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 $ .

The Diophantine equation $\displaystyle a \left(\frac{b^k - 1}{b - 1}\right) = \mathcal{U}_n - \mathcal{U}_m$

“…Studies have focused on their properties, connections with recurrent sequences, and representations as concatenations, differences, or products of other mathematical entities. For details of recent work related to this problem studied in this paper, we refer the reader to [4,5,6,8,9,10]. They contribute to the field of number theory, inspiring further research in the exploration of repdigits and their intricate connections with other mathematical entities.…”