Find the GCD of all the numbers from the set $$\{(n+2014)^{n+2014}+n^n\mid n\in \mathbb{N},n>2014^{2014}\}$$ Now I have the proof but i can't understand one thing
Lets say $d$ is the GCD.Now let $$x_n=(n+2014)^{n+2014}+n^n,\mathbb{for\space n \in N}$$ Let $p$ be prime which doesn't divide $2014$ and $n>2014^{2014}$ such that $p \mid n$.Then $p\not\mid n+2014$,so $p\not\mid x_n$.From this we can conclude that d must be divisible by one of the numbers in the set $\{2,19,53\}$
Why does $d$ be divisible by one of the numbers?And what does the $p$ have to do with this?(I mean why aren't there cases when p doesn't divide either $n$ or $n+2014$ but divides $x_n$)
The numbers 2,19 and 53 are the prime factors of 2014.
The proof shows that no other prime can divide d by showing that there is an n, such that p does not divide $x_n$. It is easy to choose n > $2014^{2014}$ such that n is a multiple of p. But if p divides n, it cannot divide n + 2014, otherwise , it would divide 2014, which was ruled out. So, p does not divide $x_n$, so p cannot divide d. If it would divide d, it would also divide $x_n$.
The last formulation is somewhat misleading. It would be clearer to say that d can only have the prime factors 2,19 and 53.
I am wondering what d really is! It was only restricted in the proof.