Tunnell's theorem

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

${\displaystyle {\begin{aligned}A_{n}&=\#\{(x,y,z)\in \mathbb {Z} ^{3}\mid n=2x^{2}+y^{2}+32z^{2}\},\\B_{n}&=\#\{(x,y,z)\in \mathbb {Z} ^{3}\mid n=2x^{2}+y^{2}+8z^{2}\},\\C_{n}&=\#\{(x,y,z)\in \mathbb {Z} ^{3}\mid n=8x^{2}+2y^{2}+64z^{2}\},\\D_{n}&=\#\{(x,y,z)\in \mathbb {Z} ^{3}\mid n=8x^{2}+2y^{2}+16z^{2}\}.\end{aligned}}}$

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 ${\displaystyle y^{2}=x^{3}-n^{2}x}$, 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. For instance, for a given n, the numbers A_n,B_n,C_n,D_n can be calculated by exhaustively searching through x,y,z in the range ${\displaystyle -{\sqrt {n}},\ldots ,{\sqrt {n}}}$.

• 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, hdl:10338.dmlcz/137483