Prime divisors of the sequence terms $a_n=a\cdot 2017^n+b\cdot 2016^n$

Solution 1:

Suppose the set of primes dividing the sequence is finite instead, and number them $p_1,...,p_K.$

Associate to each term $a_n = a \cdot 2016^n + b \cdot 2017^n$ the index $i, 1 \le i \le K,$ such that biggest prime power that divides $a_n$ is $p_i^{\beta_i(n)}$.

By the pigeonhole principle, there is an index $i_0$ and $a_n,a_m,\,m>n,m-n \le K,$ such that $a_n,a_m$ are both associated to $i_0,$ for each $n.$ Therefore, letting $l = \min(\beta_{i_0}(n),\beta_{i_0}(m)),$ we get that

$$ p_{i_0}^l | a\cdot 2016^n + b\cdot 2017^n, p_{i_0}^l | a \cdot 2016^m + b\cdot 2017^m.$$

The relations above imply, on the other hand, that

$$ p_{i_0}^l | b\cdot(2016^{m-n} - 2017^{m-n}). $$

From the fact that the number of primes is bounded, we get that $p_{i_0}^{l \cdot K} \ge 2017^n.$ On the other hand, by the equation above,

$$ p_{i_0}^l \le C \cdot 2017^K.$$

As $K$ is fixed, we see that

$$ 2017^{\frac{n}{K} - K} \le C, \text{for infinitely many } n \in \mathbb{N}.$$

This leads to a contradiction by letting $n \to \infty.$