A pythagorean triplet ("PT") consists of three natural numbers x, y and z
with x2 + y2 = z2. PT's with greatest
common divisor 1 ("PPT's") are of particular interest.
Every PT can be obtained in a unique way as a product of a PPT and a natural number k.
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.)
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 = n2 - m2, y = 2mn , z = n2 + m2
resp. 1/2(z - x) = m2,
1/2(z + x) = n2.
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: (x|y|z) --> ( x-2y+2z| 2x-y+2z| 2x-2y+3z) [or (m|n) --> (n|2n-m)]
B: (x|y|z) --> ( x+2y+2z| 2x+y+2z| 2x+2y+3z) [or (m|n) --> (n|2n+m)]
C: (x|y|z) --> (-x+2y+2z|-2x+y+2z|-2x+2y+3z) [or (m|n) --> (m|2m+n)]
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for more information about the pairs ( m | n ) and the transformations A, B, C see: number theory interactive ..
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