6 problems

Borwein–Choi conjecture.

Let a ${\mathcal B}$-number be a positive integer represented by the binary quadratic form $a^2+ab+b^2$. Suppose that such a number is written in the general form … where $x$ has n…

Let $C,D$ be distinct, nonzero rational numbers, and let $f(x,y)=Cx^2+Dy^2$ be a binary quadratic form. A rational periodic vector of type $(L,R,L,R)$ is a rational vector undergoi…

Let $k$ be a positive integer such that $4\Vert k$ and $2k-1$ is a prime power. Write … where $a$ is square-free. The form $(a^2b^2,0,2k-1)$ is the binary quadratic form…

Terai's conjecture. The equation has only the solution

Bounded-error conjecture. There is a small constant $M$ such that, for all $k$ and $d$,