Examples of finding remainders using Wilson's theorem.
Bernouilli's inequality.
Journal of Integers Sequences.
Mersenne primes (Wiki).
Generalizations of Fibonacci numbers.
Bertrand-Chebyshev theorem (Proof Wiki).
Ramanujan's proof of Bertrand's postulate.
Green-Tao theorem (Wikipedia).
Pisot-Vijayaraghavan number (Wikipedia).
Practical number (Wikipedia).
Root of Unity (Wikipedia).
A prime-representing function (W. H. Mills, Bulletin of the American Mathematical Society).
On the interval containing at least one prime number (Jitsuro Nagura).
Chebyshev functions (Wikipedia).
Albert Ingham (Wikipedia).
A remarkable collection of Babylonian mathematical texts (pdf).
Brocard's conjecture (Wiki).
Buffon's needle (Wiki).
Group Orbit.
Fountain problems.
"Coins in a fountain" problems.