gen-in-3-4-sentences-explain-diophantine-equations-state-any-key-90a8

In 3-4 sentences, explain Diophantine equations. State any key formula.

# Diophantine Equations Diophantine equations are polynomial equations where only integer solutions are sought, named after the ancient Greek mathematician Diophantus. The simplest form is the linear Diophantine equation **ax + by = c**, which has integer solutions if and only if gcd(a,b) divides c. When solutions exist, they can be expressed as **x = x₀ + (b/d)t** and **y = y₀ - (a/d)t**, where (x₀, y₀) is a particular solution, d = gcd(a,b), and t is any integer. Famous examples include Fermat's Last Theorem (xⁿ + yⁿ = zⁿ has no positive integer solutions for n > 2) and the Pythagorean equation (x² + y² = z²).