Integer equation - practice problems

An integer equation, also known as a Diophantine equation, is an equation for which only integer solutions are sought or allowed. Named after the ancient Greek mathematician Diophantus, these equations range from simple linear forms to complex polynomial expressions. Common examples include linear Diophantine equations (ax + by = c) and the famous Pythagorean equation (x² + y² = z²). Finding integer solutions often requires techniques different from standard equation solving, including modular arithmetic and number theory. Integer equations have applications in cryptography, computer science, and optimization problems with discrete variables. Some Diophantine equations, like Fermat's Last Theorem, have challenged mathematicians for centuries.