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Ask HN: Pure Math Problem

2 points by nsomaru ↗ HN
I'm reading A Course of Pure Mathematics, by G. H. (Godfrey Harold) Hardy – available free on Gutenberg. In the first chapter he presents the following problem (on rational numbers):

If λ, m, and n are positive rational numbers, and m > n, then λ(m^2 − n^2), 2λmn, and λ(m^2 + n^2) are positive rational numbers. Hence show how to determine any number of right-angled triangles the lengths of all of whose sides are rational.

I'm not asking the solution, but I'm not sure what he means by "any number of right-angled triangles"

6 comments

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ciceroqaz
should probably ask in Art of Problem Solving.
nsomaru
I did a Google search, but I'm not able to come up with anything but a non-profit that doesn't seem like a QnA site. Please advise?
sesqu
I'm guessing it means you have to come up with a generative process that produces k such shapes for any k.
benjdezi
It probably means that you need to use the given postulates and known theorems about right-angled triangles to come up with a way or formula for generating such triangles.
amair
think Pythagorean triples
Someone
"any number" means that you must be able to produce two triangles if asked for two, three if asked for three, four if asked for four, 7 trillion if asked for 7 trillion, etc. Effectively, you have to be able to generate infinitely many (technically: aleph-0) Pythagorean triangles.

Aside: I do not think that is a the nice problem, as there are way easier ways to generate aleph-0 such triangles (I think most mathematicians would agree the simplest is 3x,4x,5x for x in {1,2,...}) But that is not what this question is hinting at.