Answer to Question #227153 in Abstract Algebra for Ann

Question

Discuss Diophantine equations and expound on how it will really be valuable to students

Answer 1

In mathematics, a Diophantine equation is a polynomial equation, usually involving two or more unknowns, such that the only solutions of interest are the integer ones (an integer solution is such that all the unknowns take integer values). A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.

Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations. As such systems of equations define algebraic curves, algebraic surfaces, or, more generally, algebraic sets, their study is a part of algebraic geometry that is called Diophantine geometry.

In mathematics diophantine equations are central objects in number theory as they express natural questions such as the ways to write a number as a sum of cubes, but they naturally come up in all questions that can be reduced to questions involving discrete objects. It can also be applied in cryptography using elliptic curves in real life situations.