Diophantine equations are polynomial equations that seek integer solutions. Named after the ancient Greek mathematician Diophantus, these equations often take the form ax + by = c , where a , b , and c are integers, and x and y are the unknowns. The challenge lies in finding whole number solutions, which may not always exist. These equations can be simple, like x + y = 5 , or more complex, involving higher degrees and multiple variables. Fermat's Last Theorem is a famous example of a Diophantine equation, stating that there are no three positive integers a , b , and c that satisfy a^n + b^n = c^n for any integer n > 2 .