Tunnell's theorem

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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.

Congruent number problem[edit]

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[edit]

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.

History[edit]

The theorem is named for Jerrold B. Tunnell, a number theorist at Rutgers University, who proved it in Tunnell (1983).

Importance[edit]

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 .

See also[edit]

References[edit]

  • 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