# Except for $1$, are there any perfect powers in the sequence $1,21,321,4321,\cdots$?

Except for $$1$$, are there any perfect powers in the sequence $$1,21,321,4321,\cdots$$?

This is sequence OEIS A000422, the concatenation of positive integers from $$n$$ down to $$1$$.

If there is any perfect square in the sequence, $$n$$ must be equal to $$1\pmod{3}$$, as when $$n\equiv1\pmod{3}$$, then the number is equal to $$1\pmod{3}$$,which can be a perfect square and also possibly a perfect cube.

If $$n\cdots4321$$ is divisible by $$9$$, then $$n$$ is equal to $$8\pmod{9}$$ in order to be a perfect square.

In case $$n\cdots4321$$ is divisible by $$27$$, then $$n$$ is equal to $$26\pmod{27}$$ in order to be a perfect cube.

I don’t know about higher odd prime perfect powers.

The only sure thing is that any number(except $$1$$) in OEIS A000422 is never a perfect 5th power, due to the fact that $$21$$ isn’t a fifth power residue $$\pmod{100}$$.

Also, the only possibilities that $$n\cdots4321$$ is a perfect power is that either $$n\equiv1\pmod{3}$$, $$n\equiv8\pmod{9}$$, or $$n\equiv26\pmod{27}$$.

I tried to used Pari GP to check if there is any perfect power in the sequence other than $$1$$, but so far, I didn’t find one.

• @lulu oh right my bad Jan 10 at 13:32
• For $n\le 10^4$ , there is no perfect power. Jan 10 at 14:54
• You have edited this post, so is there any connection? Do you know whether any perfect square $b^2>1$ is of this form, or not? Jan 10 at 19:32

There are no squares in the sequence because $$10987654321$$ (the $$10$$th entry in the sequence) is not a square mod $$10^{11}$$, and the first $$10$$ entries are not squares (except $$1$$). Since the residues modulo $$10^{11}$$ are all the same after this, there cannot be a square in the sequence.
There are also no fifth powers since $$21$$ is not a fifth power mod $$100$$.
This style of proof will not show that there are not $$p$$th powers in the sequence for $$p$$ coprime to $$10$$, I'm afraid. This is because Hensel's lemma implies that an integer ending in $$1$$ is a $$p$$th power mod $$10^n$$ for any $$n$$ and $$p$$ coprime to $$10$$.