
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 results
x  y  a  b  c 
1  2  3  4  5 
1  3  6  8  10 
1  3  5  12  13 
3  4  7  24  25 
Some triples are not very original as they are just multiples of earlier triples.
Basic triples have no common factors or: GCD(a,b,c) = 1

