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.

• 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?

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?

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.

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.

Can't believe no one spoke of the Collatz conjecture yet.