Diophanttiset

Diophanttiset, also known as Diophantine equations, are polynomial equations for which only integer solutions are sought. These equations are named after the ancient Greek mathematician Diophantus of Alexandria, who studied them extensively. The most famous example is perhaps the Pythagorean equation x² + y² = z², where integer solutions correspond to the sides of a right-angled triangle (Pythagorean triples).

The study of Diophantine equations is a significant branch of number theory. Unlike equations over real or complex numbers, Diophantine equations often have no solutions, a finite number of solutions, or infinitely many solutions. Determining the existence and nature of these solutions can be a challenging problem.

Fermat's Last Theorem, which states that there are no positive integer solutions to the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than 2, is a prime example of a historically important Diophantine equation problem. The proof of this theorem by Andrew Wiles in 1994 was a landmark achievement in mathematics.

The field encompasses a wide range of types of equations, including linear Diophantine equations, quadratic Diophantine equations, and more complex polynomial forms. Techniques used to solve them include modular arithmetic, prime factorization, and algebraic number theory. The solvability of Diophantine equations is generally not guaranteed, and finding methods to determine solvability or to construct solutions remains an active area of research.