List of not-especially famous problems in undergraduate level mathematics

Question

• I know lists of problems like these have been compiled before, but most tend to collect either extremely difficult problems ( like Collatz conjecture in a question about number theory ) or very specialized and require expensive background experience to even understand the question.
• I was wondering if you had any questions that would be appropriate for a mathematics undergrad: number theory would be especially good, but other undergrad topics are great as well.
• The other condition is that the problems can’t be very famous: it’s ok if they’re on things like Open Problem Garden ( I’ve gotten a couple there myself ) but the goal is to have them come from your individual sources, or even internetwide things that might not be so popular.

Thank you !.

Answer 1

Here's an open problem that I believe satisfies your criterion:

Do there exists $n$ consecutive integers, each having either two distinct prime factors less than $n$ or a repeated prime factor less than $n$ ?

I believe the above question comes from "Unsolved Problems in Number Theory", by Richard K. Guy. More so, this book has a fantastic collection of unsolved problems in Number Theory.

As a note however: One difficult part about research is that there is no telling just how hard an open problem will be. For example, Fermat's Last Theorem is simple to understand, but it's solution is unbelievably sophisticated. On the other hand, Hilbert thought that showing ${\sqrt{2}}^{\sqrt{2}}$ is irrational will be a very difficult problem to prove (but now it's commonly proven in most introductory proof courses).

However I chose the above problem because you can attack it with a computer program to test different sequences of integers (PARI might be useful here). Maybe you'll find such a sequence, or maybe you'll verify no such sequence exists for $n<100$. The point is to get your hands dirty and struggle with a problem that you can understand. If I were you, I would definitely ask a professor to see if they can advise you in any way. Good Luck!

Answer 2

There are quite many such problems and conjectures in combinatorics. Here is a few of them:

Barnette conjecture

Any planar bipartite cubic 3-connected planar graph has a Hamiltonian cycle

Eppstein conjecture

Any cubic graph has $\leq 2^{\frac{n}{3}}$ Hamiltonian cycles

Hadwiger conjecture

If a finite simple graph $\Gamma$ does not contain $K_n$ as a minor, then its chromatic number is less than $n$.

Friendly groups problem

Do there exist two non-isomorphic finite groups $G$ and $H$, such that $Aut(G) \cong H$ and $Aut(H) \cong G$?

$D_4$ conjecture

If $G$ is a finite group and $Aut(G) \cong G$, then either $G$ is centerless or $G \cong D_4$

Jack Schmidt problem

Are there any perfect Leinster groups?

Conway-Dietrich-O'Brien conjecture

If the number of non-isomorphic groups of order $n$ is equal to $n$, then $n = 1$

Breuer-Guralnick-Kantor conjecture

If $G$ is a finite group, then the following two statements are equivalent:

1) $\forall g \in G \setminus \{e\} \exists h \in G \setminus \{g\}$ such that $G = \langle g, h \rangle$

2) All proper quotients of $G$ are cyclic

"Groupy numbers" conjecture

There exists such $N \in \mathbb{N}$, that $\forall n > N$, there exist more groups of order exactly $2^n$, than groups of order strictly less, than $2^n$.

Monstrous conjecture

Monster group is the largest finite group with $194$ conjugacy classes

Bertram conjecture

Suppose $G_k$ is the largest finite group with $k$ conjugacy classes. Then the sequence $\{|G_k|^{\frac{1}{k}}\}_{k=1}^\infty$ is bounded.

Wesolowski conjecture

The diameter of the $S_n$ generated by the transposition $(1,2)$ and both left and right rotations by $(1,2,\ldots,n)$ is equal to $\frac{n(n+1)}{2}$ for $n \neq 3$

Pyber conjecture

$S_n$ has $2^{(\frac{1}{16} + o(1))n^2}$ subgroups.

Cerny conjecture

If a complete deterministic finite automaton with $n$ states has a synchronising word, then it has a synchronising word of length $(n - 1)^2$

Primitive word problem

Is the language of primitive words context-free?

Finally, there is this list of $5$ interesting open problems by John. H. Conway (however, only $4$ of them are still open)

All terminology used in the formulation of this problems is quite easy to understand. All objects mentioned here are usually mentioned on the undergraduate lessons too, and the ones that aren't are easy googlable. However, all the aforementioned problems are still open.

Answer 3

If $C$ is a simple closed curve in $\Bbb R^2$ (i.e. $C$ is homeomorphic to the circle $S^1$ and $C\subset \Bbb R^2$) must there exists $4$ points of $C$ that are the vertices of a square? This may be an open Q even if we replace "square" with "rectangle".

Is the box-product topology on $\Bbb R^{\omega}$ a normal (i.e. $T_4$) space?

A conjecture of Lehmer: If $1<n\in \Bbb Z$ and $n$ is not prime then the totient $\phi(n)$ is not a divisor of $n-1.$

I dk whether Mother Worm's Blanket qualifies as not-so-famous.