Perrin numbers that are concatenations of two repdigits

Abstract

AbstractLet $$ (P_n)_{n\ge 0}$$ ( P n ) n ≥ 0 be the sequence of Perrin numbers defined by ternary relation $$ P_0=3 $$ P 0 = 3 , $$ P_1=0 $$ P 1 = 0 , $$ P_2=2 $$ P 2 = 2 , and $$ P_{n+3}=P_{n+1}+P_n $$ P n + 3 = P n + 1 + P n for all $$ n\ge 0 $$ n ≥ 0 . In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and the reduction procedure involving the theory of continued fractions, to explicitly determine all Perrin numbers that are concatenations of two repeated digit numbers.