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King's Maths School

Challenge 461: Double digits

Can a power of 2 end in a repeated digit?

I was passing some time recently writing down powers of 2:

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, ...

I noticed that these numbers never end in a repeated digit. By contrast, square numbers certainly can, with 144 being an example.

Is it possible for a power of 2 to end in a repeated digit? If so, what is the maximum number of repeated digits it can end in? (In other words, can it end in 3 of the same digit? 4 of the same digit? 5 of the same digit...?)

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