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It is proven by Swenson in E. Swenson, A cut point theorem for CAT(0) groups, J. Differential Geom. 53 (1999), no. 2, 327–358. that each infinite CAT(0) group contains an infinite order element.


I'm suprised that the following two haven't shown up: What is the smallest Riesel number? What is the smallest Sierpiński number? In both cases we know they exist because they are smaller than or equal to 509,203 and 78,557 respectively.


If $n$ is a perfect square, any $n$ and $k$ satisfying that condition may be a solution of the pell; $n^2-k y^2=-1$, forsome integer $y=n/k$


The constant in the Berry-Esseen theorem: If we have a bunch of i.i.d. random variables $(X_j)_{j\geq 1}$ with a finite third moment, that is $E[|X_j|^3]<\infty$ (and thus they also have some mean $\mu$ and variance $\sigma^2$), then we can prove without too much trouble that their scaled average, $A_n := \frac{(\sum_{j=1}^n X_j)-n\mu}{\sigma \sqrt{n}}$ ...


How would a basis for $\mathbb R$ as a field over $\mathbb Q$ look like? By the axiom of choice we know one must exist, but I have no idea how it would look. It would be quite odd, as it must be large enough so that every real number can be expressed as a finite sum of its elements, but it must be small enough to be linearly independent. For a simpler ...


There are many examples of objects whose existence can be proven using Probability Theory. One example is the existence of matrices with the restricted isometry property: https://en.wikipedia.org/wiki/Restricted_isometry_property. If one wants matrices of certain dimensions, the only known way to construct them is to create matrices at random using a ...


I'm currently not studying this subject in math, but I found some websites that may help. http://user.math.uzh.ch/ayoub/PDF-Files/periods-GKZ.pdf http://arxiv.org/pdf/1307.1045.pdf http://mathoverflow.net/questions/99791/on-grothendiecks-period-relations I hope I helped :)


It can be proved, for instance,that $x^2+x+1$ is irreducible in $\mathbb F_p[x]$ for $p=29$ which gives two "irrational" elements (the two roots of the polynomial) in a quadratic extension of $\mathbb F_{29}$. What kind of object is each of these two roots? Absolutely non idea. And for all finite field there are in general infinitely many of these ...


If we raise $\sqrt{2}$ to the $\sqrt{2}$ power, and raise the result to the $\sqrt{2}$ power, then we have raised an irrational number to an irrational power and gotten a rational, but we don't know in which step we did it...


Many problems that are just computationally hard. Only about 40 or so Mersenne primes are known (not a very good example, because there is no proof they are infinite, so we might know all of them. ). Take the sequence of primes p where the gap between p and the next larger prime is larger than any earlier gap between consecutive prime numbers. These two ...


What is the least infinitely repeating prime gap? See the recent work by Zhang, et. al.


ZFC proves that there exists a well-ordering of the real numbers. (Many such, in fact.) Nobody has a clue what one is like.


A function $\mathcal{G}(n)$ that, for each positive integer $n$, gives the length of the Goodstein sequence for $n$. We know such a function is well-defined and finite for all $n \in \mathbb Z^+$, but the function values get so large for small $n$ that it is difficult to compute. Moser's worm problem for a convex set. Blaschke's selection theorem ...


The functions given by the Riemann mapping theorem. A simply connected region can be mapped bijectively and holomorphically onto the open unit disk. Take some slightly weird shape, say, a square with 4 half-circles attached to the sides, and it is likely there is no nice description on how this map looks like. Similarly, there are a lot of such examples ...


A partition of the 3D ball into 5 distinct pieces such that, through only translations and rotations, the pieces can be moved and reassembled into two balls, each of equal size to the original. This is the Banach-Tarski paradox. The existence of such a partition depends on the axiom of choice, so in particuar there is no way to say exactly what the ...


The leading (decimal) digit of the ludicrously huge number $TREE(3)$. (See https://cp4space.wordpress.com/2012/12/19/fast-growing-2/ and http://www.cs.nyu.edu/pipermail/fom/2006-March/010279.html.) OK, one might object "But that's an absolutely uninteresting object!" That may be; I'd argue, though, that it is meta-interesting in the following sense. The ...


(1) From the set-theory axiom system called ZFC we can prove that the set of reals has a cardinal number $\mathfrak{c}$. But, assuming that ZFC is consistent, we don't know what $\mathfrak{c}$ is: it cannot be proven or disproven from ZFC that $\mathfrak{c}$ is the least uncountable cardinal, or the second or even the $\mathfrak{c}$-th. (2) Perhaps someone ...


Collisions in cryptographic hashes must exist due to the pigeonhole principle. Although some collisions have been found for some hash functions, we "have no idea what they are" in the sense that they aren't readily calculated.


Not sure if this satisfies the requirement that we "have no idea what they are", but the extremely strange Mill's constant seems worth mentioning here: There is supposed to be some real number $r>0$ with the property that the integer part of $$r^{3^n}$$ is prime for every natural $n$. It is not known if $r$ is rational and as far as I know not even a ...


There are a number of games, like Hex and Chomp, for which it is easy to prove a first player win by strategy stealing but we do not generally know the winning strategy.


Take objects which existence proof uses the axiom of choice, e.g: Each vector space has a basis (the standard existence proof uses Zorn's lemma). How does a concrete basis of $C[a,b]$ look like? What about $\mathbb R$ as a $\mathbb Q$-vector space? Ultrafilter, which are used in the construction of the hyperreals: Does the sequence $(0,1,0,1,0,1,\ldots)$ ...

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