Fermat numbers

If \(F_n\) denotes the \(n\)th Fermat number, then find the value of \(n\):

\[F_n=\overline{xyzabcde6}\]

for some \(\overline {xyzabcde}\).

Note: Here, \(\overline{xyzabcde6}\) is not the product of \(\overline{xyzabcde}\) and \(6\). It is in accordance with the base-10 positional system.

Bonus: Prove your answer.

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