Pythagorean Boxes with Primitive Faces

TL;DR

This paper develops methods to generate infinitely many Pythagorean Boxes with primitive faces, expanding on prior work by establishing explicit conditions and proving the non-existence of certain configurations.

Contribution

It introduces formulas and conditions for constructing Pythagorean Boxes with primitive faces and proves the impossibility of some configurations.

Findings

Derived explicit conditions for Pythagorean Boxes with primitive faces

Established methods to generate infinitely many such boxes

Proved non-existence of boxes with square base and primitive face

Abstract

In their paper "Pythagorean Boxes", Raymond A.Beauregard and E.R.Suryanarayan define the concept or notion of Pythagorean Rectangle as one with sidelengths and integer diagonal lengths(see [1]);they also introduce the concept of a Pythagorean Box as a rectangular three-dimensional parallelepiped whose edges and diagonals have integer lengths.As in that paper, the abbreviation PB will simply stand for "Pythagorean Box";also in this article,the abbreviated notion PR will stand for "Pythagorean Rectangle". In the Beauregard and Suryanarayan paper,it was shown that there exist infinitely many PB's with a square base and height equal to 1.In this paper,we present a method and formulas that generate infinitely many PB's that contain a pair of opposite(and hence congruent)PR's which are primitive;a PR is primitive if the four congruent Pythagorean triangles contained there in are primitive.There are three results in this paper.In Result1,we derive certain explicit conditions that a PB must satisfy, if it possesses two pairs of(opposite)primitive PR's.In Result2,we show that if similar conditions are satisfied then infinitely many PB's can be generated containing a pair of(opposite)PR's.In Result3, we prove that there exist no PB's with a square base and a face(and hence two faces)which is a primitive PR.