Tribonacci Numbers Close to the Sum $2^a+3^b+5^c$

Abstract

We show that there are exactly 22 solutions to the inequalities \[ 0\le T_n-2^a-3^b-5^c\le10, \] where $T_n$ denotes the $n^{\it th}$ term ($n\ge0$) of the Tribonacci sequence, and $0\le a,b\le c$ are integers. All the solutions are explicitly determined.