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In 2018, Luca and Patel conjectured that the largest perfect power representable as the sum of two Fibonacci numbers is 38642 = F36 + F12. In other words, they conjectured that the equation has no solutions with a ≥ 2 and ya > 38642 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 ya, which depends on the size of y. In this paper, we find an explicit upper bound for ya, which only depends on the Hamming weight of y with respect to the Zeckendorf representation. More specifically we prove the following: If y = Fn1 + · · · + Fnk and equation (*) is satisfied by y and some non-negative integers n, m and a ≥ 2, then Here, ϵ > 0 can be chosen arbitrarily and C(ϵ) is an effectively computable constant. [ABSTRACT FROM AUTHOR]

Let (U_n)_{n\in \mathbb {N}} be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants B and N_0 such that for any b,c\in \mathbb {Z} with b> B the equation U_n - b^m = c has at most two distinct solutions (n,m)\in \mathbb {N}^2 with n\geq N_0 and m\geq 1. Moreover, we apply our result to the special case of Tribonacci numbers given by T_1= T_2=1, T_3=2 and T_{n}=T_{n-1}+T_{n-2}+T_{n-3} for n\geq 4. By means of the LLL-algorithm and continued fraction reduction we are able to prove N_0=2 and B=e^{438}. The corresponding reduction algorithm is implemented in Sage. [ABSTRACT FROM AUTHOR]