Pillai’s problem with the Fibonacci andPadovan sequences

Abstract

Let ( 𝐹 𝑚 ) 𝑚 > 0 and ( 𝑃 𝑛 ) 𝑛 > 0 be the Fibonacci and Padovan sequences given by the initial conditions 𝐹 0 = 0 , 𝐹 1 = 1 , 𝑃 0 = 0 , 𝑃 1 = 𝑃 2 = 1 and the recurrence formulas 𝐹 𝑚 +2 = 𝐹 𝑚 +1 + 𝐹 𝑚 , 𝑃 𝑛 +3 = 𝑃 𝑛 +1 + 𝑃 𝑛 for all 𝑚, 𝑛 > 0 , respectively. In this note we study and completely solve the Diophantine