# 10 Facts About Diophantine equations

1.

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.

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2.

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

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3.

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

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4.

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 :.

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5.

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

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6.

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.

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7.

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.

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8.

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.

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9.

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

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10.

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.

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