

# Diophantine equations and quadratic forms

A Diophantine equation is a polynomial equation in which the variables are required to be integers. Quadratic forms are a special case of Diophantine equations, where the highest degree of the equation is 2.

Consider the quadratic Diophantine equation:

$$ax^2 + bxy + cy^2 + dx + ey + f = 0$$

This equation represents a quadratic form.

To solve a quadratic Diophantine equation, we can use the method of completing the square. This method involves isolating the squares and perfect squares in the equation and then manipulating the equation to find integer solutions.

## Exercise
Solve the quadratic Diophantine equation:

$$3x^2 + 5xy - 2y^2 - 7x + 4y + 10 = 0$$

The quadratic form can be written as:

$$(3x - \frac{5}{2}y - \frac{7}{6})^2 + \frac{1}{3}y^2 + \frac{4}{3}y + \frac{5}{6} = 0$$

# Modular arithmetic and quadratic residues

Modular arithmetic is a system of arithmetic for integers modulo a fixed integer. Quadratic residues are numbers that have a square root modulo a given number.

Consider the modular arithmetic of the number 7. The numbers 0, 1, 2, 3, 4, 5, and 6 are the residues modulo 7.

To find the quadratic residues of a number, we can use the Legendre symbol. The Legendre symbol is a function that takes two arguments, an integer a and a prime number p, and returns 1 if a is a quadratic residue modulo p, and -1 if a is a non-residue modulo p.

## Exercise
Find the quadratic residues of the number 7.

The quadratic residues of 7 modulo 7 are 0, 1, 2, 3, and 4.

# Factorization of quadratic forms

The factorization of a quadratic form is a way to rewrite the form as a product of smaller forms. This can be useful in solving Diophantine equations and finding integer solutions.

Consider the quadratic form:

$$x^2 + 2xy + y^2$$

This form can be factorized as:

$$(x + y)^2$$

To factorize a quadratic form, we can use the method of completing the square. This method involves isolating the squares and perfect squares in the equation and then manipulating the equation to find the factors.

## Exercise
Factorize the quadratic form:

$$2x^2 + 3xy + 2y^2$$

The quadratic form can be written as:

$$(2x + \frac{3}{2}y)^2 - \frac{1}{4}y^2$$

# The Shanks' square forms algorithm

The Shanks' square forms algorithm is a method for finding the factorization of a quadratic form. This algorithm is based on the method of completing the square and modular arithmetic.

Consider the quadratic form:

$$2x^2 + 3xy + 2y^2$$

Using the Shanks' square forms algorithm, we can find the factorization as:

$$(2x + \frac{3}{2}y)^2 - \frac{1}{4}y^2$$

The algorithm works by first finding the reduced form of the quadratic form modulo a prime number. Then, the algorithm iterates by finding the square roots of the reduced form and reducing the form further until a solution is found.

## Exercise
Find the factorization of the quadratic form:

$$3x^2 + 4xy + 3y^2$$

The quadratic form can be written as:

$$(3x + 2y)^2 - y^2$$

# Applications of Shanks' square forms factorization

Shanks' square forms factorization has applications in various fields, such as number theory, cryptography, and physics.

In number theory, Shanks' square forms factorization can be used to find the factorization of a number into its prime factors.

In cryptography, Shanks' square forms factorization can be used to find the discrete logarithm in a cyclic group.

## Exercise
Find the factorization of the number 125.

The factorization of 125 is:

$$5^2 \times 5$$

# Conclusion and further reading

In conclusion, Shanks' square forms factorization is a powerful method for finding the factorization of quadratic forms. This algorithm has applications in various fields and can be used to solve Diophantine equations and find integer solutions.

## Exercise
Find the factorization of the quadratic form:

$$4x^2 + 5xy + 4y^2$$

The quadratic form can be written as:

$$(2x + \frac{5}{2}y)^2 - \frac{1}{2}y^2$$

For further reading, consider exploring the works of John Shanks and his contributions to the field of number theory. Additionally, you can explore the applications of Shanks' square forms factorization in other fields, such as cryptography and physics.