The world largest known prime to the test!

If \({ a }_{ n }\) satisify

\[{ a }_{ n }={ a }_{ n-1 }^{ 2 }-2\]

and if \({ a }_{ 0 }=4\), find

\[{ a }_{ 80,000,000 } \pmod{{ 2 }^{ 74,207,281 }-1} \]

Hint: \({ 2 }^{ 74,207,281 }-1\) is a Mersenne prime.

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