Common Values of Padovan and Perrin Sequences

The integer sequence defined by P n + 3 = P n + 1 + P n with initial conditions P 0 = P 1 = P 2 = 1 is known as the Padovan sequence ( P n ) n ∈ Z . The Perrin sequence ( R m ) m ∈ Z satisfies the same recurrence equation as the Padovan sequence but with starting values R 0 = 3 , R 1 = 0 , and R 2 = 2 . In this note, we solve the Diophantine equation P n = ± R m with ( n , m ) ∈ Z 2 .