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