Sums of Tribonacci numbers close to powers of 2

Abstract The Tribonacci sequence ( T n ) n > 0 is a generalization of the Fibonacci sequence whose first terms are 0, 1, 1 and each term afterwards is the sum of the three preceding terms. The present paper combines Baker’s theory of linear forms in logarithms with a variant of the Baker-Davenport reduction method to determine all sums of two Tribonacci numbers that are close to powers of 2, based on a notion of closeness introduced by Chern and Cui. Specifically, it determines all nonnegative integer solutions ( n , m , a ) to the Diophantine inequality ∣ T n + T m –2 a | < 2 a /2 . This work extends a previous result of Hasanalizade who found all sums of two Fibonacci numbers close to a power of 2.