An exploration of very triangular numbers

TL;DR

This paper investigates the properties and distribution of very triangular numbers, revealing new theorems about their occurrence, gaps, and density among all triangular numbers.

Contribution

It introduces new results on the distribution, gaps, and density of very triangular numbers, including analogs of classical number theory theorems.

Findings

Twin very triangular number theorem established

Existence of arbitrarily long gaps proven

Analog of Bertrand's postulate for very triangular numbers demonstrated

Abstract

We present a collection of results concerning the location and distribution of very triangular numbers among triangular numbers, including the twin very triangular number theorem, the existence of arbitrarily long gaps between -- and an analog of Bertrand's postulate for -- very triangular numbers.