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

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98
votes
19answers
4k views

Past open problems with sudden and easy-to-understand solutions

What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved ...
81
votes
5answers
5k 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 ...
79
votes
23answers
2k views

Open mathematical questions for which we really, really have no idea what the answer is

There is no shortage of open problems in mathematics. While a formal proof for any of them remains elusive, with the "yes/no" questions among them mathematicians are typically not working in both ...
52
votes
8answers
6k views

What is the oldest open problem in geometry?

Geometry is one of the oldest branches of mathematics, and many famous problems have been proposed and solved in its long history. What I would like to know is: What is the oldest open problem in ...
45
votes
18answers
6k views

What are some things we can prove they must exist, but have no idea what they are?

What are some things we can prove they must exist, but have no idea what they are? Examples I can think of: Values of the Busy beaver function: It is a well-defined function, but not computable. It ...
30
votes
3answers
2k views

Are there open problems in Linear Algebra?

I'm reading some stuff about algebraic K-theory, which can be regarded as a "generalization" of linear algebra, because we want to use the same tools like in linear algebra in module theory. There ...
25
votes
4answers
4k 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 ...
24
votes
1answer
1k views

What are some of the major open problems in category theory?

What are some of the major open problems in category theory? Just curious - I'm interested in category theory.
23
votes
5answers
559 views

Famous Problems Where We Only Know the Elementary

Define a graph with vertex set $\mathbb{R}^2$ and connect two vertices if they are unit distance apart. The famous Hadwiger-Nelson problem is to determine the chromatic number $\chi$ of this graph. ...
23
votes
1answer
633 views

How to prove there are an infinite number of squarefree numbers of the form $2^p-1$?

How to prove there are an infinite number of squarefree numbers of the form $2^p-1$, where $p$ is prime? It is conjectured that all numbers of the form $2^p-1$ are squarefree. I've been having ...
22
votes
14answers
2k views

How can we produce another geek clock with a different pair of numbers?

So I found this geek clock and I think that it's pretty cool. I'm just wondering if it is possible to achieve the same but with another number. So here is the problem: We want to find a number ...
22
votes
4answers
2k views

Can someone explain the ABC conjecture to me?

I am an undergrad and I know that the conjecture may have been proven recently. But in reading about it, I am entirely confused as to what it means and why it is important. I was hoping some of you ...
22
votes
2answers
668 views

Odd perfect squares whose decimal representation consist only of 1's and o's

Are there any odd perfect squares (apart from the trivial 1), whose decimal representations only uses 1 and 0? Working modulo 8, we can get that the last 3 digits must be 001. However, since $4251^2 ...
21
votes
2answers
15k views

Has anyone found a “pattern” in prime numbers?

Yesterday I was having some fun trying to look for some patterns in primes; and I think I found something interesting (to me at least). I still have not found any lists of patterns already found, ...
20
votes
2answers
933 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} + ...
20
votes
1answer
630 views

Any proof to $\pi^{e}$'s irrationality?

I've searched for this for a while but get nothing... There are plenty of proofs to irrationality of $e$,$\pi$,$e^{\pi}$. However, I can't find a proof for $\pi^e$. More, when searching for this I ...
19
votes
5answers
3k 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 ...
19
votes
1answer
926 views

Why are these geometric problems so hard?

I was surprised to learn that both for the Moving Sofa Problem and Packing 11 Squares solutions have been proposed, but in either case the optimality of the proposed solution is, as of yet, only ...
18
votes
1answer
546 views

How many points can you find on $y=x^2$, for $x \geq 0$, such that each pair of points has rational distance?

Open problem in Geometry/Number Theory. The real question here is: Is there an infinite family of points on $y=x^2$, for $x \geq 0$, such that the distance between each pair is rational? The ...
17
votes
2answers
658 views

Are there any infinite sets that are not known to be either countable or uncountable?

Are there any known examples of sets that are definitely infinite, but where we don't know whether or not they're countable? I haven't heard of anything like this before, but it seems that there ...
16
votes
5answers
2k views

What does proving the Collatz Conjecture entail?

From the get go: i'm not trying to prove the Collatz Conjecture where hundreds of smarter people have failed. I'm just curious. I'm wondering where one would have to start in proving the Collatz ...
16
votes
1answer
597 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 ...
16
votes
2answers
404 views

$\tan (n) > n$ for infinitely many positive integers

I heard the following problem is open: $ \tan(n ) > n $ for infinitely many positive integers in radians. Does anyone know if it is still open or if any progress has been made on this ...
14
votes
2answers
1k views

What does the Hodge conjecture mean?

I read from the Internet that according to the Hodge conjecture, a certain harmonic differential form in a projective, non-singular algebraic variety is a rational linear combination of the cohomology ...
14
votes
2answers
1k views

Why is Hodge more difficult than Tate?

There are strong connections between the Hodge and the Tate conjectures, mainly at the level of similarities and analogies. To quote from an answer of Matthew Emerton on MathOverflow: "[...] we ...
14
votes
2answers
860 views

Status of the classification of non-finitely generated abelian groups.

From the Wikipedia on abelian groups: By contrast, classification of general infinitely-generated abelian groups is far from complete. How far are we from a classification exactly? It seems ...
14
votes
1answer
373 views

The four runner problem/conjecture

I've recently read here the following problem, called « four-runner problem » : Suppose four runners (represented by labeled spheres) run around a circular track. Their speeds are constant ...
13
votes
12answers
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 ...
13
votes
2answers
182 views

Prove that $\lim_{n\rightarrow \infty} \frac{\log_{10}\lfloor\text{Denominator of } H_{10^n}\rfloor+1 }{10^n}=\log_{10} e$

In short, my question is asking to prove that the $$\lim_{n\to\infty}\frac{\text{number of digits in the denominator of} \sum_{k=1}^{10^n} \frac 1k}{10^n}=\log_{10} e$$ I know that the number of ...
13
votes
4answers
617 views

When is $\left\lfloor \frac {7^n}{2^n} \right\rfloor \bmod {2^n} \ne 0\;$?

Is $$\left\lfloor \frac {7^n}{2^n} \right\rfloor \bmod{2^n} \ne 0\;$$ always true when $n \ge 3$. Baker's theorem on transcendental numbers that provide bounds for diophantine equations may be ...
13
votes
1answer
308 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 ...
12
votes
2answers
872 views

Thoughts on the Collatz conjecture; integers added to powers of 2

I've had a thought about the Collatz conjecture (the 3n+1 problem). Suppose some number, C, diverges under the iteration. We first note that C must be odd because if C were even it would be halved ...
11
votes
4answers
594 views

The value of $\sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+\cdots\sqrt{1-\sqrt{1+1}}}}}}$?

How to find value of $\sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+\cdots\sqrt{1-\sqrt{1+1}}}}}}$ ? I've calculated it by MATLAB for some finite terms and I've got : $0.3001 - 0.4201i$, but I don't know how to ...
11
votes
1answer
3k views

Fractional oblongs in unit square via the Paulhus packing technique

Oblongs of size $ \frac{1}{1} \times \frac{1}{2}$, $ \frac{1}{2} \times \frac{1}{3}$, $ \frac{1}{3} \times \frac{1}{4}$, $ \frac{1}{4} \times \frac{1}{5}$, ... have a total area of 1. ...
11
votes
2answers
2k 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 ...
10
votes
3answers
641 views

Solving P vs NP with computer

Is it possible to build a computer program that would (eventually) bring a solution to the P vs. NP question?
10
votes
3answers
224 views

likely open number theory problem: finite sum of $\zeta(2)$ equal to a square of rationals

Which $n$ can let $S=1+\frac14+\frac19+\cdots+\frac1{n^2}$ be a square of a rational number? Obviously, $1$ and $3$ work, but how to prove they are the only ones? I think this problem is really hard. ...
10
votes
1answer
233 views

Independence results that cannot be established by forcing.

I read the Wikipedia article on Absoluteness recently and found mention of Shoenfield’s Absoluteness Theorem, which states that if $ \phi $ is any $ \Sigma^{1}_{2} $- or $ \Pi^{1}_{2} $-sentence of ...
10
votes
2answers
184 views

If $3^x$ and $5^x$ are both integers, is $x$ an integer?

Does the following statement hold? $$x\in \mathbb{R}^+ \text{and} \ 3^x, 5^x \in \mathbb{Z} \implies x \in \mathbb{Z}$$ In words: If $x>0$ is a real number, and $3^x$ and $5^x$ are ...
10
votes
1answer
234 views

Biggest Little Polyhedron

The Biggest Little Polygon problem asks for the polygon with greatest area where the largest diameter is 1. Let's add a dimension and find the largest volumes. What is the biggest little polyhedron ...
10
votes
0answers
107 views

Can anyone improve on this work and find a closed form of $\zeta(3)$?

This was something I and another user came across independently, although he decided to post it on reddit. So while its already online, let me reproduce it here with the hope that someone will be able ...
10
votes
1answer
135 views

Any book on major (recent) math discovery (results) in an easy understanding way?

All: Can anyone recommend a book which illustrate major (recent) math discoveries (results) in an easy understanding way ? For "recent discoveries", I meaning something discovered in last 50 years. ...
10
votes
0answers
231 views

What turmite runs the longest before becoming predictable?

When looking at 2D Turing machines, many of them eventually become predictable. For example, Langton's Ant, the champion 2-color 1-state turmite, develops a highway after 10,000 steps. Predictable ...
9
votes
2answers
368 views

Open problems in Mathematical Tomography?

Since I feel that Tomography can be applied to a wide range of sciences, I was wondering what the current open problems in Tomographic Reconstruction are. Furthermore, I am curious as to how these ...
9
votes
1answer
420 views

Proving the Riemann Hypothesis and Impact on Cryptography

I was talking with a friend last night, and she raised the topic of the Clay Millennium Prize problems. I mentioned that my "favorite" problem is the Riemann Hypothesis; I explained what it posits ...
9
votes
0answers
237 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 ...
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 ...
8
votes
3answers
732 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 ...
8
votes
1answer
282 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 $1$ ...
8
votes
2answers
530 views

Why Goldbach's conjecture is difficult to prove?

Why Goldbach's conjecture is still non-solved and is difficult to prove? What makes the mathematicians fail when trying to prove it?