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# All the same digits!

One day I was playing with my calculator, I pushed $$6^5$$ and then it came out 7776. I found it very interesting, because except the units digit, all the other digits are 7!

Then a problem came up in my mind:

How many integers in the form $$a^b$$ where $$a$$ and $$b$$ are positive integers greater than 1 which except the units digits, the other digits are all the same?

This problem is equivalent to finding integer solutions that satisfy $a^b=\overline{\underbrace{mmm\ldots mm}_{x~m\text{'s}}n}$ Here, $$x>1$$.

There is another number I found: $$21^2=441$$

Any ideas?

If you have a solution, comment and list out the steps.

Note by Tan Kenneth
2 years, 10 months ago

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A computer search shows that for numbers up to $$60$$ digits long, the only ones that have this property are $$225, 441, 7776$$. If there exists another, it's going to be longer than $$60$$ digits. · 2 years, 10 months ago