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I have learnt this theorem in my class, however, "elementary" examples are very limited. (focusing more on analytic machinery)

Are there any interesting applications to elementary number theory that you can share with me?

This will greatly benefit me in my studies!

user147794
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  • Some time ago I invented the following problem: Given three (non-zero) positive integers $a_1,a_2,a_3$. Define the sequence $(a_n)n$ such that $a{n+3}$ is the remainder of $a_n^{a_{n+1}}$ when divided by $a_{n+2}$, for all $n>0$ as long as $a_{n+2}$ is not $0$.
    A) Prove that we always have $a_k=0$ for some $k>3$, i.e. the sequence is finite. B) Prove that for every $n>3$, there exist integers $a_1,a_2,a_3$ such that $a_n=0$, i.e. the sequence contains exactly $n$ numbers. I hope you enjoy this problem. P.S: A is easier than B ;-)     – Bart Michels Jun 06 '14 at 18:53

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