2020? But it's still 2015!

Algebra Level 4

f(x+1)=(52x1+1)(x+1)2f(x)\large f(x+1) = (5^{2^{x-1}} + 1)(x+1)^{2}f(x)

Suppose we define a function of xx as described above for xx is a non-negative integer with f(0)=51f(0)=\sqrt{5} - 1.

If the value of f(2020)f(2020) can be written in form (5p1)(q!)2(5^{p} - 1)(q!)^{2}, for positive integers pp and qq, find 2qp\frac{2^{q}}{p}.

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