Originally Posted by

**padsinseven** I have some Pythagorean Triples proofs that I kind of have a hard time getting started on. I am just looking for a little insight to help me get started. Any help at all would be greatly appreciated. Here goes....

(b) Prove that there are infinitely many pythagorean triples (a,b,c) for which c -b=1. (Suggestion: direct proof by constructing them explicitly. Start by finding some small ones and look for a pattern. )

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Hi,

there are infintely many Pythagorean triples:

a) if (a, b, c) is a Pythegorean triple then $\displaystyle (ta, tb, tc), t\ \in \ \mathbb{N}$ is a Pythagorean triple too;

b) let $\displaystyle \boxed{\begin{array}{l}a=u^2-v^2 \\b=2uv\\c=u^2+v^2\end{array}}$ and $\displaystyle u>v~\wedge~u-v\text{ is odd}$

then you have a Pythagorean triple.

Examples Code:

u v | (a, b, c)
-------------------------
2 1 | (3, 4, 5)
3 2 | (5, 12, 13)
4 1 | (15, 8, 17)
.. ... | ........