CoCalc Public Fileswww / simuw06 / notes / keith_conrad.txtOpen in CoCalc with one click!
Author: William A. Stein
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Dear William,
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Hi. On page 5 of your notes
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http://sage.math.washington.edu/simuw06/notes/notes.pdf
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the correspondences between (a,b,c) and (x,y) are not as good as they
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could be. If instead you use
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f(a,b,c) = (n(a+c)/b, 2n^2(a+c)/b^2)
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and
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g(x,y) = ((x^2 - n^2)/y, 2xn/y, (x^2+n^2)/y),
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which are inverses of each other, then there is sign-preservation: a,
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b, and c are all positive if and only if x and y are both positive
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(from y^2 = x^3 - n^2x = x(x^2 - n^2), if x and y are positive then so
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is x^2 - n^2, so the first coordinate in g(x,y) is indeed positive).
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Since this is a problem about lengths of sides of a triangle, it is
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nice to have a correspondence which produces positive data from
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positive data in both directions. In your correspondence, the (3,4,5)
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right triangle is attached to the point (-3,9) on y^2 = x^3 - 36x. In
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the above correspondence, the (3,4,5) right triangle is attached to
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(12,36) instead.
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The connection between the correspondences above and the ones in your
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notes is this: the two sets each have an involution: (a,b,c) -->
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(-a,-b,c) and (x,y) --> (-n^2/x, n^2y/x^2). If you apply f and g
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above and then compose with these involutions, you get the
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correspondences in your notes.
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Best,
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Keith
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--
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Oh, I was thinking you might change the notes! But perhaps you just
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want to leave them as they really were. Rather than linking to my
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email, consider instead a link to the wikipedia entry
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http://en.wikipedia.org/wiki/Congruent_number, which I edited to
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include this correspondence. It comes out of the intro to Tunnell's
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paper (check it yourself). The only thing I really noticed is that
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you can take out the absolute value signs in the correspondence as in
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Tunnell's paper by just subtracting in the right order.
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