# 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

${\displaystyle {\begin{matrix}A_{n}&=&\#\{(x,y,z)\in \mathbb {Z} ^{3}|n=2x^{2}+y^{2}+32z^{2}\}\\B_{n}&=&\#\{(x,y,z)\in \mathbb {Z} ^{3}|n=2x^{2}+y^{2}+8z^{2}\}\quad \\C_{n}&=&\#\{(x,y,z)\in \mathbb {Z} ^{3}|n=8x^{2}+2y^{2}+64z^{2}\}\\D_{n}&=&\#\{(x,y,z)\in \mathbb {Z} ^{3}|n=8x^{2}+2y^{2}+16z^{2}\}.\end{matrix}}}$

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 ${\displaystyle y^{2}=x^{3}-n^{2}x}$, 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 1983.

## 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 An,Bn,Cn,Dn can be calculated by exhaustively searching through x,y,z in the range ${\displaystyle -{\sqrt {n}},\ldots ,{\sqrt {n}}}$.