On some congruence with application to exponential sums (Q1431067)

The author denotes by \(\omega_g(m)\) the order of \(g\) modulo \(m\), where \(g\) and \(m>0\) are coprime integers, and proves the following main result. For any integers \(n>2\) and \(g\) odd, (a) if \(g\not\equiv \pm 1\) (mod \(2^n\)), then \(g^{\omega_g(2^n)/2} \equiv 2^{n-1}+1 \text{ (mod }2^n\))