Sum and Difference of Powers of Two Fibonacci Numbers

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-n(x) +/- F-m(x) = p(k) for 0 <= m < n, where F-n and F-m are the n-th and m-th Fibonacci numbers, respectively. For m not equal 0, the gcd( F-n, F-m) = 1 and F-n(x) + F-m(x) = p(k), where x is not a power of 2.