# Math Help - gaps between square free

1. ## gaps between square free

Show that there are arbitrarily long gaps between
square free integers. That is,if k is any positive integer,there
is a number N so that none of N + 1, N + 2, ..., N + k are
square free.

2. Originally Posted by bigb
Show that there are arbitrarily long gaps between
square free integers. That is,if k is any positive integer,there
is a number N so that none of N + 1, N + 2, ..., N + k are
square free.
this claim is true only for $k=1,2,3.$ because if $k \geq 4,$ then for any $N$ at least one of the elements in $N + 1, N + 2, \cdots , N + k,$ will divide $4$.