Generating Triples When One Side Is Known
The following method is a direct algebraic manipulation of the Euclid equations.
Start with any even integer, and use the relation from the Euclid formula. Identify all factor-pairs (m,n) of and use the Euclid equations to calculate the remaining sides of the triple.
Examples: Let (e.g. the known side is even)
so that . The factor pairs (m,n) of 12 are (12,1), (6,2) and (4,3). The three possible triples are therefore:
Let (e.g. the known side is odd)
The two unknown sides could also be calculated by making use of the relation . This would be a factoring exercise in finding the difference of two squares, but a simpler approach is to multiply the known side by two and continue as before:
- so that
The factor pairs (m,n) of 35 are (35,1), (7,5).
The two triples are therefore (note that is necessary to remove the factor of 2 which was introduced):
Read more about this topic: Formulas For Generating Pythagorean Triples
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