Elementary Number Theory Section 3.2 Binary Quadratic Forms
Definition 3.2.1 (a) A monomial axk11xk22· · · xknn, a 6= 0 is said to have degree k1+ k1+· · · + kn.
(b) The degree of a polynomial is the maximal of the degrees of the mono- mial terms in the polynomial.
(c) A polynomial is called a form, or is said to be homogeneous if all its monomial terms have the same degree.
(d) A form of degree 2 is called a quadratic form.
(e) A form in two variables is called binary.
(f) The discriminant of a binary quadratic form f = ax2+ bxy + cy2 is the quantity d = b2 − 4ac.
Remark 3.2.2 Let f = ax2+ bxy + cy2. Then 4af (x, y) = (2ax + by)2− dy2. Theorem 3.2.3 Let f (x, y) = ax2+ bxy + cy2 be a binary quadratic form with integral coefficients and discriminant d.
(a) If d is a square, then the equation f (x, y) = 0 has infinitely many solutions inZ.
(b) If d 6= 0 and d is not a perfect square, then a 6= 0, c 6= 0, and the only solution of the equation f (x, y) = 0 in integers is given by x = y = 0.
Definition 3.2.4 (a) A form f (x, y) is called indefinite if it takes on both positive and negative values.
(b) The form is called positive (or negative) semidefinite if f (x, y)≥ 0 (or f (x, y) ≤ 0) for all integers x, y.
(c) A semidefinite form is called definite if in addition the only integers x, y for which f (x, y) = 0 are x = 0, y = 0.
Example 3.2.5 (a) x2− 2y2 is indefinite. (b) x3− 2xy + y2 is positive semidef- inite. (c) x2+ y2 is positive definite.
Theorem 3.2.6 Let f (x, y) = ax2+ bxy + cy2 be a binary quadratic form with integral coefficients and discriminant d.
(a) If d > 0 then f (x, y) is indefinite.
(b) If d = 0 then f (x, y) is semidefinite but not definite.
(c) If d < 0 then a and c have the same sign and f (x, y) is either positive definite or negative definite according as a > 0 or a < 0.
Theorem 3.2.7 Let d ∈ Z. There exists at least one binary quadratic form in Z[x, y] with discriminant d, if and only if d ≡ 0 or 1(mod 4).
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Definition 3.2.8 (a) We say that a quadratic form f (x, y) represents an integer n if there exist integers x0 and y0 such that F (x0, y0) = n.
(b) Such a representation is called proper if (x0, y0) = 1; otherwise it is improper.
Remark 3.2.9 The representations of n by f may be found by determining the proper representations of gn2 for those g such that g2|n.
Theorem 3.2.10 Let n, d be integers with n 6= 0. There exists a binary qua- dratic form of discriminant d that represents n properly if and only if the con- gruence x2 ≡ d(mod 4|n|), has a solution.
Corollary 3.2.11 Suppose that d ≡ 0 or 1(mod 4). If p is an odd prime, then there is a binary quadratic form of discriminant d that represents p, if and only if p|d or (d
p
)= 1.
Theorem 3.2.12 Let M =
[M11 m12
m21 m22
]
be a matrix with real entries and put [u
v
]= M [x
y
]
. The the following are equivalent:
(i) The linear transformation defines a permutation of lattice points;
(ii) the matrix M has integral coefficients and det(M ) = ±1.
Definition 3.2.13 (a) The group of 2×2 matrices with integral coefficients and determinants 1 is denoted by Γ, and is called the modular group.
(b) The quadratic forms f (x, y) = ax2 + bxy + cy2 and g(x, y) = Ax2 + Bxy + Cy2 are equivalent, and we write f ∼ g, if there is an M = [mij] ∈ Γ such that g(x, y) = f (m11x + m12y, m21x + m22y). In this case we say that M takes f to g.
(c) Let f = ax2 + bxy + cy2. Let F = [a b2
b 2 c
]
. If X = [x
y
]
, then XtF X = f (x, y). F is called the matrix associated with f .
Remark 3.2.14 Let f, g be binary quadratic forms and F, G be the matrices associated with F and G, respectively.
(a) If M takes f to g, then MtF M = G. Moreover, if M =[m11 m12
m21 m22
], then A = f (m11, m12), C = f (m21, m22), B = 2am11m12+ 2cm21m22 + b(m12m21+ m11m22).
(b) f ∼ g if and only if there exists M ∈ Γ such that MtF M = G.
(c) The relation ∼ is an equivalence relation.
Theorem 3.2.15 Let f and g be equivalent quadratic forms.
(a) For any given integers n, the representation of n by f are in one -to-one correspondence with the representation of n by g.
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(b) Also, the proper representation of n by f are in one-to-one correspon- dence with the proper representation of n by g.
(c) The discriminant of f and g are equal.
Definition 3.2.16 Let f be a binary quadratic form whose discriminant d is not a perfect square.
(a) If d is not a square, we call f reduced if −|a| < b ≤ |a| < |c| or if 0≤ b ≤ |a| = |c|.
(b) If d is a square, we call f reduced if c = 0 and 0 ≤ a < |b|.
Theorem 3.2.17 Let d be a given integer which is not a perfect square. Each equivalent class of binary quadratic forms of discriminant d contains at least one reduced form.
Example 3.2.18 Find a reduced form equivalent to the form 133x2 + 108xy + 22y2.
Theorem 3.2.19 Let f be a reduced binary quadratic form whose discriminant d is not a perfect square.
(a) If f is indefinite, then 0 <|a| ≤ 12√ d (b) If f is positive definite then 0 < a≤√
−d 3 .
(c) In either case, the number of reduce forms of a given nonsquare dis- criminant d is finite.
Definition 3.2.20 If d is not a perfect square then the number of equivalence classes of binary quadratic forms of discriminant d is called the class number of d, denoted H(d).
Example 3.2.21 An odd prime p can be written in the form p = ax2 − 2y2 if and only if p≡ ±1(mod 8).
Lemma 3.2.22 Let f (x, y) = ax2 + bxy + cy2 be a reduced positive definite form. If for some pair of integers x and y we have (x, y) = 1 and f (x, y)≤ c, then f (x, y) = a or c, and the point (x, y) is one of the six points±(1, 0), ±(0, 1), ±(1, −1).
Moreover, the number of proper representation of a by f is
2 if a≤ c,
4 if 0 ≤ b < a = c, 6 if a = b = c.
Theorem 3.2.23 Let f (x, y) = ax2+ bxy + cy2 and g(x, y) = Ax2+ Bxy + Cy2 be two equivalent reduced positive definite forms. Then f = g.
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Definition 3.2.24 Let f (x, y) be a positive definite binary quadratic form. A matrix M ∈ Γ is called an automorph of f if M takes f into itself. The number of automorphs of f is denoted by w(f ).
Theorem 3.2.25 (a) Let f and g be equivalent positive definite binary quadratic forms. Then w(f ) = w(g), there are exactly w(f ) matrices that takes f to g, and there are exactly w(g) matrices that takes g to f .
(b) The only values of w(f ) are 2,4,and 6. If f is reduced then
w(f ) = 4 if a = c and b = 0, w(f ) = 6 if a = b = c,
w(f ) = 2 otherwise.
