# 2020? But it's still 2015!

Algebra Level 5

$\large f(x+1) = (5^{2^{x-1}} + 1)(x+1)^{2}f(x)$

Suppose we define a function of $$x$$ as described above for $$x$$ is a non-negative integer with $$f(0)=\sqrt{5} - 1$$.

If the value of $$f(2020)$$ can be written in form $$(5^{p} - 1)(q!)^{2}$$, for positive integers $$p$$ and $$q$$, find $$\frac{2^{q}}{p}$$.

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