Prove that powers of any fixed prime p contain arbitrarily many consecutive equal digits.
It is an intuitive re-statement of Baltic Way 2012 (I think there are shortlists in Baltic Way every year and this is a part of the 2012 shortlist):
Prove that, for every prime p and positive integer a, there exists a positive integer n such that pn contains a consecutive equal digits.
It is a tough one and I haven't found a solution on the Internet.
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