Expert: Vijilant - 11/10/2007

Question
QUESTION: Sir,
I am Mahima and I have 2 questions in this time. Please answer .
1. Is there a formula to figure out different Pythagorean Triples?
2. How many Pythagorean triples are there?
Please explain breiefly.
Thanking you,
mahima.

ANSWER: Hello Mahima
1.  Yes there is a formula for all Pythagorean Triples.
A primitive triple is one where there is no common factor of all three.  Every such triple can be written as:
x = a^2 + b^2,  y = a^2 - b^2, z = 2ab, where (a,b) = 1 , a>b and exactly one of a,b is even.
Non primitive triples are formed by multiplying each member of a primitive triple by any number > 1.
2. There are of course an infinite number of such triples.

Best wishes
vijilant


---------- FOLLOW-UP ----------

QUESTION: Sir,
Thank you very much. I want a proof for bothe the questions. please give me sir.
I will wait for your reply.
thank you.
,mahima,

Answer
Hello Mahima
This is a standard question, and there is a proof in almost all number theory text books.  There are also many different proofs.
Here is one that you should find easy enough to understand.
http://planetmath.org/encyclopedia/ProofOfPythagoreanTriples.html

Since there are arbitrarily many pairs of values of m,n, there are an infinite number of such triples.

Best wishes
vijilant