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Computational number theory front page

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[QUICK DESCRIPTION]
Number theory tells us many beautiful results, but it does not always do so explicitly. For example, if $p$ is a prime number, then the multiplicative group mod $p$ must be cyclic: that is, there must exist some $a$ such that $a^{p-1}\equiv 1$ mod $p$ but $a^r\not\equiv 1$ whenever $1\leq a<p-1$. But this result is proved by a counting argument that gives no clue about how to find such an $a$ or how to establish that a given number $a$ has that property. Such questions are the domain of ''computational'' number theory. There are a number of beautiful tricks in the area that make it particularly well suited to being discussed in the Tricki.

[PREREQUISITES]
Elementary number theory, and especially modular arithmetic.

===Links to articles===
Some of the tricks, though clever, are quite simple to explain. As a result, some of the following articles are quite short.

[[To work out powers mod n, use repeated squaring|To work out powers mod $n$, use repeated squaring]]

[[To establish that n is composite, show that Fermat's little theorem does not hold for n|To establish that $n$ is composite, show that Fermat's little theorem does not hold for $n$]]

[[To find a factor of n, find some m such that (m,n) is not 1|To find a factor of $n$, find some $m$ such that $(m,n)\ne 1$]]

[[To factorize n, find a non-trivial square root of 1 mod n|To factorize $n$, find a non-trivial square root of $1$ mod $n$]]

[[To find a rational with low denominator near a given real, use continued fractions]]

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