Perrin numbers that are concatenations of two repdigits

Abstract

Let $$ (P_n)_{n\ge 0}$$ ( 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.

$k$-Pell-Lucas numbers which are concatenations of two repdigits

“…Consequently, Ddamulira searched all Tribonacci and Padovan numbers which are concatenations of two repdigits in [8] and [9] respectively. Batte et al [4] searched for only Perrin numbers which are concatenations of two repdigits.…”

Padovan numbers which are concatenations of three repdigits

“…Ddamulira searched all Padovan and Tribonacci numbers which are concatenations of two repdigits in [5], [6], respectively. Batte et al [3] proved that the only Perrin numbers which are concatenations of two repdigits are {10, 12, 17, 22, 29, 39, 51, 68, 90, 119, 277, 644}. In [14], Rayguru and Panda showed that 35 is the only balancing number which is concatenation of two repdigits.…”