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2

The problem of asymptotic behavior of the maximal cardinality of a cap sets in $\mathbb{Z}/3\mathbb{Z}^r$ as $r$ to infinity gives rise to the following yes/no question that is open. A cap set, here, is a set with no three points on an affine line. This is equivalent to the existence of $x,y,z$ such that $x+y+z = 0$, or the existence of a 3-term arithmetic ...


0

Feasibility of reformulating all of math in only well-defined ultrafinitistic terms From the point of view of physics there is something strange about the way math is used. By the Church–Turing–Deutsch principle all physical processes have (quantum) computable descriptions, but the way we do math invokes non-computable concepts such as the uncountable ...


3

I don't think anyone has a clear idea whether there exist classical solutions to the Navier-Stokes equation. http://www.claymath.org/millenium-problems/navier%E2%80%93stokes-equation Most attempts have focused on trying to prove it is true. However Leray gave a suggestion for looking for a counterexample. It was later shown that his proposed ...


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Existence of rectangular cuboid with all edges, all faces' diagonals, and the main diagonal being integers.


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The Higman Conjecture concerns the number of conjugacy classes of $UT_n(\mathbb{F}_q)$, the group of unipotent upper-triangular matrices with entries in a finite field with $q$ elements. The conjecture is that for a fixed $n$ the number of conjugacy classes of $UT_n(\mathbb{F}_q)$ is given by a polynomial in $q$. This has been proven up to $n=13$, but beyond ...


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There is some "natural" axiom that added to ZFC decides CH? Interesting discussion in Introduction to Set Theory, Third Edition, Revised and Expanded.


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Hilbert's 10th problem over over $\mathbb{Q}$/Mazur's conjecture. These are two open problems that point in opposite directions, and I think experts really aren't sure which way to guess. Hilbert's 10th problem over $\mathbb{Q}$ Is there an algorithm which, given a collection of polynomial equations with rational coefficients, do they have a rational ...


3

Does there exist a finitely presented, infinite torsion group? A torsion group is a group where every element has finite order. Burnside's problem (1902) asked if there exists a finitely generated, infinite torsion group. Such a group of unbounded exponent was constructed by Golod and Shafarevich in 1964, while Novikov and Adian did it for bounded ...


10

From a number theoretic perspective, there are a few famous problems related to ranks of elliptic curves, which a lot of modern research in the area is geared towards solving. For example, Manjul Bhargava recently received the Fields medal partly for his work on bounding average ranks of elliptic curves (and proving that the Birch and Swinnerton Dyer ...


2

The existence of projective finite planes. All the known examples have order prime power. Quote: The existence of finite projective planes of other orders is an open question. The only general restriction known on the order is the Bruck-Ryser-Chowla theorem that if the order $N$ is congruent to 1 or 2 $\text{mod}$ 4, it must be the sum of two squares. ...


5

An easy-to-understand open problem involves the first counterexample to Euler's Sum of Powers conjecture: Q: Does $x_1^5+x_2^5+x_3^5+x_4^5 = x_5^5$ have infinitely many primitive integer solutions? (Primitive being the $x_i$ have no common factor.) Only three are known so far and nobody has given a good heuristic argument that the list is finite, or ...


3

Do all compact smooth manifolds of dimension $\geq 5$ admit Einstein metrics? (An Einstein metric is a Riemannian metric with constant Ricci curvature.) A list of fundamental open problems in differential geometry and geometric analysis can be found at the end of Yau's excellent survey, Review of Geometry and Analysis. It was written in 2000, so is ...


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I think a proper answer to this question are examples of questions where numerical evidence is extremely difficult to obtain. So for example, we don't know anything interesting about the Collatz conjecture but at least we know that it's true for a huge number of cases. As an example of something we don't know at all, consider $S_n$ the symmetric group, and ...


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Taken from http://arxiv.org/abs/0902.3961 through this question http://mathoverflow.net/questions/21003/polynomial-bijection-from-mathbb-q-times-mathbb-q-to-mathbb-q . Is $f(x,y)=x^7+3y^7$ injective on $\Bbb Q \times \Bbb Q \ $ ?


9

The smooth Poincare conjecture in dimension 4 has already been mentioned, so I'll mention the smooth Schönflies Problem in that dimension. The question is whether there is a diffeomorphism of $S^4$ taking any smoothly embedded copy of $S^3$ in $S^4$ to the standard equatorial $S^3\subset S^4$. This is true in all other dimensions, but $4$ is such an unusual ...


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I believe whether or not the Thompson group $F$ is amenable is such question. The paper/article "WHAT is... Thompson's Group" mentions that at a conference devoted to the group there was a poll in which 12 said it was and 12 said it was not. There are in fact papers claiming (at least at the time) to have proofs for both sides. Here are some posts to get an ...


8

The first proof that many people learn is that there are infinitely many primes. (If not the first, then it's often second to the fact that $\sqrt 2$ is irrational). A natural generalization of this was considered by Dirichlet, who showed that as long as the arithmetic progression $a, a+d, a+2d, a+3d, ...$ doesn't have a trivial reason for not having many ...


5

From what I can tell, neither the existence nor non-existence of the Moore Graph of degree 57 and diameter 2 is strongly attested. Most of the work to date on the subject revolves around the various properties such a graph (should it exist) must or must not possess, but none of these seem to give a strong indication to lean one way or the other. Also, the ...


9

In the theory of dynamical systems, problems involve limit cycles in general are always very difficult. The second part of Hilbert's sixteenth problem is my personal "favorite". The upper bound for the number of limit cycles of planar polynomial vector fields of degree $n$ remains unsolved for any $n>1$. For example, can quadratic plane vector fields ...


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A more or less elementary example I'm quite fond of is the Erdős conjecture on arithmetic progressions, which asserts the following: If for some set $S\subseteq \mathbb{N}$ the sum $$\sum_{s\in S}\frac{1}s$$ diverges, then $S$ contains arbitrarily long arithmetic progressions. I've never seen a heuristic argument one way or the other - I believe ...


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In 4 dimensions, it is an open question as to whether there are any exotic smooth structures on the 4-sphere.


0

The following occurred to me just now. This is not a complete answer to my question, but I merely wanted to collect my thoughts into a single location: Since $\sigma(q^k) \equiv k + 1 \pmod 4$, $k \equiv 1 \pmod 4$ and $n$ is odd, then $\sigma(q^k) \neq n$. We consider two cases: $$\sigma(q^k) < n$$ and $$n < \sigma(q^k).$$ The first implies ...


2

I've verified there are only 2 cases for the first 100000 partial sums. Here's the Mathematica code I used: Count[ ParallelTable[Sqrt[HarmonicNumber[r, 2]] ∈ Rationals, {r, 100000}], True ] (* 2 *)


3

I do not see anything intuitive about the first problem, so I put it on computer. I normalized by having the denominator at step $n$ be $(n!)^2.$ A human being would probably use $\operatorname{lcm}(1,2,3,\ldots,n)^2 $ which is much smaller, but the computer does not care. There are some reasons to expect the conjecture to be true. The primes $2,3,7$ are ...


2

The first problem, which essentially asks whether $\sqrt{H_n^{(2)}}$, where $H_n^{(2)}$ is a Generalized Harmonic Number, can ever be rational for $n\ge4$, is very hard. I don't see a way to approach it. The Second Problem Let $k_n\in\mathbb{Z}$ be so that $2^{k_n}\le n\lt2^{k_n+1}$. The only integer not exceeding $n$ divisible by $2^{k_n}$ is $2^{k_n}$. ...



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