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 integerss can be the area of a right triangle with all three sides rational. Tunnell's theorem relates this to the number of integral solutions to a few fairly simple Diophantine equations. The theorem is named for Jerrold B. Tunnell, a number theorist at Rutgers University, who proved it in 1983.
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.
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