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.
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 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 .
- Koblitz, Neal (1984). Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics, no. 97, Springer-Verlag. ISBN 0-387-97966-2.
- Tunnell, Jerrold B. (1983). "A classical Diophantine problem and modular forms of weight 3/2". Inventiones Mathematicae. 72 (2): 323–334. doi:10.1007/BF01389327.