On the congruent number problem: which integers are the area of a rational right triangle
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
formula_1
Tunnell's theorem states that supposing "n" is a congruent number, if "n" is odd then 2"A""n" = "B"n and if "n" is even then 2"C""n" = "D""n". Conversely, if the Birch and Swinnerton-Dyer conjecture holds true for elliptic curves of the form formula_2, 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 formula_3, the numbers formula_4 can be calculated by exhaustively searching through formula_5 in the range formula_6.