Are there infinitely many pythagorean triples? I believe these questions are all asking different things, but:


*

*Are there infinitely many (integer) solutions to the pythagorean theorem?


*Is every positive integer part of a solution to the pythagorean theorem?

Also, is there a difference in multiplying the pythagorean triple by a constant factor, let's say $k$, on both sides and multiplying each number $a, b, c$ by a constant $k$?

That should be the easiest one:
Let n be a factor of every Integer in a pythagorean triple:
n* 3sq + n* 4sq = n* 5sq
There are infinitely many positive integers and so also infinitely many n´s. Now we just need to show that the pythagorean triples are all correct with the factor n:
n* 3sq + n* 4sq = n* 5sq         (factoring out n)
n(3sq + 4sq) = n* 5sq            (:n)
3sq + 4sq = 5sq
This shows that the factor can be "ignored" and every pythagorean triple of that sort works and that there are infinitely many of them.
