Is every primorial number ( a number of the form $p$#$\pm 1$ ) squarefree ?
According to my calculation, there is no prime $q\le 270,000$, such that $q^2$ can be a divisor of $p$#$-1$ or $p$#$+1$. The PARI/GP program is as follows
? p=1;gef=0;while(gef==0,p=nextprime(p+1);if(Mod(p,10^4)==1,print(p));q=p^2;n=1; forprime(s=1,p,n=n*s;n=component(Mod(n,q),2);if(Mod(n^2-1,q)==0,gef=1;print(s," ",q)))) 70001 90001 150001 160001 180001 270001
Have I programed correctly ?
Considering that no small squares can divide a primorial number, I strongly conjecture that they are all squarefree. But can it be proven ?