Exercise 2.3.18 (For any $k$, there are $k$ consecutive integers not square free)

Question

Given any positive integer $k$, prove that there are $k$ consecutive integers each divisible by a square $>1$.

Richard Ganaye · Accepted Answer

\begin{proof} 
Write $p_n$ the $n$-th prime number : $p_1 = 2,\ p_2= 3,\  p_3 = 5,\ldots$

Since $p_1^2,\ldots,p_k^2$ are relatively prime by pairs, the Chinese Remainder Theorem shows that there is some integer $x$ such that
\begin{align*}
&x \equiv -1\pmod {{p_1}^2},\
&x \equiv -2 \pmod {{p_2}^2},\
&\ldots\
&x \equiv -k \pmod {{p_k}^2}.\
\end{align*}
Then $p_1^2 \mid x + 1,\ p_2^2 \mid x+2,\ldots,p_k^2 \mid x+k$, thus each of the $k$ consecutive integers $x+1, \ldots,x+k$ is divisible by a square greater than $1$ (they are not square free).
\end{proof}

Exercise 2.3.18 (For any $k$, there are $k$ consecutive integers not square free)

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Exercise 2.3.18 (For any $k$, there are $k$ consecutive integers not square free)

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