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The Congruent Number Problem
- A relevant webpage [aimath.org]
Connection with Elliptic Curves (proof by computer)
Explicit Bijection
In fact, there is a bijection between
$$
A = \left\{(a,b,c) \in \mathbf{Q}^3 \,:\, \frac{ab}{2} = n,\, a^2 + b^2 = c^2\right\}
$$
and
$$
B = \left\{(x,y) \in \mathbf{Q}^2 \,:\, y^2 = x^3 - n^2 x, \,\,\text{with } y \neq 0\right\}
$$
given by the maps
$$
f(a,b,c) = \left(-\frac{nb}{a+c},\,\, \frac{2n^2}{a+c}\right)
$$
and
$$
g(x,y) = \left(\frac{n^2-x^2}{y},\,\,-\frac{2xn}{y},\,\, \frac{n^2+x^2}{y}\right).
$$
Define bijection between rational right triangles with area and points on the elliptic curve with .
Use computer to verify that this is a bijection.
By working in the quotient polynomial ring and avoiding fractions we get that the composition is the identity map.
So we know that the claimed bijections are valid.
Tunnell's Criterion:
First congruent number
This year isn't congruent: