Sums of Fibonacci numbers close to a power of 2

Elchin Hasanalizade (University of Lethbridge, Canada)

The Fibonacci sequence \(F(n) : (n\geq 0) is the binary recurrence sequence defined by

$$F(0) = F(1) = 1 \qquad \mbox{and} \\ F(n+2) = F(n+1) + F(n) \qquad \forall n \geq 0.$$

There is a broad literature on the Diophantine equations involving the Fibonacci numbers. In this talk, we will study the Diophantine inequality

$$\left\lvert F(n) + F(m) − 2a\right\rvert < 2a/2$$

in positive integers n,m and a with $n \geq m$. The main tools used are lower bounds for linear forms in logarithms due to Matveev and Dujella-Petho version of the Baker-Davenport reduction method in Diophantine approximation.