Diophantine Equations - Examples and Constructions

Special classes of Diophantine equations exhibit fascinating patterns and connections to other areas of mathematics. From sums of squares to Mordell curves, these examples reveal deep arithmetic structure.

DefinitionSums of Two Squares

The equation $x^2 + y^2 = n$ asks which integers can be written as sums of two squares.

Fermat proved that an odd prime $p$ is a sum of two squares if and only if $p \equiv 1 \pmod{4}$ . More generally, $n$ is a sum of two squares if and only if every prime $p \equiv 3 \pmod{4}$ appears in the prime factorization of $n$ to an even power.

ExampleSums of Two Squares

$5 = 1^2 + 2^2$ (since $5 \equiv 1 \pmod{4}$ )
$13 = 2^2 + 3^2$ (since $13 \equiv 1 \pmod{4}$ )
$45 = 3^2 + 6^2$ (factorization: $45 = 3^2 \cdot 5$ , both allowed)
$21 = 3 \cdot 7$ cannot be written as sum of two squares (both $3, 7 \equiv 3 \pmod{4}$ appear to odd powers)
$50 = 1^2 + 7^2 = 5^2 + 5^2$ (multiple representations possible)

DefinitionMordell's Equation

For an integer $k$ , Mordell's equation is: $y^2 = x^3 + k$

The number of integer solutions depends heavily on $k$ . Some values have no solutions, others finitely many, while the general theorem (Mordell's Conjecture, proved by Faltings) states that there are only finitely many solutions for any fixed $k \neq 0$ .

ExampleMordell's Equation Solutions

For $y^2 = x^3 + 1$ :

$(x, y) = (0, \pm 1)$ : $1 = 0 + 1$ ✓
$(x, y) = (2, \pm 3)$ : $9 = 8 + 1$ ✓
$(x, y) = (-1, 0)$ : $0 = -1 + 1$ ✓

These are all integer solutions (verified by Mordell).

For $y^2 = x^3 - 2$ :

$(x, y) = (3, \pm 5)$ : $25 = 27 - 2$ ✓

This is the only integer solution.

DefinitionFermat's Last Theorem

For $n \geq 3$ , the equation: $x^n + y^n = z^n$ has no positive integer solutions.

This conjecture, posed by Fermat in 1637, remained unsolved for over 350 years until Andrew Wiles proved it in 1995 using advanced techniques from algebraic geometry and modular forms.

ExampleSpecial Cases of Fermat

$n = 3$ : $x^3 + y^3 = z^3$ has no positive integer solutions (proven by Euler)
$n = 4$ : $x^4 + y^4 = z^4$ has no positive integer solutions (proven by Fermat using descent)

These special cases were known long before the general theorem.

DefinitionCatalan's Conjecture (Mihăilescu's Theorem)

The only solution to $x^p - y^q = 1$ in natural numbers $x, y > 0$ and $p, q > 1$ is: $3^2 - 2^3 = 1$

This was conjectured by Catalan in 1844 and proved by Preda Mihăilescu in 2002.

ExampleNear-Solutions to Catalan

While $(3, 2, 2, 3)$ is the only solution, there are some "near misses":

$2^5 - 5^2 = 32 - 25 = 7$ (difference 7, not 1)
$3^3 - 5^2 = 27 - 25 = 2$ (difference 2, not 1)
$13^2 - 2^7 = 169 - 128 = 41$ (difference 41, not 1)

The uniqueness of $3^2 - 2^3 = 1$ is remarkable.

Remark

Modern approaches to Diophantine equations use tools from algebraic geometry (elliptic curves, abelian varieties), transcendence theory (Baker's theorem), and computational methods. The classification of rational points on varieties remains an active area of research.

These examples illustrate how diverse and deep the theory of Diophantine equations becomes, touching nearly every branch of modern mathematics.