Summary: Let \(p\) be a prime number and let \(x,k>1\) be integers. We find all nonnegative integer solutions \((n,m,x,p,k)\) to the Diophantine equations \(F^x_n\pm F^x_m=p^k\) for \(0\le m<n\), where \(F_n\) and \(F_m\) are the \(n\)-th and \(m\)-th Fibonacci numbers, respectively. For \(m\ne 0\), the gcd\((F_n,F_m)=1\) and \(F^x_n+F^x_m=p^k\), where \(x\) is not a power of 2.