Triangle Area Numbers and Solid Rectangular Numbers

TL;DR

This paper investigates triangle area numbers with integer sides and rational areas, proving they are multiples of 6, and provides a list of such numbers up to 999, along with related divisibility conditions.

Contribution

The work establishes that all triangle area numbers with integer sides and rational areas are multiples of 6 and enumerates all such numbers up to 999.

Findings

Every triangle area number is an integer multiple of 6.

Derived divisibility conditions for side lengths.

Listed all triangle area numbers ≤ 999.

Abstract

In this work, we define a triangle area number to be the area number of a triangle whose sides have integer lengths, and whose area is a rational number. In Result 3, on page 17, we prove that every triangle area number is in fact an integer which is a multiple of 6. Certain divisibility and other conditions and formulas are also derived, which the three integer sidelengths must satisfy. On pages 20 and 21, we list all the triangle area numbers not exceeding 999.