
Pythagorean Triples
A pythagorean triplet ("PT") consists of three natural numbers x, y and z
with x^{2} + y^{2} = z^{2}. PT's with greatest
common divisor 1 ("PPT's") are of particular interest.
Theorem 1
Every PT can be obtained in a unique way as a product of a PPT and a natural number k.
Theorem 2
In every PPT ( x  y  z ) one of the numbers x or y ("legs") is even and the other one is odd.
(Let x always be the odd leg.)
Theorem 3
For every PPT ( x  y  z ) there exists one and only one pair ( m  n ) of relatively prime natural numbers
of different parity (ie.: one of the numbers is even and the other is odd) with m < n, such that:
x = n^{2}  m^{2}, y = 2mn , z = n^{2} + m^{2}
resp. ^{1}/_{2}(z  x) = m^{2},
^{1}/_{2}(z + x) = n^{2}.
Theorem 4
Every PPT can in a unique way be obtained from ( 3  4  5 ) by application of a "sequence" of transformations
A, B or C defined by:
A: (xyz) > ( x2y+2z 2xy+2z 2x2y+3z) [or (mn) > (n2nm)]
B: (xyz) > ( x+2y+2z 2x+y+2z 2x+2y+3z) [or (mn) > (n2n+m)]
C: (xyz) > (x+2y+2z2x+y+2z2x+2y+3z) [or (mn) > (m2m+n)]
to other interesting mathematical topics ...
for more information about the pairs ( m  n ) and the transformations A, B, C see: number theory interactive ..
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