Algebraic factors of $5^n+6^n+10^n$?

Question

I run currently a factoring project on $$f(n)=5^n+6^n+10^n$$ This expression has only no small prime factor , if $6\mid n$ holds. A prime occurs for $n=3168$ (not a proven prime, but extremely probably a prime) and no other $n\le 50\ 000$. I search a prime factor for the cases $$n=1038,1188,1230,1470,1842,2022,2778,3030,3090,3294,3318$$ Note that the current year is in the wanted-list. My question now is

Are there forced factors (algebraic/aurifeullian or similar kinds of factors) for some $n$ ?

This would explain why this expression is prime so rarely.

Answer 1

Proof for prime $7$ when $n\equiv\pm1\pmod3$

►First for $n\equiv1\pmod3$

$f(n)=5^{3n+1}+6^{3n+1}+10^{3n+1}$.

$f(n)\equiv(-2)^{3n+1}+(-1)^{3n+1}+(-4)^{3n+1}\pmod7$

By induction, it is true for $n=0$. Let it be true for $n$ so for $n+1$ we have $$f(n+1)\equiv(-2)^3(-2)^{3n+1}+(-1)^3(-1)^{3n+1}+(-4)^3(-4)^{3n+1}\pmod7$$ Thus $$f(n+1)\equiv-f(n)\equiv0\pmod7$$

Similarly $f(n)\equiv0\pmod7$ when $n\equiv-1\pmod3$