determinant is ±1. We conclude that if
the quadratic forms (a, b, c) and(a ′ ,b ′ ,c ′ ) are related by a change of variables
as in (5.1), then they represent the same set of integers if αδ − βγ = ±1.