Prove or disprove :
There exist 7 consecutive odd integers that are each divisible by a perfect cube greater than 1.
it is exist statement so if it is true we have to provide example and if it is false then we have to prove it
I see it is false statement because I do not find sequence like 7 consecutive odd integers that are each divisible by a perfect cube greater than 1.
let the 7 consecutive odd integers look like $2k+1 , 2k+3 , 2k+5,2k+7 ,2k+9 , 2k+11 $, and $2k+13$ are odd so if we select any positive integer $k$ there is at least one of odd integer will be prime
so , we cannot have cubic divisor
is my work correct or any suggestion on that? thanks