The happy ending problem, is concerned with determining, for $ n \geq 3 $, the smallest number of points in the plane, each in general position (i.e., not collinear), such that any arrangement of these points will always include at least one set of $ n $ points forming the vertices of a convex polygon with $ n $ sides. ...

The conjecture known as the Birch–Tate conjecture within the field of algebraic K-theory, and was jointly proposed by Bryan John Birch and John Tate.

Let $K$ represent a totally real number field, $\zeta_K$ denote the Dedekind zeta function, and $K_2(\mathcal{O}_K)$ be the algebraic $K_2$-group. ...

Goldbach's original conjecture, referred to as the "ternary" Goldbach conjecture, was communicated in a letter to Euler on June 7, 1742. [1]. It suggests that:

Every number greater than 2 can be expressed as the sum of three primes. ...

The text below is excerpted from [1], with minor modifications in wording.

Birkhoff's celebrated theorem states that any two-sphere equipped with a Riemannian metric invariably harbors a closed geodesic. An enduring puzzle posed by V.I. Arnold in 1981 [2] is as follows: ...

Consider a vector space $V$. The projective space $\mathbb{P}$ consists of all 1-dimensional subspaces of $V$. The projective dual space $\mathbb{P}^*$ is obtained by projectivizing $V^*$, where $V^*$ denotes the vector space containing all linear functions over $V$. For every subspace $A$ in $V$, we can associate its annihilator $A^*$ in $V^*$. ...

The Brennan conjecture, introduced by James E. Brennan in 1978, [1] is a significant hypothesis within the realm of complex analysis. It revolves around estimating, under certain conditions, the integral powers of the moduli of the derivatives of conformal maps into the open unit disk. Specifically, the conjecture deals with a simply connected open subset $W$ of the complex plane, with at least two boundary points in the extended complex plane, and a conformal map $\psi$ of $ W$ onto the open unit disk. The Brennan conjecture posits that $$\int_W |\psi'|^q\,dxdy < \infty, \qquad \hbox{ whenever } 4/3 < q < 4.$$ While Brennan provided a proof for $4/3 < q < q_0$ for some constant $q_0 > 3$, it remains unresolved for the entire range. In 1999, Bertilsson demonstrated that the conjecture holds true when $4/3 < q < 3.422$, but the complete resolution of the conjecture is yet to be achieved. ...

The following problem appears in : [1] ...