Triangular Numbers With a Single Repeated Digit

TL;DR

This paper revisits the classification of triangular numbers with a single repeated digit in their decimal form, providing a new elementary proof to correct previous flawed arguments and confirm the complete list of such numbers.

Contribution

It offers a new, elementary proof that accurately characterizes all triangular numbers with a repeated digit, correcting earlier flawed proofs from the 1970s.

Findings

Confirmed the complete list of triangular numbers with a single repeated digit

Provided a corrected and elementary proof of the classification

Clarified the validity of previous results in the literature

Abstract

The question of which triangular numbers have a decimal representation containing a single repeated digit seamed to be settled since at least the 1970s: Ballew and Weger provided a complete list and a proof that these are the only numbers of this kind. This assertion is referenced by other authors in the field. However, their proof is flawed. We present a new and elementary proof of the statement, which corrects the error.