Wikipedia number-theory item 104: Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem…

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A positive rational $q$ is congruent if there exist $a,b,c\in\mathbb Q_{>0}$ such that $a^2+b^2=c^2$ and $ab/2=q$, so that $q$ is the area of a rational right triangle. Multiplying triangle sides by $t\in\mathbb Q_{>0}$ multiplies its area by $t^2$. If $q=u/v$ is in lowest terms, then $v^2q=uv$ is an integer; removing square factors of $uv$ reduces the question to a positive squarefree integer. Equivalently, the elliptic curve $E_q:y^2=x^3-q^2x$ has a rational point with $y\ne0$.

Statement recorded in the supplied source.

Classify all positive rational numbers $q$ which are congruent, giving a necessary and sufficient characterization of the existence of the displayed rational right triangle (equivalently, of a rational point on $E_q$ with nonzero $y$). [S2]

Short English statement

Determine precisely which positive rational areas occur for right triangles with all three sides rational.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400104 (AMR-093-0104). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Chandan Singh Dalawat, Congruent numbers, elliptic curves, and the passage from the local to the global: An update, arXiv:2201.11071v2 (2022) Section on the sign of the functional equation, p. 2, recalling Proposition 6. https://arxiv.org/pdf/2201.11071.
• S3 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Wikipedia number-theory item 104: Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem…. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.