What are some of the open problems that can be suitably introduced in a calculus course?

Question

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 $v_icdot v_j=0$ 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?

Gerald Edgar · Answer

You probably get Euler's constant $gamma$ when you do the integral test comparing $sumfrac1n$ to $intfrac{dx}{x}$.  Then you can remark that it is unknown whether $gamma$ is rational.

Alexander Woo · Answer

It's still not known whether $$zeta(5) = sum_{n=1}^infty frac{1}{n^5}$$ is a rational number.

Chris Cunningham · Answer

This is a bit obvious I think, but when you introduce sequences and their notation in either an algebra or calculus class, you should certainly show students the Collatz Conjecture as one of the examples.

Answered by Chris Cunningham on September 6, 2021

Joseph O'Rourke · Answer

It takes a lot of browsing to find problems somehow related to calculus or analysis, but this is a great MathOverflow list: Not especially famous, long-open problems which anyone can understand. Here are a few from
that list:

Are there an infinite number of primes $p$ such that the repeating
part of the decimal expansion of $1/p$ has length $p-1$?
Link.
Does there
exist a point in the unit square whose distance to each of the four
corners is rational? Link.
Is the sequence $(3/2)^n bmod 1$ dense in the unit interval? Link.

What are some of the open problems that can be suitably introduced in a calculus course?

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