10 Facts About Diophantine equations

In mathematics, a Diophantine equations equation is a polynomial equation, usually involving two or more unknowns, such that the only solutions of interest are the integer ones.

An exponential Diophantine equations equation is one in which unknowns can appear in exponents.

Simplest linear Diophantine equations equation takes the form, where, and are given integers.

The Chinese remainder theorem asserts that the following linear Diophantine equations system has exactly one solution such that, and that the other solutions are obtained by adding to a multiple of :.

Homogeneous Diophantine equations equation is a Diophantine equations equation that is defined by a homogeneous polynomial.

Degree three, there are general solving methods, which work on almost all Diophantine equations that are encountered in practice, but no algorithm is known that works for every cubic equation.

Diophantine geometry, which is the application of techniques from algebraic geometry in this field, has continued to grow as a result; since treating arbitrary equations is a dead end, attention turns to equations that have a geometric meaning.

The central idea of Diophantine geometry is that of a rational point, namely a solution to a polynomial equation or a system of polynomial equations, which is a vector in a prescribed field, when is not algebraically closed.

Depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as equivalently described as recursively enumerable.

In other words, the general problem of Diophantine equations analysis is blessed or cursed with universality, and in any case is not something that will be solved except by re-expressing it in other terms.