Effective results for linear equations in members of two recurrence sequences

Abstract

Let (U n ) ∞ n=0 and (V m ) ∞ m=0 be two linear recurrence sequences. For fixed positive integers k and ℓ, fixed k-tuple (a 1 , . . . , a k ) ∈ Z k and fixed ℓ-tuple (b 1 , . . . , b ℓ ) ∈ Z ℓ we consider the linear equationin the unknown non-negative integers n 1 , . . . , n k and m 1 , . . . , m ℓ . Under the assumption that the linear recurrences (U n ) ∞ n=0 and (V m ) ∞ m=0 have dominant roots and under the assumption of further mild restrictions we show that this equation has only finitely many solutions which can be found effectively.

On sums of Fibonacci numbers with few binary digits

“…These problems are special cases of the following more general problem: Find all numbers that have few digits with respect to both the Zeckendorf and the binary representation. The second author proved a result about linear equations in members of two recurrence sequences [15], which implies that such problems have only finitely many solutions. Moreover, he combined the above-mentioned results and completely solved the problem for the bound M = 5 of total number of digits.…”

Collisions of digit sums in bases 2 and 3

“…This statement is, at least heuristically, close to Furstenberg's conjecture (1). This result was extended by Stewart [56], Mignotte [40], Schlickewei [48,49], Pethő-Tichy [45], and Ziegler [59].…”