Questions on problems that have yet to be completely solved by current mathematical methods.

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7
votes
1answer
478 views

Any serious work on Lychrel numbers/$196$-Algorithm?

I've been googling around a little lately and have stumbled across the so called Lychrel problem. For a natural number $x$, let $Rx$ denote the number obtained by reversing the base-$10$ digits of $x$ ...
5
votes
2answers
806 views

Undergraduate Research Problems

I'm a third year undergrad with this summer off so would appreciate some material to look at. I took courses in Galois theory, Topology, Complex analysis, ... My main interest is in Analysis / ...
1
vote
0answers
134 views

Panel structure on epi F**

It is known that, generically, the convex hull of a hypersurface embedded in $\mathbb{R}^n$ has a panel structure of simplices. Of course one can construct embeddings where this is not the case, but ...
7
votes
1answer
240 views

What is the importance of 3n in the Collatz Conjecture?

I'm not mathematician, so forgive me if I make wrong assumptions. I was wondering what the importance of the $3n$ is in the Collatz Conjuncture. If you just do $n + 1$, it seems you'll end up at $1$ ...
19
votes
5answers
2k views

The main attacks on the Riemann Hypothesis?

Attempts to prove the Riemann Hypothesis So I'm compiling a list of all the attacks and current approaches to Riemann Hypothesis. Can anyone provide me sources (or give their thoughts on possible ...
1
vote
0answers
212 views

Is there a ccc but not separable space $X$ with a zeroset-diagonal, that isn't submetrizable?

Is there a ccc but not separable space $X$ with a zeroset-diagonal, that isn't submetrizable? separable = $X$ has a countable dense subset. A space $X$ has a zeroset-diagonal when there is a ...
3
votes
1answer
143 views

Can 7 Hoffman-Singleton graphs cover $K_{50}$?

Three copies of the Clebsch Graph can cover $K_{16}$, the complete graph on 16 vertices. This is part of the demonstration that $\mathrm{Ramsey}(3,3,3) > 16$. The Hoffman–Singleton graph is a ...
2
votes
0answers
516 views

Social Golfer Problem - Quintets

I wrote an article on the Social Golfer Problem, which has questions like: Each day, 16 people play Munchkin in foursomes simultaneously. How many days can they play with no two people playing with ...
4
votes
0answers
151 views

Is the maximal temperature of the curlicue fractal acheived by $e\times\gamma$?

The Curlicue Fractal is defined as follows: Choose an irrational number $s$ and a horizontal unit segment with angle $\phi_0 = 0$. Define $\theta_{n+1} = \theta_{n} + 2 \pi s \pmod{2 \pi}$, with ...
20
votes
2answers
838 views

Complex solutions for Fermat-Catalan conjecture

The Fermat-Catalan conjecture is that $a^m + b^n = c^k$ has only a finite number of solutions when $a, b, c$ are positive coprime integers, and $m,n,k$ are positive integers satisfying $\frac{1}{m} + ...
5
votes
0answers
219 views

Can the 57-cell be made in vZome without strut crossings?

Here's the 57-cell in vZome with lots of strut crossings: Is it possible to construct the 57-cell in vZome without any strut crossings? That is, 57 nodes, 171 struts, in the 57-cell / Perkel graph ...
-5
votes
2answers
2k views

Is the Birch and Swinnerton-Dyer conjecture solved?

I read today that in 2010 Manjul Bhargava with Arul Shankar proved the conjecture basing upon the work of Kolyvagin. Is it right? Does it satisfy for all elliptic curves, or is it limited to some ...
1
vote
1answer
141 views

What is this variation of the traveling salesman problem called?

I've encountered a description of a well-known, hard to compute problem. Let us consider $n$ sites that need to be connected in the shortest possible way. If I am only allowed to connect the sites I ...
8
votes
0answers
204 views

Weak version of Fortune's conjecture

Let $p\#=2\cdot3\cdot5\cdots p$ denote the primorial and $N(x)$ the smallest prime greater than or equal to $x$. Then Fortune's conjecture is that $N(p\#+2)-p\#$ is prime for all $p$. (Heuristic: to ...
76
votes
4answers
4k views

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif ...
20
votes
4answers
3k views

Primes of the form $n^2+1$ - hard?

I met a student that is trying to prove for fun that there are infinitely many primes of the form $n^2+1$. I tried to tell him it's a hard problem, but I lack references. Is there a paper/book ...
4
votes
1answer
179 views

Source of the problem: does there exist k,n>2, such that $\sum_{j=1}^k j^n = (k+1)^n $?

In my other question two days ago I asked for confirmation, whether one step of an attempt to that problem does there exist $k,n>2$ , such that $\sum_{j=1}^k j^n = (k+1)^n $ ? was ...
11
votes
3answers
1k views

Why has the Perfect cuboid problem not been solved yet?

Why hasn't Perfect Cuboid Problem been solved yet, whereas (possibly) more nontrivial ones such as FLT and Sphere packing have been solved? I understand that calling some problems more nontrivial ...
16
votes
1answer
487 views

Status of a conjecture about powers of 2

I recently saw a conjecture on a blog ( http://blog.tanyakhovanova.com/?p=311 ) which the author refers to as the 86 conjecture. The conjecture claims that all powers of 2 greater than $2^{86}$ have a ...
2
votes
3answers
243 views

Large-scale Coordinated Research in Mathematics

I've long dreamt of an occasion where hundreds or thousands of mathematicians work on a single problem in a truly coordinated way. Well, now that I'm nearing my doctoral defense in mathematics, I've ...
6
votes
1answer
428 views

Best place to find open questions / latest research

Is there a central wiki or something where open questions (and relevant research on them) takes place?
8
votes
5answers
1k views

How valid is the proposed P=NP solution for the 3-SAT problem?

Source link: http://romvf.wordpress.com/2011/01/19/open-letter/ (Referred from slashdot) The fact of existence of the polynomial algorithm for 3-SAT problem leads to a conclusion that P=NP. Is ...
4
votes
3answers
3k views

Proof of Collatz conjecture?

I was "playing" on Project Euler and passed across the Collatz Problem, and it mentioned it was still unproven. I immediately noticed two things: After the $3n+1$ the result is always even, leading ...
2
votes
0answers
247 views

Invariant Subspace Problem

Louis de Branges had a paper on his homepage claiming a solution for the Invariant Subspace Problem. But I don't see that paper anymore, though he still has a "proof" of Riemann Hypothesis on his ...
8
votes
3answers
633 views

Branches of mathematics not having a general method to solve

I studied applied math, so each course (except abstract algebra) was dedicated to solution of a similar problems. After those courses it seems that every branch of mathematics has a developed theory ...
12
votes
10answers
1k views

Favourite open problem?

Do you have any favorite open problem? Let me mention one of my favorites. Let $A(\mathbb{T})$ be the Wiener algebra, that is, the linear space of absolutely convergent Fourier series on the unit ...
12
votes
1answer
296 views

Proving that this Game on Polygons Ends

About two years ago, I started thinking about the following problem: You're given an $N$ and an $S$, positive integers. You start with an $N$-gon that has positive integer labels at each vertex, such ...