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

Abstract

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 $ .