Tunnell's Theorem

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

:$\begin A_n & = \#\, \\ B_n & = \#\, \\ C_n & = \#\, \\ D_n & = \#\. \end$

Tunnell's theorem states that supposing ''n'' is a congruent number, if ''n'' is odd then 2''A''_''n'' = ''B''_n and if ''n'' is even then 2''C''_''n'' = ''D''_''n''. Conversely, if the Birch and Swinnerton-Dyer conjecture holds true for elliptic curves of the form $y^2 = x^3 - n^2x$, 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 .

Importance

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 $-\sqrt,\ldots,\sqrt$.

See also

*Birch and Swinnerton-Dyer conjecture
*Congruent number

References

*
* {{citation
, last = Tunnell
, first = 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
, url = http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002099403, hdl = 10338.dmlcz/137483
, hdl-access = free

Theorems in number theory
Diophantine equations