
Three natural numbers (a,b,c) are called a "Pythagorean triple" if a^{2} + b^{2} = c^{2}
Note: calculations using the Pythgoras lemma in most cases result in answers with roots.
The most wellknown Pythagorean triple is (3,4,5) because 3^{2} + 4^{2} = 5^{2}
Question: are there more triples?
Let's explore
a^{2} + b^{2} = c^{2}
b^{2} = c^{2}  a^{2}
b^{2} = (c  a)(c + a)
Now define:
c  a = x^{2}
c + a = y^{2}.............so
b^{2} = x^{2}y^{2}
b = xy
next we solve the equations:
The original formula a^{2} + b^{2} = c^{2} may be rewritten as:
summary:
a = y^{ 2} − x^{ 2}
b = 2 x y
c = y^{ 2} + x^{ 2}
For each pair of natural numbers (x,y) now a,b en c may be calculated.
Conclusion
There exists an infinite number of Pythagorean triples.
Some triples are not very original as they are just multiples of earlier triples.
Basic triples have no common factors or: GCD(a,b) = 1
Some results
A small Delphi program calculates the triples:
download program (zip format)
download Delphi7 project

