What is your idea about a problem in number theory that says:

$\frac{p^q-1}{p-1}$ never divides $\frac{q^p-1}{q-1}$ if $p,q$ are distinct primes.

This is a $\textbf{conjecture}$ and the validity of this conjecture would simplify the proof of solvability of groups of odd order, (W. Fiet, J. G. Thompson, $\textit{Pacific J. Math.}$, 13, no.3 (1963), 775-1029), rendering unnecessary the detailed use of generators and relations.

An other interesting application of number theory is in real world. Many years ago, cables used for communication. A lot of cables must be gathered near to each other for more efficiency. But for blocking the noises of each cable to the other cable, an a special arrange of cables needed. For this arrangement and neighboring of cables, scientist used reminder theorem and number theory. I think first time, Bell company's scientists invented it.

Also, I think this relation between group theory, graph theory and number theory is very nice example: Suppose the order of group $G$, $|G|$, is $n=p_1^{k_1}p_2^{k_2}\ldots p_s^{k_s}$. We fix two prime numbers $p_i$ and $p_j$ of divisors of $n$ and define a graph $\Gamma(G)$ as fallow:

The vertices of $\Gamma(G)$ are the elements of group $G$ and two vertices $g_1$ and $g_2$ are adjacent if and only if $o(g_1g_2)=p_ip_j$, where $o$ means the order of element $g_1g_2$ as a group element. These graphs have very nice structures and well defined as $\textit{Prime Graph}$ of group.

3 added 450 characters in body

What is your idea about a problem in number theory that says:

$\frac{p^q-1}{p-1}$ never divides $\frac{q^p-1}{q-1}$ if $p,q$ are distinct primes.

This is a $\textbf{conjecture}$ and the validity of this conjecture would simplify the proof of solvability of groups of odd order, (W. Fiet, J. G. Thompson, $\textit{Pacific J. Math.}$, 13, no.3 (1963), 775-1029), rendering unnecessary the detailed use of generators and relations.

An other interesting application of number theory is in real world. Many years ago, cables used for communication. A lot of cables must be gathered near to each other for more efficiency. But for blocking the noises of each cable to the other cable, an special arrange of cables needed. For this arrangement and neighboring of cables, scientist used reminder theorem and number theory. I think first time, Bell company's scientists invented it.

2 added 22 characters in body

What is your idea about a problem in number theory that says:

$\frac{p^q-1}{p-1}$ never divides $\frac{q^p-1}{q-1}$ if $p,q$ are distinct primes.

This is a conjecture $\textbf{conjecture}$ and the validity of this conjecture would simplify the proof of solvability of groups of odd order, (W. Fiet, J. G. Thompson, Pacific $\textit{Pacific J. Math.Math.}$, 13, no.3 (1963), 775-1029), rendering unnecessary the detailed use of generators and relations.