I once played the following game at an IMO training session for the advanced candidates. I simply asked the students one by one, to state theorems related to number theory. This turn out to be an excellent game to get the students involved and they are really actively engaged. According to my memory, here are the theorems stated:
1) Fermat’s little theorem
2) Euler’s theorem
3) Wilson’s theorem
4) Fermat’s Last Theorem
5) Fibonacci power theorem
6) q-th power lemma
7) Fermat’s two-square theorem
Lagrange’s four-square theorem
9) The prime number theorem
10) Hasse-Weil bound
11) Formula for primitive pythagorean triples
12) Solution to Pell’s equation
13) Quadratic reciprocty
14) Dirichlet’s theorem on infinitude of primes in arithmetic progression
15) Iwaniec’s theorem on the largest prime of n^2 + 1
16) Chen’s theorem
17) Iwaniec’s theorem on the primes and almost primes of n^2 + 1
18) Catalan’s theorem
I think there are a few more, but I do not remember them. Please add to the list if you have interesting theorems to share.
has no integer solutions.
. If 
is even then we have 
, which is absurd. Hence 
, then 
, which is absurd and thus 
Since 
There exists a prime 
and 
divides 
This implies 
and this has no solution since 
is a square mod 
iff 
Therefore the original equation has no solution in integers.
are roots of 
