Odd Repdigits to Small Bases Are Not Perfect

Abstract

We demonstrate, by considering each base in the range 2 through 9, that no odd repdigit with a base in that range is a perfect number.

Perfect repdigits

“…By this we mean (X 1 , Y 1 ) is the smallest pair of positive integers such that X 2 1 − DY 2 1 = ε holds with some ε ∈ {±1}. This always exists since there always exists a solution in positive integers of equation (21) for which ε = 1.…”

“…This always exists since there always exists a solution in positive integers of equation (21) for which ε = 1. If a solution with ε = −1 exists, then the minimal one, namely (X 1 , Y 1 ), has corresponding ε = −1 and the smallest solution with ε = 1 in this case is (2X 2 1 + 1, 2X 1 Y 1 ). From the theory of the Pell equations, we know that equation (2.1) always has infinitely many positive integer solutions (X, Y ).…”

“…In this paper, we present an algorithm to compute all perfect repdigits in base g. As an illustration, we extend the computations from [2] to all bases g ∈ [2,333]. As a byproduct of our work we also get some theoretical bounds such as: Theorem 1.1.…”

“…Until now, no explicit upper bound on the largest solution as a function of g has been computed. In [2], perfect repdigits to bases g have been computed for all g ∈ {2, . .…”

“…, 10}. The method of [2] reduces in most cases to using modular constraints or solving several particular exponential type Diophantine equations.…”