I'm studying P. Erdős & J. Surányi's book Topics in Theory of Numbers. The Exercise 2.20b) reads:
Prove that for an arbitrary positive integer $N$, there exists $N$ consecutive integers, none of which is a power of an integer (where the exponent is greater than one).
Context: The section in which the exercise is uses the Chinese remainder theorem to prove that for any integer $N>0$ there exists a prime $p$ such that $p-1,\ldots, p-N$ and $p+1,\ldots,p+N$ are all composite. As a matter of fact, exercise 2.20a asks for a proof that there are $N$ consecutive, non squarefree numbers, which I have solved.
Thoughts & Tries: With the idea of using ChRT in mind, I have looked for the appropiate moduli. I have thought on using squares, something like $x_j\equiv j\pmod {p_j^2}$, but the fact that at a number is not the power of $N$ primes does not imply that it is not a power of the next one...
I have had some other ideas, like beginning at $2^{N!}$ or $N^N$, but none of them seems to work.
I still want to solve the problem, so I prefer hints than complete solutions.
you can do this easily with the chinese residue theorem.
Just take $n$ prime numbers $p_0,p_1,\dots,p_{n-1}$ and find a number that is a solution to:
$x\equiv 0 \bmod p_0^2$
$x\equiv p_1 -1 \bmod p_1^2$
$x\equiv p_2-2 \bmod p_2^2$
$\dots$
$x\equiv p_{n-1}-(n-1)\bmod p_{n-1}^2$
None of the numbers $x,x+1,\dots,x+n-1$ is a perfect square, as $p_j| x+j$ but $p_j^2 \nmid x+j$.