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Suppose, $a,b\ge 2$ are integers. Then $$N=a^b$$ is a perfect power. Assuming that $a$ is not divisible by $10$, are there infinite many such perfect powers consisting of at most $2$ distinct decimal digits ?

The first few perfect powers with the desired property are :

? for(j=1,10^7,if(Mod(j,10)<>0,if(ispower(j)>0,if(length(Set(digits(j)))<=2,prin
t1(j," ")))))
4 8 9 16 25 27 32 36 49 64 81 121 144 225 343 441 484 676 1331 1444 7744 7776 11
881 29929 44944 55225 69696 9696996
?

Additionally, I found the square $$6661661161$$

Are there more examples, and if yes, are there infinite many examples ?

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  • $\begingroup$ Note: If you want empiric evidence, find all 3 digit numbers n with only two different digits in the last three digits. To find the 4 digit numbers with only two different digits in the last four digits, examine n, n+1000, n+2000, ..., n+9000. You can examine all numbers up to 10^k in very roughly 2^n operations. $\endgroup$ – gnasher729 Dec 28 '18 at 20:44
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    $\begingroup$ Quote from oeis.org/A018885 : Is 6661661161 the largest term not of the form 10^k, 4*10^k or 9*10^k? Any larger ones must have >= 22 digits. $\endgroup$ – Oldboy Dec 29 '18 at 11:37

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