An exponential equation involving k-Fibonacci numbers

Abstract

For k ≥ 2, consider the k -Fibonacci sequence (F(k) n )n≥2−k having initial conditions 0, . . . , 0, 1 (k terms) and each term afterwards is the sum of the preceding k terms. Some well-known sequences are special cases of this generalization. The Fibonacci sequence is a special case of (F(k) n )n≥2−k with k = 2 and Tribonacci sequence is (F(k) n )n≥2−k with k = 3. In this paper, we use Baker’s method to show that 4, 16, 64, 208, 976, and 1936 are all k -Fibonacci numbers of the form (3a ± 1)(3b ± 1), where a and b are nonnegative integers