On a Diophantine equation involving powers of Fibonacci numbers

On the equation ∑j=1kjFjpAltassan1, Luca2 2020Journal of Number TheorySelf Cite2000View full textAdd to dashboardCite

“…We assume that k < X 1/4 and we search for an acceptable X 0 . From Lemma 4 in [4], we have that F j /F j+1 ≤ 2/3 for all j ≥ 2. In particular, F k−j /F k ≤ (2/3) j for all j = 1, 2, .…”

“…These equations have been solved in [4] and they have no solutions with k ≥ 3. So, we record what we have proved.…”

“…In [10], the authors found all solutions when {p, q} ⊆ {1, 2} by ad-hoc methods. A general method to find all solutions of the title equation when p and q are given was proposed in [4]. As an application, the authors of [4] confirmed Conjecture 1.1 for the range of the exponents {p, q} ⊂ [1,10].…”

“…Conjecture 1.1. The only nontrivial solutions of the title equation are given by (k, n, p, q) = (3, 4, 1, 2), (3,4,3,3), (4,8,1,1).…”

“…A general method to find all solutions of the title equation when p and q are given was proposed in [4]. As an application, the authors of [4] confirmed Conjecture 1.1 for the range of the exponents {p, q} ⊂ [1,10]. None of the above papers addressed the question of whether the title equation has only finitely many positive integers solutions and whether the current methods allow us to bound them or even compute them.…”