Sums of Fibonacci numbers close to a power of $2$

Abstract: The Fibonacci sequence $(F_n)_{n \geq 0}$ is the binary recurrence sequence defined by $F_0 = F_1 = 1$ and $$ F_{n+2} = F_{n+1} + F_n \text{ for all } n \geq 0. $$ There is a broad literature on the Diophantine equations involving the Fibonacci numbers. In this talk, we will study the Diophantine inequality $$ | F_n + F_m - 2^a | < 2^{a/2} $$ 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-Pethö version of the Baker-Davenport reduction method in Diophantine approximation.

combinatoricsnumber theory

Audience: researchers in the topic

Lethbridge number theory and combinatorics seminar