Sunday, 10 November 2013

number theory - Prove that powers of any fixed prime p contain arbitrarily many consecutive equal digits.


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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