An integer $x$ is said to be square if it can be written in the form $x=n^2$. I'm trying to find the largest number of consecutive square-free positive integers. Now I know my approach is wrong but I will state it so my question would be clear.
At first I tried to take the square of the positive integers so we get: $1, 4, 9, 16, 25, 36, 49..$
Then counting the numbers between 1 and 4 , we would get 2 square-free integers. Between 4 and 9 we would have 4 square-free integers. And 6 square-free integers between 9 and 16. So we could assume that there are at least $2n$ square-free integers.
(Note that I assumed that the missing integers between 1 and 4 namely 2 and 3 would be square-free).
However, If you look at this question it seems that a square-free number is number where the primes in its decomposition has only exponents equal to 1 or 0. And you can see the proof followed there.
Does this mean that for example $50=5^2\cdot2$ is a square number? Am I mixing up two completely different definitions? If not, how can for example 8 be a square number?