There is given $n$ different odd prime numbers: $p_{1},...,p_{n}$.
When following system of equation is solvable?
$$\begin{cases} x-p_{1}=p_{2}^{a_{2}}\\x-p_{2}=p_{3}^{a_{3}}\\...\\x-p_{n}=p_{1}^{a_{1}}\end{cases}$$
Where exponents are natural numbers greater than 1.
I tried to do this for $n=2$ and it seems that it has no solutions
I will be gratefull if someone will help me.
Regards.