Harman Patil (Editor)

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.

Contents

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

A n = # { ( x , y , z ) ∈ Z 3 | n = 2 x 2 + y 2 + 32 z 2 } B n = # { ( x , y , z ) ∈ Z 3 | n = 2 x 2 + y 2 + 8 z 2 } C n = # { ( x , y , z ) ∈ Z 3 | n = 8 x 2 + 2 y 2 + 64 z 2 } D n = # { ( x , y , z ) ∈ Z 3 | n = 8 x 2 + 2 y 2 + 16 z 2 } .

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 y 2 = x 3 − n 2 x , 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 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 n, the numbers An,Bn,Cn,Dn can be calculated by exhaustively searching through x,y,z in the range − n , … , n .

References

Tunnell's theorem Wikipedia
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