Tunnell's theorem

Short description: 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

[math]\displaystyle{ \begin{align} A_n & = \#\{ (x,y,z) \in \mathbb{Z}^3 \mid n = 2x^2 + y^2 + 32z^2 \}, \\ B_n & = \#\{ (x,y,z) \in \mathbb{Z}^3 \mid n = 2x^2 + y^2 + 8z^2 \}, \\ C_n & = \#\{ (x,y,z) \in \mathbb{Z}^3 \mid n = 8x^2 + 2y^2 + 64z^2 \}, \\ D_n & = \#\{ (x,y,z) \in \mathbb{Z}^3 \mid n = 8x^2 + 2y^2 + 16z^2 \}. \end{align} }[/math]

Tunnell's theorem states that supposing n is a congruent number, if n is odd then 2A_n = B_n and if n is even then 2C_n = D_n. Conversely, if the Birch and Swinnerton-Dyer conjecture holds true for elliptic curves of the form [math]\displaystyle{ y^2 = x^3 - n^2x }[/math], 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 (Tunnell 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 [math]\displaystyle{ n }[/math], the numbers [math]\displaystyle{ A_n,B_n,C_n,D_n }[/math] can be calculated by exhaustively searching through [math]\displaystyle{ x,y,z }[/math] in the range [math]\displaystyle{ -\sqrt{n},\ldots,\sqrt{n} }[/math].

See also

• Birch and Swinnerton-Dyer conjecture
• Congruent number

References

• Koblitz, Neal (2012), Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics (Book 97) (2nd ed.), Springer-Verlag, ISBN 978-1-4612-6942-7 
• 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, http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002099403