I feel it may be a good idea to introduce some related open problems in a calculus course. Surely I am not expecting my students to solve any one of them, though I cannot say it is absolutely impossible; but I think it is good to let the students see some unsolved problems which may motivate them to love mathematics or even to start some undergraduate research. I am only interested in open problems that:
- the statement of which can be understood by an average calculus student;
- It is related to some material in calculus.
As an example, after teaching the scalar product of vectors, I may introduce the following (open as of 22/06/2020) problem:
Does there exist $668$ vectors $v_1,\ldots,v_{668}$ in $\mathbb{R}^{668}$ such that each coordinate of each vector is $1$ or $-1$ and $v_i\cdot v_j=0$ for every distinct $i,j$? The number $668$ can be replaced by some other numbers including $716, 892, 1004, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964$.
This is related to Hadamard Matrix and I have rephrased the problem so that it is understandable to an average calculus III student.
Any other examples of open problems that can be suitably introduced in a calculus course?