

Ask HN: Pure Math Problem - nsomaru

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):<p>If λ, m, and n are positive rational numbers, and m &#62; 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.<p>I'm not asking the solution, but I'm not sure what he means by "any number of right-angled triangles"
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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.

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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.

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sesqu
I'm guessing it means you have to come up with a generative process that
produces k such shapes for any k.

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ciceroqaz
should probably ask in Art of Problem Solving.

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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?

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amair
think Pythagorean triples

