Greatest common divisor of consecutive square free numbers

Question

I guess that every prime number occurs as the greatest common divisor of two consecutive square free numbers, which I don't expect a proof of.

But I've done some experiments indicating that:

If $m, n$ are consecutive square free numbers then $\gcd(m, n)$ is not composite.

Is that true and can it be proved?

Answer 1

Barring miscalculation we have $$\gcd(28331962460555993122305,28331962460555993122290)=15$$

And these two numbers are consecutive square free integers. Indeed we can obtain the relevant factorings via WA.

This example was constructed out of the Chinese Remainder Theorem, using $$\text {ChineseRemainder}[(0,1,2,3,4,5,6,7,8,10,11,14),$$$$(15,4,7^2,9,11^2,25,13^2,17^2,19^2,23^2,29^2,31^2)]$$ in WA