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Is Yaranaikian sequence power-free?

The following sequence $69,696,6969,69696,....$ is called Yaranaikian. To clarify, each term is formed by writing 6 and 9 alternately(starting off with 6 from the left) . It was found that 69696 is actually a perfect square, its square root being 264. That was when my curiosity kicked in. Is there anymore perfect square in the sequence?. After some observation and works, I managed to prove that Yaranaikian sequence is sqaure-free , except for 69696 of course; the proof can be found on my fb's note section,if you're interested. However, it didn't stop there. A more challenging question popped up, as stated below;

Is Yaranaikian sequence excluding 69696 power-free, that is it doesn't contain a term of the form $a^n$ where $a,n$ are positive integers greater than 1?

It's still an open question , but I managed to crack some of the cases as described in the papers on my fb's note section. I sincerely appreciate any attempts or suggestions from anyone.

One may slightly generalise the problem by substituting 6 and 9 by any digits.

Note by ComradeGeneral Poon Thongsai
2 years, 11 months ago

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