43

Yes, with the appropriate definition of "circle". Namely, define a circle of radius $R$ centered at $x$ on manifold $M$ to be the set of points which can be reached by a geodesic of length $R$ starting at $x$. This seems pretty reasonable, and reproduces the usual definition in Euclidean space. It's not hard to see that concentric circles on a torus or ...


42

The situation is impossible if we make the following assumptions: Each circle has exactly one center, which is a point. Concentric circles have the same center. Each circle has exactly one radius, which is a number. (We make no assumptions, besides those listed, about the meaning of the word "number.") If two circles intersect each other at a point, then ...


31

Consider the geometry in which the plane is identified with $(\mathbb Z / 4 \mathbb Z)^2$. Define the circle with center $C(a, b)$ and radius $r$ as the locus of all points $P(x, y)$ such that $(x - a)^2 + (y - b)^2 = r^2$. Let $C = (0, 0)$ and consider the two circles centered at $C$ with radii $0$ and $2$. Since $2^2 = 0$ in $\mathbb Z / 4 \mathbb Z$, the ...


24

Historically, a very important, computationally intensive problem arising from physics was lattice QCD (LQCD). LQCD is a theoretical framework for computing basic quantities like the mass of the proton, and it was introduced by Ken Wilson back in the 70's. However, after some initial successes, this approach stagnated due to a lack of computer power. The ...


24

If you are including games as part of “math”, chess provides some nice unsolved problems due to computational limits. The game of chess itself cannot even be weakly solved (https://en.m.wikipedia.org/wiki/Solved_game#Overview). But strong solutions are known for a subset of chess positions, those with seven or fewer (total) pieces on the board. These are ...


21

Packing problems come to mind, i.e. how to achieve the densest packing of some kind of geometric objects, such as spheres or dodecahedrons. The interesting thing is that this is not a discrete problem, as there are uncountably many irregular, non-periodic packings that need to be checked. Still, the original proof of the sphere packing problem managed to ...


15

Is $e^{e^{e^{79}}}$ an integer? See this question for some background. Many other problems of this type are also technically unsolved, although the answer is almost definitely "no". This can be verified by a finite computation, but the sheer size of the numbers involved means that this is not feasible at the moment. Note: as pointed out by @ruakh, if $e^{e^{...


11

It is strongly believed that the second Hardy-Littlewood conjecture is false, because it contradicts the first Hardy-Littlewood conjecture, which has the backing of not only the probabilistic heuristic but also a lot of recent work. The second link even states that if the first conjecture (also called the prime $k$-tuples conjecture) holds, then there are in ...


10

Not with the usual definition of a circle as the set of points at a fixed distance $r$ from a center $C$. If circles are concentric that means they have the same center. If they intersect at one point then they have the same radius. That means they are the same circle. That argument works in any geometry where distance is defined. If the circles need not ...


10

"Rearranging" objects is a procedure that is not inherently related with order. The relation with order is more due to how we usually list things than the essence of permutation itself. For instance, I have three pockets in my pants, on which I put my cell phone, keys and wallet, one on each pocket. I sometimes rearrange them; driving with the wallet on the ...


8

Optimal sorting networks for $n>10$. For small, fixed numbers of inputs n, optimal sorting networks can be constructed, with either minimal depth (for maximally parallel execution) or minimal size (number of comparators)... The following table summarizes the known optimality results: $$ \begin{array}{l|ccccccccccccccccc|} \hline n & 1&...


8

"A bijective map from a set to itself" does not require the set to be ordered, but when applied to an ordered set, this map acts to reorder the set. This definition is therefore a generalization of the idea of "reordering an ordered set" to a more general setting. Often, in mathematics, a name lifts with a generalization.


7

Euler's conjecture that it takes $n$ $n$th powers to sum to an $n$ power is true for $n=3$ but proven false for $n=4,5$, for example, $$27^5+ 84^5+110^5+ 133^5= 144^5\qquad\text{(found in 1966)}$$ $$95800^4 + 217519^4 + 414560^4 = 422481^4\qquad\text{(found in 1988)}$$ but nobody knows if it is false for any or all $n\geq6$. There are heuristics that ...


6

The order of every finite projective plane is a prime power. If this is false, a counterexample can be constructed by exhaustive search of all non-prime powers in increasing order. This has been done by hand for $n=6$ and by computer for $n=10$, but as far as I know, $n=12$ is still out of reach, or at least, it hasn't been done.


5

I quite like this very old question, and I think there is room for another answer. I'll propose the theorem underlying Dehn's Algorithm as having one of the greatest impacts on infinite group theory: Theorem: For $g \geq 2$, let $\Gamma_g$ be the fundamental group of a closed genus $g$ surface, with generators $S_g=\{a_1,b_1,\ldots,a_g,b_g\}$ and with ...


5

The rationale is that $\frac{7}{2}$ is the multiplicative inverse of $\frac{2}{7}$ over the field $\Bbb Q$ i.e. $$\frac{2}{7} \times \frac{7}{2} = \frac{7}{2} \times \frac{2}{7} = 1$$ This explains the "flip" aspect of the "flip and multiply" maxim. And in any field (like $\Bbb Q$), division is just a short-hand notation for multiplication by the ...


5

If I were grading this proof, the questions I would have are: It looks like you are proceeding by induction. Have you established the base case? How do you know $d=\text{gcd}(d', a_n)$? Note, I am not concerned about the existence of $y_1,\dots,y_{n-1}$, since that follows from the induction hypothesis. How do you know such $x$ and $y$ exist? (This is the ...


4

Littlewood proved in 1914 that there exists a number $n\in\mathbb{N}$ (called Skewes' number) such that: $$ \pi(n) > \operatorname{li}(n), $$ where $\pi(n)$ is the amount of primes below $n$ and $\operatorname{li}(n)$ denotes the logarithmic integral $\displaystyle \int_0^n \frac{dt}{\ln t}$. It is conjectured that $n$ is a huge number, recent analysis ...


4

The Klein bottle and the torus. The first isn't orientable, so its $2$nd integral homology group isn't isomorphic to $\Bbb Z$. The torus is orientable however... Meanwhile if you use $\Bbb Z_2$ coefficients, their $2$nd homology groups are the same ($\Bbb Z_2$).


3

The number of distinct magic squares, for deceptively small sizes A magic square of order $n$ is a square grid of $n \times n$ boxes where each box contains one distinct integer from the interval $[1 .. n^2]$, so that the sums of the numbers on each row, on each column and on each of the two diagonals are equal to each other. They have been studied for ...


3

There was the question: Are there m consecutive positive integers from k to k+m-1 which contain more primes than the m integers from 2 to m+1? The problem itself is unsolved, but there is a hypothesis with the twin-prime hypothesis as the simplest special case: Given n ≥ 2, and n integers $0 = k_1 < k_2 < ... < k_n$, and for every prime p ≤ n the ...


3

Hint: Using the binomial expansion, $\lim\limits_{n\to\infty}\left(1+\dfrac xn\right)^n=\lim\limits_{n\to\infty}\left(1+x+\dfrac{n(n-1)}{n^2}\dfrac{x^2}2+\dfrac{n(n-1)(n-2)}{n^3}\dfrac{x^3}6+...\right).$


3

I don't think there's a lot of sense in learning things that will be in your high school classes. You might want to take in some elementary set theory. Once you understand that, you can start plunging into any of the modern formal math subjects -- axiomatic linear algebra, group theory, graph theory, and so on. You don't have to dive too deeply into any ...


2

The simplest thing may be to just say what you mean. That is, if your theorem is that some statement is false, then just make that the theorem you're claiming. The only reason to display the false statement as you want is if there is going to be a protracted development before it is refuted or perhaps before you even begin to refute it. If you are going to ...


2

Here's another proof (copied from brilliant.org) of the infinite series, but for arbitrary $r<1$. I wonder if it can be adapted for the finite case by thinking about a trapezoid like this instead of a triangle....


2

See this images: This are a graphic explaination of the sum of the geometric progression of ratio $\frac{1}{2}$.


2

Let's define $$F = \sum_{i=1}^k F_i = \sum_{i=1}^k A_i^2 4^{B_i}$$ Since $A_i$ and $B_i$ are positive, maximum of $F$ will occur when each individual value is as big as possible, so we can convert the inequalities to equalities $$\sum_{i=1}^k A_i = A$$ $$\sum_{i=1}^k B_i = B$$ We can use Lagrange multipliers to try to find the extrema of $F$ under these ...


1

Draw a segment one unit long. Tick at the first third from the left. On the right side, tick at the first third from the left. On the right side, tick at the first third from the left. On the right side, tick at the first third from the left. … When you are done, you have the infinite sum for $a=\frac13,r=\frac23$. If you stop before infinity, the ...


1

I'll make a guess: Think about streets, a "circle" is a path in which you can go from point $A$ and return to itself and you can define a "center" in some reasonable way. As the structure of the streets can be very messy (in certain streets, you can go in only one direction, in others, you can go both ways, etc), I guess you can construct different "circles" ...


1

What you have are a collection of $relations$. A relation can be thought of as a set of ordered pairs, the left element of each belonging to one set, the right to some (other or possibly the same) set. Let me give you some examples: Let $P$ be the set of people, and $H$ be the set of houses, and say we have relations $N : (H,H)$ with the semantic meaning ...


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