CoCalc Public Fileswww / simuw06 / notes / keith_conrad.txt
Author: William A. Stein
1Dear William,
2
3Hi.  On page 5 of your notes
4
5http://sage.math.washington.edu/simuw06/notes/notes.pdf
6
7the correspondences between (a,b,c) and (x,y) are not as good as they
8could be. If instead you use
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10f(a,b,c) = (n(a+c)/b,  2n^2(a+c)/b^2)
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12and
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14g(x,y) = ((x^2 - n^2)/y, 2xn/y, (x^2+n^2)/y),
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16which are inverses of each other, then there is sign-preservation: a,
17b, and c are all positive if and only if x and y are both positive
18(from y^2 = x^3 - n^2x = x(x^2 - n^2), if x and y are positive then so
19is x^2 - n^2, so the first coordinate in g(x,y) is indeed positive).
20Since this is a problem about lengths of sides of a triangle, it is
21nice to have a correspondence which produces positive data from
22positive data in both directions.  In your correspondence, the (3,4,5)
23right triangle is attached to the point (-3,9) on y^2 = x^3 - 36x.  In
24the above correspondence, the (3,4,5) right triangle is attached to
26
27The connection between the correspondences above and the ones in your
28notes is this: the two sets each have an involution: (a,b,c) -->
29(-a,-b,c) and (x,y) --> (-n^2/x, n^2y/x^2).  If you apply f and g
30above and then compose with these involutions, you get the
31correspondences in your notes.
32
33 Best,
34 Keith
35
36--
37
38Oh, I was thinking you might change the notes!  But perhaps you just
39want to leave them as they really were.  Rather than linking to my
40email, consider instead a link to the wikipedia entry
41http://en.wikipedia.org/wiki/Congruent_number, which I edited to
42include this correspondence.  It comes out of the intro to Tunnell's
43paper (check it yourself).  The only thing I really noticed is that
44you can take out the absolute value signs in the correspondence as in
45Tunnell's paper by just subtracting in the right order.
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