Diophantine equation

Some problems only make sense with whole answers: you cannot buy 2.7 chickens or pay with 1/3 of a coin. A Diophantine equation is one where we insist the solutions be integers, named after the ancient Greek mathematician Diophantus. The same equation that has endlessly many real solutions may have only a handful of integer ones, or none at all.

The simplest important case is the linear Diophantine equation a x + b y = c in two unknowns x and y. It has integer solutions if and only if gcd(a, b) divides c. When solutions exist, you find one using the extended Euclidean algorithm and then generate all the rest by sliding x and y in opposite directions in steps tied to b and a.

Beyond the linear case the difficulty explodes. Pythagorean triples like 3, 4, 5 solve x^2 + y^2 = z^2 in integers, but Fermat's claim that x^n + y^n = z^n has no positive integer solutions for n greater than 2 stood unproven for over 350 years. In general there is no single method that decides whether an arbitrary Diophantine equation has integer solutions — a fact that is itself a celebrated theorem.