john mangual
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 12h comment Kloosterman Sums and Lattice Hyperbolas @GregMartin this is how I feel about the blog I am reading :-( Aug 26 comment Large regions of the plane $(x,y) \in \mathbb{Z}^2$ with no relatively prime points: $\mathrm{gcd}(x,y) > 1$ this is perfect. the number you get using Chinese Remainder Theorem get pretty large, but all I asked is some way to assure me the existence of large holes. Aug 26 comment Large regions of the plane $(x,y) \in \mathbb{Z}^2$ with no relatively prime points: $\mathrm{gcd}(x,y) > 1$ What do you mean pairwise distinct? We have infinitely many prime numbers, so it's good. I am guessing you mean distinct along each row or each column ? Aug 25 comment Groups of order $8n$ have at least five distinct conjugacy classes @JasonDeVito OK. There could be any odd number of Sylow 2-groups, all conjugate to each other, which intersect each other in interesting ways. Aug 25 comment Groups of order $8n$ have at least five distinct conjugacy classes @JasonDeVito Third Sylow theorem says the maximal Sylow $p$-group is normal. I stand corrected. Aug 24 comment Intuition behind the construction of Young Symmetrizer @anon sorry I have been occupied. Aug 23 comment Intuition behind the construction of Young Symmetrizer hep.caltech.edu/~fcp/math/groupTheory/young.pdf Aug 16 comment Area of the lattice generated from $(n, n\sqrt{2} \mod 1)$ @LeeMosher I start out with the line $(t, t \sqrt{2}) \in \mathbb{R}^2$ and mod the y-coordinate by $1$, $(x,y) \mapsto (x, y \mod 1)$ so it's wrapping around a cylinder $\mathbb{R}\times S^1$. Additionally $t \in \mathbb{Z}$ so although I started with a line in the plane, it really looks like a 2D lattice on the cylinder. Aug 16 comment Moebius band not homeomorphic to Cylinder. @ThomasAndrews sure I can; I am taking the closure of a subset of the cylinder/Mobius band in the relative topology. Aug 15 comment Proof of $\sum_{d|n} {\tau}^3(d)=\left(\sum_{d|n}{\tau}(d)\right)^2$ (not standard proof) Also $\sum k^3 = \left( \sum k \right)^2$. Concidence? Aug 15 comment Moebius band not homeomorphic to Cylinder. @PyRulez My constructions are fine. If the two spaces where homeomorphic, we could map one meridian circle to the other $S^1 \subset X \leftrightarrow S^1 \subset Y$. Since both spaces fiber over the circle, we can take a closed interval over each point in $S^1$. The result is a closed cylinder on the one hand and a closed Möbius band on the other. Aug 15 comment Area of a Random Polygon I don't think the exact formula is a very exciting thing to compute. There are some interesting formulas in the large $n$ limit. See the papers of John Pardon. arXiv:1110.5656 Central limit theorems for uniform model random polygons. Aug 15 comment Without using Heegner-Stark-Baker, $\mathbb{Q}(\sqrt{-11})$ has class number $1$. Possibly related Dense Packings from Algebraic Number Fields and Codes Shantian Cheng Aug 13 comment Does $\cos (\pi/5)$ belong to $\mathbb{Q} (\sin(\pi/5))$? Somehow we should exploit: $\cos \frac{\pi}{5} = \sqrt{1 - \sin^2 \frac{\pi}{5}}$ or something. I been looking at these proofwiki.org/wiki/Quintuple_Angle_Formulas Aug 11 comment Number of irreducible quadratic polynomials over a finite field I like your use of "invisible ink" - it's great pedagogical tool. Aug 11 comment Number of irreducible quadratic polynomials over a finite field In $\mathbb{F}_2$, there are only 4 possibilities $x^2 , x^2 + 1=(x+1)^2, \color{blue}{x^2 + x + 1}, x^2 + x = x(x+1)$ Jul 29 comment Can the numbers $2^m 3^n$ have an infinitely long arithmetic sequence? See it's not so easy? Dirichlet's theorem was my motivation for this problem. As for the comments above, proving the natural density is 0 might also be hard. Jul 25 comment Can $O(\sqrt{x})$ be considered $o(x)$? @user251257 Oh my. What an important question. I guess $\boxed{x \to \infty}$, but $x \to 0$ could be important. Jul 19 comment Product of complex numbers $m+in$ with $0 < m,n \leq N$ @zardo if you can find Stirling approximation in that case then please write an answer Jul 18 comment Explaining Mathematical Modelling to a nonmathematician have you considered Math Educators Stackexchange