Pythagorean Triplets

ab

We seek to identify all positive integers a, b, and c so that a2+b2=c2. A triple (a,b,c) satisfying this equation is called a pythagorean triplet. Since any multiple of a pythagorean triplet is again one, we only seek primitive ones where a, b, and c have no common factor. By setting and , we can first seek rational numbers x and y so that x2+y2=1. =2in
Now, if (x,y) is such a pair of rational numbers, the line from (x,y) to (-1,0) will have rational slope - call it t. Along this line, y=t(x+1), so x2 + t2(x+1)2=1 telling us that

(1+t2)x2 + 2t2 x = 1-t2

or

Complete the square on the above equation giving

Taking square roots gives

Write with u, v positive integers having no common factor. This gives

Using and we see that

for some rational number k. By thinking about the highest power pr of an odd prime appearing in the denominator of k (written in lowest terms), one sees that pr divides uv, u2+v2, -u2+v2, 2u2, and 2v2. This means divides both u and v contradicting u and v having no common factor, UNLESS r=0. Similarly by noting exactly one of a and b can be odd, and choosing b to be the even number, we can eliminate powers of 2 in the denominator of k. This shows that k is a whole number and we obtain: Theorem: The pythagorean triplets with b even are exactly the triples of the form

where u and v are positive whole numbers with no common factor.