Abstract
Let (F-n)(n >= 0) and (P-n)(n >= 0) be the Fibonacci and the 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 Fn+2 = Fn+1 + F-n, Pn+3 = Pn+1 + Pn for all n >= 0, respectively. In this note we study and completely solve the Diophantine equation
P-n + P-m = F-l
in non-negative integers (n, m, l).