Integer number m is square-free if m=rad(m).
Hypothesis. (Papava). 22^n-1=rad(22^n-1).
Hypothesis.(Open problem). Fermat number Fn=22^n+1 is square-free.
Result. If Fermat number is square-free then 22^n-1 is square-free.
P r o v e
Mathematical induction:
22^1-1=3 is square-free.
Let 22^k-1 is square-free when k<=n then 22^(k+1)-1=(22^k-1)(22^k+1) is square-free, because Fk=22^k+1 square-free and 22^k-1 is square-free when k<=n, also (22^k-1, 22^k+1)=1.
Or 22^n-1=Fn-2=F0F1F2F3...Fn-1, (Fi, Fj)=1, i<>j.
Emzari Papava
|