Diophantine Equations - Core Definitions

Diophantine equations, named after the ancient Greek mathematician Diophantus of Alexandria, are polynomial equations for which we seek integer or rational solutions. These equations form one of the oldest and most fascinating branches of number theory.

DefinitionDiophantine Equation

A Diophantine equation is a polynomial equation of the form: $P(x_1, x_2, \ldots, x_n) = 0$ where $P$ has integer coefficients, and we seek solutions where all variables take integer values (or sometimes rational values).

Diophantine equations can be classified by degree, number of variables, and special forms.

The study of Diophantine equations asks fundamental questions: Does a solution exist? If so, are there finitely or infinitely many? Can we find all solutions explicitly?

DefinitionLinear Diophantine Equations

A linear Diophantine equation in two variables has the form: $ax + by = c$ where $a, b, c \in \mathbb{Z}$ and we seek integer solutions $(x, y)$ .

This equation has solutions if and only if $\gcd(a, b) \mid c$ .

ExampleLinear Diophantine Equation

Solve $6x + 9y = 15$ .

Since $\gcd(6, 9) = 3$ and $3 \mid 15$ , solutions exist. Dividing by $3$ : $2x + 3y = 5$

Find a particular solution using the extended Euclidean algorithm. Since $\gcd(2, 3) = 1$ : $3 - 2 \cdot 1 = 1 \implies 2 \cdot (-1) + 3 \cdot 1 = 1$

Multiply by $5$ : $2 \cdot (-5) + 3 \cdot 5 = 5$ .

So $(x_0, y_0) = (-5, 5)$ is a particular solution.

General solution: $x = -5 + 3t, \quad y = 5 - 2t$ for $t \in \mathbb{Z}$ .

DefinitionPythagorean Triples

A Pythagorean triple is a solution $(x, y, z)$ in positive integers to: $x^2 + y^2 = z^2$

A triple is primitive if $\gcd(x, y, z) = 1$ .

ExamplePythagorean Triples

Famous examples include:

$(3, 4, 5)$ : $9 + 16 = 25$
$(5, 12, 13)$ : $25 + 144 = 169$
$(8, 15, 17)$ : $64 + 225 = 289$
$(7, 24, 25)$ : $49 + 576 = 625$

All primitive Pythagorean triples are generated by: $x = m^2 - n^2, \quad y = 2mn, \quad z = m^2 + n^2$ where $m > n > 0$ , $\gcd(m, n) = 1$ , and $m - n$ is odd.

DefinitionPell's Equation

For a positive non-square integer $d$ , Pell's equation is: $x^2 - dy^2 = 1$

This equation always has infinitely many positive integer solutions, with all solutions generated from the fundamental solution.

ExamplePell's Equation for d=2

For $x^2 - 2y^2 = 1$ :

The fundamental solution is $(x, y) = (3, 2)$ since $9 - 8 = 1$ .

From this, we generate infinitely many solutions:

$(17, 12)$ : $289 - 288 = 1$
$(99, 70)$ : $9801 - 9800 = 1$
$(577, 408)$ : $332929 - 332928 = 1$

Each new solution is obtained by the recursion involving the fundamental solution.

Remark

Diophantine equations range from simple linear equations solvable by elementary methods to extremely deep problems like Fermat's Last Theorem ( $x^n + y^n = z^n$ has no positive integer solutions for $n \geq 3$ ), proven by Andrew Wiles in 1995 after centuries of effort.

The diversity of Diophantine equations reflects the richness of number theory itself, connecting elementary arithmetic to the deepest reaches of modern mathematics.