17 problems

Selmer conjecture. The ranks $r(n)$ and $s(n)$ have the same parity.

The weak Birch–Swinnerton-Dyer conjecture. The integers $r_n$ and $R(n)$ are equal.

Let $D$ be a positive square-free integer and let $E_D:y^2=x^3-D^2x$ be the corresponding congruent number elliptic curve. A positive integer is a congruent number when it is the a…

For nonzero rational $D$, let $E_D$ be the elliptic curve … over ${\bf Q}$. Write $r_{\rm an}(E_D/{\bf Q})$ for the order of vanishing at $s=1$ of $L(E_D/{\bf Q},s)$. In the case w…

Birch and Swinnerton-Dyer parity prediction. Integers $n\equiv 5,6,7\pmod 8$ should be congruent numbers.

Let $n>5$ be a square-free integer. For a real angle $\theta$ with rational cosine, a positive integer is $\theta$-congruent if it is the area parameter of a rational $\theta$-tria…

A positive integer $n$ is a congruent number if it is the area of a right triangle with rational side lengths. The Congruent Number Conjecture. For any $n\equiv 5,6,7\pmod{8}$, $n$…

Congruent number conjecture. All integers $n\equiv 5,6,7\pmod{8}$ are congruent numbers, while $100\%$ of integers $n\equiv 1,2,3\pmod{8}$ are not.

Congruent number conjecture. Every positive integer $n$ satisfying

A positive integer is a reflecting congruent number if it is a $(2,2)$-reflecting number, meaning that there is a positive rational number $t$ such that $n-t^2$ and $n+t^2$ are bot…

Tunnell's conjecture. The integer $n$ is a congruent number if and only if

Yoshida's conjecture.

Let $n$ be a positive integer power of $2$, so that $n\in 2^{\mathbb{N}}$. A positive integer is congruent if it is the area of a right triangle with rational side lengths. Noncong…

Bounded-rank conjecture. The ranks in the family of congruent number curves $E_t$ are bounded.

Unique hypotenuse multiplicity conjecture. There do not exist two dissimilar primitive right triangles with the same hypotenuse and whose squarefree parts of the areas are equal to…

Alter–Curtz–Kubota conjecture. If $[?]$ is an integer congruent to $5$, $6$, or $7$ modulo $8$, then $[?]$ is a congruent number. This concerns the classical problem of determining…

Let $n$ be a positive integer. An integer $n$ is π/3-congruent if it occurs as the area of a rational triangle having angle $π/3$. Yoshida's conjecture. The following congruence co…