Here is another easy to state problem which is 140 years old but not very famous. Consider the potential of finitely many positive charges: $$u(x)=\sum_{j=1}^n\frac{a_j}{|x-x_j|},\quad x,x_j\in R^3,\quad a_j>0$$ How many equilibrium points can this potential have? Equilibrium points are solutions of $\nabla u(x)=0$.

First conjecture: it is always finite.

Second conjecture: when finite, it is at most $(n-1)^2$. This estimate is stated by Maxwell in his Treatease on Electricity and Magnetism, vol. I as something known. The editor (J. J. Thomson) wrote a footnote that he "could not find a source of this statement".

Nobody can justify this statement to this time. This is unknown even in the simplest case when all $a_j=1$ and $n=3$.

Here is another easy to state problem which is 140 years old but not very famous. Consider the potential of finitely many positive charges: $$u(x)=\sum_{j=1}^n\frac{a_j}{|x-x_j|},\quad x,x_j\in R^3,\quad a_j>0$$ How many equilibrium points can this potential have? Equilibrium points are solutions of $\nabla u(x)=0$.

First conjecture: it is always finite.

Second conjecture: when finite, it is at most $(n-1)^2$. This estimate is stated by Maxwell in his Treatease on Electricity and Magnetism, vol. I, section 113, as something known. The editor (J. J. Thomson) wrote a footnote that he "could not find any place where this result is proved".

Nobody could find this place to this time. This is even unknown in the simplest case when all $a_j=1$ and $n=3$.