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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

273.0. "Pythagorean abc=rst" by HARE::STAN () Wed May 01 1985 17:19

Can two different primitive Pythagorean triangles with sides (a,b,c)
and (r,s,t) be such that abc=rst?

[question proposed by Ernest J. Eckert as problem 994 in
 Crux Mathematicorum, 1984(10)318.]
T.RTitleUserPersonal
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273.1HARE::STANThu May 02 1985 20:412
To clarify: A Pythagorean Triangle (a,b,c) is a right triangle
with integer sides.  The lengths of the sides are a, b, and c.
273.2TURTLE::GILBERTTue Jun 11 1985 23:5012
Since all primitive Pythagorean triples are of the form (2mn,m^2-n^2,m^2+n^2),
with m > n > 0, the problem is to find a non-trivial integral solution to

            4   4        4   4
	mn(m - n ) = xy(x - y )

I evaluated the left-hand side of this expression (modulo 2^32) for all m,n
with 256 >= m > n > 0, found the duplicates, and checked their values (this
time modulo 2^64).  No solutions were found.

Perhaps someone would like to search further, or try proving that no such
solution exists.