<s>[INST] How prove this $3^{\frac{5^{2^n}-1}{2^{n+2}}}\equiv (-5)^{\frac{3^{2^n}-1}{2^{n+2}}}\pmod {2^{n+4}}$ Question:

show that:
$$3^{\frac{5^{2^n}-1}{2^{n+2}}}\equiv (-5)^{\frac{3^{2^n}-1}{2^{n+2}}}\pmod {2^{n+4}},n\geq 1$$

My idea: since
I have prove
$$5^{2^n}-1\equiv 0\pmod {2^{n+2}}$$
$$3^{2^n}-1\equiv 0\pmod {2^{n+2}}$$
so let
$$\frac{5^{2^n}-1}{2^{n+2}}=k_{1},\frac{3^{2^n}-1}{2^{n+2}}=k_{2},k_{1},k_{2}\in N$$
so we must prove
$$3^{k_{1}}-(-5)^{k_{2}}\equiv 0\pmod {2^{n+4}}$$
and How prove it?
Thank you
 [/INST] This is not even true. Just try with n=0
$3^1 = (-5)^{1/2}$ (mod 16) makes no sense.
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