TITLE: Favourite open problem?
QUESTION [15 upvotes]: 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.

REPLY [3 votes]: 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.