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I work towards a formula to predict $2^n$ for any $n$. I don't expect to complete it, but I have found the last four digits for any $n \in \mathbb{N}$.  Update: I have found a relationship between the sequences that in theory should allow the calculation of any digit of $2^n$ by storing only $4$ digits... Update II: The sequence fails for some numbers as it cycles in a non prime manner, numbers would either need to be stored, or a prime cycling sequence discovered which seems unlikely.