Misc,
Pythagorean Boxes with Primitive Faces
(2010)cite arxiv:1005.0084 Comment: 14 pages.
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.
