Tunnell's Theorem gives a partial resolution to the ancient congruent number problem, which asks which whole numbers can be the area of a right triangle with rational side lengths. It supplies a simple, explicitly checkable numerical criterion for whether a given number is a congruent number: failing the criterion unconditionally proves a number is not congruent, while showing the criterion is also sufficient depends on the truth of the Birch and Swinnerton-Dyer conjecture, which remains unproven in general.
Facts
StatementTunnell's theorem gives a partial resolution to the congruent number problem, and a full resolution conditional on the Birch and Swinnerton-Dyer conjecture. 1 Classification
Statement FormCharacterization Theorem 1 Connections
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Source Tunnell's theorem (Wikipedia)
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1. Tunnell's theorem (Wikipedia)
Lead paragraph
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.
References, Tunnell 1983 entry
Tunnell, Jerrold B. (1983), A classical Diophantine problem and modular forms of weight 3/2, Inventiones Mathematicae, 72 (2): 323-334
In Branch: Number Theory, Lead sentence
In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton
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