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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:

  1. the statement of which can be understood by an average calculus student;
  2. 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?

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    $\begingroup$ moving sofa problem en.wikipedia.org/wiki/Moving_sofa_problem $\endgroup$ – Reinstate Monica Jun 23 at 17:22
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    $\begingroup$ I'm confused by the example. The question is about calculus, but the example is about linear algebra. $\endgroup$ – Ben Crowell Jun 23 at 19:39
  • $\begingroup$ @BenCrowell you are right. On the other hand, scalar product may be introduced in Calculus III, or Linear algebra, or both. I am using a textbook in which there is one session on scalar product. $\endgroup$ – Zuriel Jun 24 at 0:15
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    $\begingroup$ This seems much too broad to be a good fit for the SE format. There is a vast number of open problems that can be stated in such a way as to be understandable to a calculus student. Goldbach's conjecture, Latin squares, Collatz conjecture, the list goes on and on. $\endgroup$ – Ben Crowell Jun 24 at 2:32
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It's still not known whether $$\zeta(5) = \sum_{n=1}^\infty \frac{1}{n^5}$$ is a rational number.

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  • $\begingroup$ Thanks! This is interesting! May I know the reference? $\endgroup$ – Zuriel Jun 23 at 1:46
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    $\begingroup$ This function is the famous Riemann zeta function found in the Riemann hypothesis. It's defined in the same way for any positive real number, not just 5, and can be extended to almost all complex numbers by analytic continuation. It's a famous piece of work by Euler to show that, for an even positive integer, this sum gives a power of pi divided by a Bernoulli number. It's an almost equally famous piece of work by Apery to show that zeta(3) is irrational. $\endgroup$ – Alexander Woo Jun 23 at 3:15
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    $\begingroup$ en.wikipedia.org/wiki/…) $\endgroup$ – A. Rex Jun 23 at 18:39
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    $\begingroup$ It’s not even known whether $e^{e^{e^{e^e}}}$ is an integer. (At least that MO link claims so.) $\endgroup$ – Roman Odaisky Jun 23 at 19:40
  • $\begingroup$ Probably you should phrase the oroblem the way it is expected to turn out: the series is expected to be irrational but this is still an unsolved problem. $\endgroup$ – KCd Jun 25 at 6:29
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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.
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  • $\begingroup$ The question asks about the context of a calculus course. None of these problems seem to relate to the subject matter in a calculus course. $\endgroup$ – Ben Crowell Jun 23 at 19:40
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    $\begingroup$ @BenCrowell: I interpreted "related to some material in calculus" broadly, including analysis. $\endgroup$ – Joseph O'Rourke Jun 24 at 0:26
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You probably get Euler's constant $\gamma$ when you do the integral test comparing $\sum\frac1n$ to $\int\frac{dx}{x}$. Then you can remark that it is unknown whether $\gamma$ is rational.

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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.

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