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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 integers]]s 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 equation]]. The theorem is named for [[Jerrold B. Tunnell]], a number theorist at [[Rutgers University]], who proved it in 1983.

==Theorem==
For a given square-free integer ''n'', define

:<math>\begin{matrix}
A_n & = & \#\{ (x,y,z) \in \mathbb{Z}^3 | n = 2x^2 + y^2 + 32z^2 \} \\
B_n & = & \#\{ (x,y,z) \in \mathbb{Z}^3 | n = 2x^2 + y^2 + 8z^2 \} \quad \\
C_n & = & \#\{ (x,y,z) \in \mathbb{Z}^3 | n = 8x^2 + 2y^2 + 64z^2 \} \\
D_n & = & \#\{ (x,y,z) \in \mathbb{Z}^3 | n = 8x^2 + 2y^2 + 16z^2 \}.
\end{matrix}</math>

Tunnell's theorem states that supposing ''n'' is a congruent number, if ''n'' is odd then 2''A''<sub>''n''</sub> = ''B''<sub>n</sub> and if ''n'' is even then 2''C''<sub>''n''</sub> = ''D''<sub>''n''</sub>.  Conversely, if the [[Birch and Swinnerton-Dyer conjecture]] holds true for [[elliptic curve]]s of the form <math>y^2 = x^3 - n^2x</math>, 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 ''A''<sub>''n''</sub>,''B''<sub>''n''</sub>,''C''<sub>''n''</sub>,''D''<sub>''n''</sub> can be calculated by exhaustively searching through ''x'',''y'',''z'' in the range <math>-\sqrt{n},\ldots,\sqrt{n}</math>.

==References==
* {{cite book
  | author = Koblitz, Neal
  | authorlink = Neal Koblitz
  | title = Introduction to Elliptic Curves and Modular Forms
  | publisher = Graduate Texts in Mathematics, no. 97, Springer-Verlag
  | year = 1984
  | isbn = 0-387-97966-2}}
* {{cite journal
  | author = Tunnell, Jerrold B.
  | authorlink = Jerrold B. Tunnell
  | title = A classical Diophantine problem and modular forms of weight 3/2
  | journal = [[Inventiones Mathematicae]]
  | volume = 72
  | issue = 2
  | pages = 323–334
  | year = 1983
  | doi = 10.1007/BF01389327}}

[[Category:Theorems in number theory]]

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