Nonnegative Integer Pairs

Question

Find all pairs $(a,b)$ of non-negative integers such that $2017^a=b^6-32b+1$.

What I tried:$2016*(2017^{a-1}+2017^{a-2}+...+2017+1)=b(b-2)(b^4+b^3*2+b^2*4+b*8+16)$

I also tried to force write them

$2017^a + 255 = (b-16)^2 = 2017^a \pmod 5$

Answer 1

This has been published for the solutions of the 48th Austrian Mathematical Olympiad National Competition, Problem 4.

Answer: The two solutions are $(0,0)$ and $(0, 2)$. The problem and solution are due to Walther Janous.