Two Problems on Narayana Numbers And Repeated Digit Numbers

Abstract

This work aims to solve two problems in the Diophantine equation of the Narayana sequence. In the first question it’s proven that there are only 177 solutions of expressing the product of two Narayana numbers as b repdigits numbers, for base 2 ≤ b ≤ 50. It’s also proven that the Narayana numbers can not be factored as a product of two repdigits numbers for base 2 ≤ b ≤ 50, except in two cases. The proofs use some number-theoretic techniques, including Baker’s method of linear forms in logarithms height, and some reduction techniques.