Arithmetics pp 75-124 | Cite as

# Algebra and Diophantine Equations

## Abstract

In this chapter, we address some classical problems in number theory, such as finding integer solutions to polynomial equations. The examples that we will look at cover three large topics.

1) The decomposition of an integer *n* into the sum of two, three or four squares, in other words, the search for solutions of the equation \(n=x_{1}^{2}+x_{2}^{2}+\dots+x_{k}^{2}\).

2) “Fermat’s last theorem” (proven by Andrew Wiles in 1995): the only solutions to the equation *x* ^{ n }+*y* ^{ n }=*z* ^{ n } for *n*≥3 are the trivial ones (i.e., *xyz*=0).

3) Solutions to the Pell’s equation *x* ^{2}−*dy* ^{2}=1 (or more generally *x* ^{2}−*dy* ^{2}=*n*). The study of congruences—the theme of Chap. 1—gives us necessary conditions for the existence of solutions to such an equation. The methods introduced in this chapter are the use of rings more general than **Z** and also results about rational approximations.

### Keywords

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