Search

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 ≥ 0 and { V m } m ≥ 0 be two linear recurrence sequences. We establish an asymptotic formula for the number of integers c in the range [ - x , x ] which can be represented as differences U n - V m . In particular, the density of such integers is 0. [ABSTRACT FROM AUTHOR]