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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]

In this paper we completely solve the family of parametrised Thue equations where Fn is the n-th Fibonacci number. In particular, for any integer n ≥ 3 the Thue equation has only the trivial solutions (±1; 0); (0;∓1);∓(Fn; 1);∓(2n; 1). [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of consecutive pairs and consecutive triples of multiplicatively dependent integers. A theorem by LeVeque on Pillai's equation implies that the only consecutive pairs of multiplicatively dependent integers larger than 1 are (2 , 8) and (3 , 9). For triples, we prove the following theorem: If a ∉ { 2 , 8 } is a fixed integer larger than 1, then there are only finitely many triples (a , b , c) of pairwise distinct integers larger than 1 such that (a , b , c) , (a + 1 , b + 1 , c + 1) and (a + 2 , b + 2 , c + 2) are each multiplicatively dependent. Moreover, these triples can be determined effectively. [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]