Padovan or Perrin numbers that are concatenations of two distinct base b repdigits

Abstract Let { P n } n ⩾0 be the Padovan sequence with initial conditions P 0 =0, P 1 =1, and P 2 =1 and the recurrence relation P n +3 = P n +1 + P n . Its companion sequence is known as the Perrin sequence { E n } n ⩾0 that satisfies the same above recurrence relation with the initial conditions E 0 =3, E 1 =0 and E 2 =2. In this paper, we determine all Padovan and Perrin numbers that are concatenations of two distinct base b repdigits with 2 ⩽ b ⩽ 9. As corollary, we prove that the largest Padovan and Perrin numbers which can be representable as a concatenations of two distinct base b repdigits are P 26 = 816 = 2244 ‾ 7 $ P_{26}=816=\overline{2244}_7 $ and E 24 = 853 = 31111 ‾ 4 $ E_{24}=853=\overline{31111}_4 $ .