On sums of two Fibonacci numbers that are powers of numbers with limited hamming weight

In 2018, Luca and Patel conjectured that the largest perfect power representable as the sum of two Fibonacci numbers is 3864(2) = F-36 + F-12. In other words, they conjectured that the equation (*) y(a) = F-n + F(m)has no solutions with a >= 2 and y(a) > 3864(2). While this is still an open problem, there exist several partial results. For example, recently Kebli, Kihel, Larone and Luca proved an explicit upper bound for y(a), which depends on the size of y. In this paper, we find an explicit upper bound for y(a), which only depends on the Hamming weight of y with respect to the Zeckendorf representation. More specifically, we prove the following: If y = F-n1 + <middle dot> <middle dot> <middle dot> + F-nk and equation (*) is satisfied by y and some non-negative integers n, m and a >= 2, then y(a )<= exp (C(epsilon) <middle dot> k((3+epsilon)k2)) . Here, epsilon > 0 can be chosen arbitrarily and C(epsilon) is an effectively computable constant.