Tunnell's theorem

In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution.

Congruent number problem
The congruent number problem asks which positive integers can be the area of a right triangle with all three sides rational. Tunnell's theorem relates this to the number of integral solutions of a few fairly simple Diophantine equations.

Theorem
For a given square-free integer n, define


 * $$\begin{align}

A_n & = \#\{ (x,y,z) \in \mathbb{Z}^3 \mid n = 2x^2 + y^2 + 32z^2 \}, \\ B_n & = \#\{ (x,y,z) \in \mathbb{Z}^3 \mid n = 2x^2 + y^2 + 8z^2 \}, \\ C_n & = \#\{ (x,y,z) \in \mathbb{Z}^3 \mid n = 8x^2 + 2y^2 + 64z^2 \}, \\ D_n & = \#\{ (x,y,z) \in \mathbb{Z}^3 \mid n = 8x^2 + 2y^2 + 16z^2 \}. \end{align}$$

Tunnell's theorem states that supposing n is a congruent number, if n is odd then 2An = Bn and if n is even then 2Cn = Dn. Conversely, if the Birch and Swinnerton-Dyer conjecture holds true for elliptic curves of the form $$y^2 = x^3 - n^2x$$, these equalities are sufficient to conclude that n is a congruent number.

History
The theorem is named for Jerrold B. Tunnell, a number theorist at Rutgers University, who proved it in.

Importance
The importance of Tunnell's theorem is that the criterion it gives is testable by a finite calculation. For instance, for a given $$n$$, the numbers $$A_n,B_n,C_n,D_n$$ can be calculated by exhaustively searching through $$x,y,z$$ in the range $$-\sqrt{n},\ldots,\sqrt{n}$$.