On December 11 2002 I posted a message to the number theory list that contained the following:
A paper by Andrew Bremner and Richard Guy published January 1988, "A
Dozen Difficult Diophantine Dilemmas" has:
A) E.T. Prothro xy(x^4 - y^4) = 2zw(z^4-w^4) (1) and,
B) Ernest J. Eckert xy(x^4 - y^4) = zw(z^4-w^4) (2)
C) "Computer searches over a small range failed to find any solution to
(1) or (2)."
In the usual notation for exponentiation these equations are:
xy(x4 - y4) = 2zw(z4-w4)
(1) and,
xy(x4 - y4) = zw(z4-w4)
(2)
and as will shortly be obvious (1) expresses the following question.
Are there two distinct Pythagorean triples, side-lengths of right
triangles, so that a product of the three (lengths of each triangle's
sides) is the same? Two individuals responded. Their correspondence led me to seek the
definition of primitive right triangles and Pythagorean
triples; Google search led to these online references, and the
hardcopy version, Eric W. Weisstein, World of Mathematics, 1999,
CRC Press LLC. A series of reading steps led to the following statment
in a paper
by Olga Taussky, "The Many Aspects of the Pythagorean Triangles,"
Linear Algebra and its Applications43:285-295, 1982:
"All of the Pythagorean triples are known to us. They are given by the
expressions
l(m2 - n2),
l2mn,
l(m2 + n2).
Here
l, m, n are whole numbers."
Verification of matrix transformation of (3, 4, 5) into other
Pythagorean triples Mathematica computations: (5, 12, 13); (21, 20, 29); and
(15, 8, 17).
"For any Pythagorean triple, the product of the two nonhypotenuse legs (i.e.,
the two smaller numbers) is always divisible by 12, and the product of
all three sides is divisible by 60.
It is not known if there are two
distinct triples having the same product. The existence of two such
triples corresponds to a nonzero solution to the Diophantine equation