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.

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.

For a given square-free integer n, define 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

, these equalities are sufficient to conclude that n is a congruent number.

The theorem is named for Jerrold B. Tunnell, a number theorist at Rutgers University, who proved it in Tunnell (1983).

The importance of Tunnell's theorem is that the criterion it gives is testable by a finite calculation.