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 circle $\mathbb{T}$ $(=\mathbb{R}/2\pi\mathbb{Z}=(-\pi,\pi])$ carrying the norm 
$$ f\mapsto \|f\|=\sum_{n\in\mathbb{Z}}|\hat{f}(n)|<\infty$$
where $\hat{f}(n)=\int_{-\pi}^\pi f(t)e^{-int}dt/2\pi$ is the $n$:th Fourier coefficient of $f$. In fact $A(\mathbb{T})$ is a unitary commutative Banach algebra. By absolute convergence it follows that each $f\in A(\mathbb{T})$ is continuous on $\mathbb{T}$. Moreover, if $f(t)\not=0$ for all $t\in\mathbb{T}$ then obviously $1/f$ is also continuous on $\mathbb{T}$ - a famous theorem of Norbert Wiener (The Wiener Lemma) states that we also have $1/f\in A(\mathbb{T})$. 
Next consider a possible quantitative refinement of the Wiener lemma: 
Given $\delta>0$  let $$C_\delta = \sup_{A_\delta}\|1/f\|$$
where $A_\delta={f\in A(\mathbb{T}):|f(t)|>\delta,\ \|f\|\leq1}$.
Problem: Find 
$$\delta_\inf=\inf_{\delta>0} \ C_\delta<\infty$$.
Remark 1: It is known that $\delta_\inf\leq 1/\sqrt{2}$ and that $\delta_\inf\geq 1/2$ (see [1,2]).
Remark 2: The problem can be treated in any commutative Banach algebra we unit.
[1] N. Nikolski, In search of the invisible spectrum, Annales de l'institut Fourier, 49 no. 6 (1999), p. 1925-1998
[2] H.S. SHAPIRO, A counterexample in harmonic analysis, in Approximation Theory, Banach Center Publications, Warsaw (submitted 1975), Vol. 4 (1979), 233-236.
 A: I like this one that is simple to state but likely very difficult to prove or disprove: The irrationality of $\zeta (5)$.

The irrationality of $\zeta (3)$  was proved by Roger Apéry only in 1979. Despite considerable effort the picture is rather incomplete about $\zeta (s)$ for the other odd integers, $s=2t+1\gt 5$.

-- Martin Aigner and Günter Ziegler, Proofs from THE BOOK.
A: The (ir)rationality of the Euler-Mascheroni constant has always intrigued me, as it's one of those constants that is "obviously" irrational and yet nobody has much of a clue on how to prove that. 
A: Can't believe no one spoke of the Collatz conjecture yet.
A: It is open whether or not there exists an algorithm which will determine whether an integer sequence satisfying a linear recurrence with integer coefficients (such as the Fibonacci sequence) has only nonzero terms.  See Terence Tao's excellent blog post on the subject.
A: I am quite fond of the abc conjecture: 
Define $q(n)$ to be the product of all distinct primes of n. The conjecture states, that for any $\epsilon > 0$, there exist only finitely many triples of coprime positive integers $(a, b, c)$, satisfying $a + b = c$ for which $c > q(a b c)^{1 + \epsilon}$.
This conjecture has an impressive list of consequences.
A: Not that practical, but finding the smallest $\varepsilon$ in the statement "Matrix multiplication/inversion takes $O(n^{2+\varepsilon})$ flops" is still a nice open problem.
A: I find Brocard's problem very interesting. It asks for integer solutions to the equation
$$m!+1=n^2.$$
Ramanujan considered, but could not solve, the problem. While very few solutions have been found: $(4,5),(5,11)(7,71)$, we do not yet know whether these are the only solutions, there are more solutions, or if infinitely many exist. Curiously, it follows from the $abc$ conjecture that, if the conjecture is true, that there are only finitely many solutions. 
A: *

*Is $\pi \cdot e$ rational?

*What is the minimal number of people in a party, such that there are necessarily either at least 5 mutual strangers or at least 5 mutual acquaintances?

*Is there a positive non-integer $x$ such that both $2^x$ and $3^x$ are integers?

*Does every closed curve in the plain contain 4 vertices of a square?

*Can you factor an integer in polynomial time?

*Can you recognize the unknot in polynomial time?

A: Let ${^n a}$ denote tetration: ${^n a} = \underbrace{a^{a^{.^{.^{.^a}}}}}_{n \text{ times}}$ or, defined recursively, ${^1}a=a$, ${^{n+1}a}=a^{({^n a})}$.
These are open problems:


*

*Is there an integer $n>1$ and a non-integer positive rational $q$ such that ${^n q}$ is an integer?

*Is there an integer $n$ such that ${^n \pi}$ is an integer?

*Is there an integer $n$ such that ${^n e}$ is an integer?

A: Does favorite allow me two - both about convex 3-dimensional polyhedra?
(Shephard's Conjecture)
Is it possible to cut along the edges of some spanning tree of any convex 3-dimensional polyhedron and unfold the cut open polyhedron so that the faces do not overlap (and one gets a simple polygon where the uncut edges indicate the faces of the original polyhedron in the resulting "net" in the plane)?
http://en.wikipedia.org/wiki/Net_%28polyhedron%29
(Barnette's conjecture)
Does every 3-valent convex 3-dimensional polyhedron all of whose faces have an even number of sides admit a hamiltonian circuit?
http://en.wikipedia.org/wiki/Barnette%27s_conjecture
A: I think there is some name attached to this conjecture (Higman perhaps?). Let $U_n(\mathbb{F}_q)$ be the group of $n \times n$ upper triangular matrices with entries taken from the finite field $\mathbb{F}_q$ and all $1$s on the diagonal. Then it is conjectured that the number of conjugacy classes in $U_n(\mathbb{F}_q)$ is a polynomial in $q$ with integer coefficients. From what I understand, this is known for $n \leq 13$.
A: Here are a two of my favourite problems that are currently open (and notoriously difficult):


*

*Kakeya Problem: Define a Kakeya/Besicovitch set as a set containing a unit line segment in every direction. The conjecture states that the Kakeya/Besicovich set has Hausdorff/Minkowski dimension $n$.

*Iwaniec Conjecture: Define the Beurling-Ahlfors transform as the principal-value singular integral
\begin{equation*}
Bf(z)=-\frac{1}{\pi} p.v.\int_{\mathbb{C}}\frac{f(w)}{(z-w)^2}dm(w)
\end{equation*}
on $L^p(\mathbb{C}),~1<p<\infty$.
This conjecture states that the $L^p$ norm of the Beurling-Ahlfors transform is $p^*-1,~p^*=\max\{ p,\frac{p}{p-1}\}$.
