  After the introduction, in section 2 we state the well known parametric
formulas that describe the entire family of Pythagorean triples. In section 3,
we list four well known results from number theory, used later in the paper. in
section 2, we prove two propositions. Proposition 1 says that the diophantine
equation z^2=x^4+4y^4, has no solutions in positive integers x,y, and z. We
then use Proposition 1, to prove the insolvability of a certain four-variable
diophantine system, which is Proposition 2. In Section 5, we present some
examplesof pairs of Pythagorean triangles with a common hypotenuse. In Section
6, we use Proposition 2 to prove Theorem 1: There exists no pair of Pythagorean
triangles such that the leg of largest length in the first triangle is the
hypotenuse of the second triangle; and in addition, with the leg of smallest
length in the first triangle; having the same length as a leg in the second
triangle. in Section 7, we present information on Pythagorean boxes, and we
describe how to generate infinitely many Pythagorean boxes with a face diagonal
of integral length. Finally, in Section 8, we apply Proposition 2 to prove
Theorem 2: There exists no Pythagorean box with a pair of congruent (or
opposite) faces being squares. And with the four diagonals of equal length in
another pair of opposite faces, also having integral length.
