Show that the term $xy+1$ is a perfect square

If $ A=\{F_{2n}, \ F_{2n+2}, \ F_{2n+4}, \ 4F_{2n+1}F_{2n+2}F_{2n+3}\}$ then I want to show that for $ x,y\in A$ with $ x\neq y$ the term $ xy+1$ is a perfect square.

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We have that $ $ F_M=2^{2^M}+1 \ \text{ and } \ F_N=2^{2^N}+1$ $ Then $ $ F_MF_N+1=\left (2^{2^M}+1\right )\cdot \left (2^{2^N}+1\right )+1=2^{2^M+2^N}+2^{2^M}+2^{2^N}+1$ $

Do we have to substitute the values for $ M$ and $ N$ to see that we get a perfect square or do we have to do something else?