The question reads
What is the largest number of consecutive square-free positive integers?
Square-free means that the prime number decomposition of, say $n$, is
$$
n = p_1^{\alpha(1)}p_2^{\alpha(2)}\dots p_k^{\alpha(k)},
$$
where every $\alpha(j)$ is either zero or one. Digging in the book, I have a possible answer: $(2*3**5*7*11*13,2*3**5*7*11*13+1)$, but I don't know how to prove the statement.
Best Answer
Since one of the numbers $n,n+1,n+2,n+3$ must be divisible by $4$, $3$ consecutive squarefree numbers is the maximum. However, there can be arbitary many consecutive numbers that are NOT squarefree.