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23h
comment How to prove that the only integer solutions to ${r}^{3}/\left({r-1}\right)$ are $r\in \left\{{0,2}\right\}$
For a general technique, see my answer to this MO question 4+ years ago: mathoverflow.net/questions/66873/…
2d
comment The Weyl group of E8 versus $O_8^+(2)$
If I remember right the index-2 normal subgroup of $O_{2k}^\pm(2)$ consists of the transformations that have an even-dimensional fixed subspace.
Feb
3
comment The sequence $a_{n+1}=\left\lceil \frac{-1+\sqrt{5}}{2}a_{n}-a_{n-1} \right\rceil$ is periodic
The eigenvalues of the corresponding linear transformation of ${\bf R}^2$ are $\exp \pm 2\pi i/5$ so it's a 72-degree rotation for an invariant quadratic form. (It had already been noted that this linear transformation is a 5th root of the identity.)
Feb
3
answered What's in the genus of the cubic lattice?
Feb
3
comment Height function on 2-torus with only 3 critical points
Another construction of such a function $f$: construct the torus as the quotient of ${\bf R}^2$ by the lattice $L = A_2$, and consider the Green's function with charges of $+1$ and $-1$ at the two nontrivial points of $A_2^* \, / \, A_2^{\phantom.}$. The degenerate saddle is at the origin. The logarithmic singularities can be capped artificially, or smoothed by applying a heat kernel. I don't know whether this helps construct an immersion with height function $f$.
Feb
2
comment The sequence $a_{n+1}=\left\lceil \frac{-1+\sqrt{5}}{2}a_{n}-a_{n-1} \right\rceil$ is periodic
Neat. You might still rotate the y-axis by 18(?) degrees to retain the approximate fivefold symmetry.
Feb
1
awarded  Nice Answer
Jan
31
revised The sequence $a_{n+1}=\left\lceil \frac{-1+\sqrt{5}}{2}a_{n}-a_{n-1} \right\rceil$ is periodic
grammar fix (whose removal, not which removal)
Jan
31
comment The sequence $a_{n+1}=\left\lceil \frac{-1+\sqrt{5}}{2}a_{n}-a_{n-1} \right\rceil$ is periodic
Pictures corrected: I had originally implemented $\lceil x \rceil$ as ${\rm round}(x + \frac12)$, but that gave $x+1$ for $x \in \bf Z$ $-$ resulting in a somewhat different recursion, though it seems periodic too. Here's the previous screenshot: math.harvard.edu/~elkies/mo229714-.png
Jan
31
answered The sequence $a_{n+1}=\left\lceil \frac{-1+\sqrt{5}}{2}a_{n}-a_{n-1} \right\rceil$ is periodic
Jan
26
awarded  Nice Answer
Jan
24
awarded  Good Answer
Jan
22
comment Curve with given Frobenius polynomial
. . . and once you have an example $C_0$ of genus 2, any curve $C$ that admits a non-constant map to $C_0$ defined over the prime field will inherit $C_0$'s eigenvalues of Frobenius.
Jan
20
comment How do i show that:$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$ without using properties of Riemann zeta function?
See the comment thread for David Speyer's answer: he refrained from sending it in to the Monthly because a very similar argument for $\zeta(4) = (2/5) \zeta(2)^2$ had already been given by Zagier.
Jan
20
awarded  Enlightened
Jan
20
awarded  Nice Answer
Jan
19
answered How do i show that:$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$ without using properties of Riemann zeta function?
Jan
19
awarded  Nice Answer
Jan
19
answered Infinitely many $k$ such that $[a_k,a_{k+1}]>ck^2$
Jan
19
comment Infinitely many $k$ such that $[a_k,a_{k+1}]>ck^2$
$[m,n]$ = least common multiple of $m$ and $n$?