“Tunnell's theorem”的意思、由来-开放百科全书

词条

Tunnell's theorem

释义

Congruent number problem
Theorem
History
Importance
See also
References

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

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 , 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 {{Harvtxt|Tunnell|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 A_n,B_n,C_n,D_n can be calculated by exhaustively searching through x,y,z in the range .

References

{{citation

| last = Koblitz
| first = Neal
| authorlink = Neal Koblitz
| title = Introduction to Elliptic Curves and Modular Forms
| edition = 2nd
| series = Graduate Texts in Mathematics (Book 97)
| publisher = Springer-Verlag
| year = 2012
| isbn = 978-1-4612-6942-7}}

{{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}}

2 : Theorems in number theory|Diophantine equations

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Congruent number problem

Theorem

History

Importance

See also

References