Pythagorean Boxes with Primitive Faces
Konstantine Zelator

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…
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Taxonomy
TopicsMathematics and Applications · History and Theory of Mathematics
