8
$\begingroup$
  • 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 !.

$\endgroup$
  • $\begingroup$ Do you want open problems or just challenging elementary problems? If you only want the latter there are many, many great problem books at which you might want to look. $\endgroup$ – Mortified Through Math Jun 8 at 14:21
  • $\begingroup$ Just looking for open problems right now. That or problem books would be great as wel. $\endgroup$ – Raymond Iacobacci Jun 8 at 21:42
  • $\begingroup$ Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, use MathJax. $\endgroup$ – dantopa Jun 9 at 1:48
2
$\begingroup$

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!

$\endgroup$
2
$\begingroup$

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 $2^n$.

Speyer conjecture

If a group $G$ has more that $(1-\frac{p-1}{p^2})|G|$ elements of order $p$ for some prime $p$, then $G$ has exponent $p$.

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 $n \neq 3$

Dedekind Problem

Is there some sort of closed formula for the number of monotonous boolean functions on $n$ variables?

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

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.

$\endgroup$
  • $\begingroup$ These are all cool. $\endgroup$ – BalancedTryteOperators Jun 11 at 19:47
  • $\begingroup$ These are all cool. $\endgroup$ – BalancedTryteOperators Jun 11 at 19:47
  • $\begingroup$ Yes, but not truly at undergraduate level... $\endgroup$ – Jean Marie 2 days ago
0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Not-so-famous is not rigorously definable. And in Canada we have many artists who are "world-famous in Canada". $\endgroup$ – DanielWainfleet Aug 2 at 18:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.