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

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mrow> <m:mn>26</m:mn> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>816</m:mn> <m:mo>=</m:mo> <m:msub> <m:mover> <m:mn>2244</m:mn> <m:mo accent="false">‾</m:mo> </m:mover> <m:mn>7</m:mn> </m:msub> </m:math> P₂₆=816=2244̄₇ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>E</m:mi> <m:mrow> <m:mn>24</m:mn> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>853</m:mn> <m:mo>=</m:mo> <m:msub> <m:mover> <m:mn>31111</m:mn> <m:mo accent="false">‾</m:mo> </m:mover> <m:mn>4</m:mn> </m:msub> </m:math> E₂₄=853=31111̄₄ .