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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.

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I don't actually wish to vote to close, but I think this is not a good question for a SE Q&A site in that it is overly broad. The answers are likely to be completely unrelated to each other, and it's hard to know how much background or sophistication to assume when describing your favorite problem to a general mathematical audience. Finally, this is so subjective that I don't see the point of voting on and ranking the answers (and note that indeed currently none has more than one upvote). –  Pete L. Clark Sep 1 '10 at 8:38
    
@Pete L. Clark: I agree in a certain sense. I'll "close" the thread by accepting the first answer - even though all of them are true (I guess). –  AD. Sep 9 '10 at 13:34
    
Yeah, seems like this belongs on mathematicians.SE or something... –  SamB Oct 21 '10 at 0:32

10 Answers 10

up vote 4 down vote accepted

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.

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  • 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?
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"Is there a positive non-integer x such that both 2x and 3x are integers?" Do you have a link on this? –  Bananas Dec 13 '11 at 1:53
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@Collman: mathoverflow.net/questions/17560/… –  Huy Dec 13 '11 at 9:45

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.

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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.

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As an aside, the value of $\gamma$ is approximately equal to that of $\frac1{\sqrt3}$ . –  Lucian Oct 15 '13 at 4:17

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?
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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.

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

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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.

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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

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I will not stop you! Is it conjectured that the answers are "yes"? –  AD. Sep 1 '10 at 4:41

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 $n \leq 13$.

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