Problem
NT-B2-M06-P020 Powers of \(2\) in \(3^{2^m}-1\)
#20
★★★★★ Level 5 of 5
Prove that for \(m\ge1\), \(v_2(3^{2^m}-1)=m+2\).
Apply the \(p=2\) LTE formula with \(a=3\), \(b=1\), \(n=2^m\).
\(v_2(3^{2^m}-1)=v_2(3-1)+v_2(3+1)+v_2(2^m)-1=1+2+m-1=m+2\).
Final problem reinforces the special formula for \(2\).