NOTE The OP didn't state "Preferably, suggest some open problems so that new results can be obtained." when this was answered.
I can provide you with Burton's Elementary Number Theory. It has a series of historical introductions and great examples you'll probably find worth of a research project. He has information and obivously theory about results from Fermat, Euler, Diophantus, Wilson, Möbius, and others. I can also provide you with the three volumes of the History of Number Theory, which might be a great source.
A few examples are
Fermat's Little Theorem If $p\not\mid a$ then$a^{p-1} \equiv 1 \mod p$
Wilson's Theorem If $p$ is a prime then
$({p-1})! \equiv -1 \mod p$
Möbius Inversion Formula If we have two arithmetical functions $f$ and $g$ such that
$f(n) = \sum_{d \mid n} g(d)$
Then
$g(n) = \sum_{d \mid n} f(d)\mu\left(\frac{n}{d}\right)$
Where $\mu$ is the Möbius function.
Maybe so interesting as the previous,
The $\tau$ and $\sigma$ functions
Let $\tau(n)$ be the number of divisors of $n$ and $\sigma(n)$ its sum. Then if $n=p_1^{l_1}\cdots p_k^{l_k}$
$\tau(n)=\prod_{m=1 }^k(1+l_m)$
$\sigma(n)=\prod_{m=1 }^k \frac{p^{l_m+1}-1}{p-1}$
Legendre's Identity
The multiplicty (i.e. number of times) with which $p$ divides $n!$ is
$\nu(n)=\sum_{m=1}^\infty \left[\frac{n}{p^m} \right]$
However odd that might look, the argument is somehow simple. The multiplicity with which $p$ divides $n$ is $\left[\dfrac{n}{p} \right]$, for $p^2$ it is $\left[\dfrac{n}{p^2} \right]$, and so forth. To get that of $n!$ we sum all these values to get the above, since each of $1,\dots,n$ is counted $l$ times as a multiple of $p^m$ for $m=1,2,\dots,l$, if $p$ divides it exactly $l$ times. Note the sum will terminate because the least integer function $[x]$ is zero when $p^m>n$.
Perfect numbers
A number is called a perfect number is the sum if its divisors equals the number, this means
$\sigma(n) =2n$
Euclid showed if $p=2^n-1$ is a prime, then $\frac{p(p+1)}{2}$ is always a perfect number
Euler showed that if a number is perfect, then it is of Euclid's kind.
$n$ - agonal or figurate numbers.
The greeks were very interested in numbers that could be decomposed into geometrical figures. The square numbers are well known to us, namely $m=n^2$. But what about triangular, or pentagonal numbers?
Explicit formulas have been found, namely
$t_n=\frac{n(n+1)}{2}$
$p_n=\frac{n(3n-1)}{2}$
You can try, as a good olympiadish excercise, to prove the following:
${t_1} + {t_2} + {t_3} + \cdots + {t_n} = \frac{{n\left( {n + 1} \right)\left( {n + 2} \right)}}{6}$
We can arrange the numbers in a pentagon as a triangle and a square:
${p_n} = {t_{n - 1}} + {n^2}$