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Aug
1
comment Does there exist a cubic polynomial $f(x)$ such that $f(x)\equiv0 \pmod p $ has no integer solutions if $p\equiv 3\pmod 4$?
Oh, you're right, for finite fields the Galois group can't be $S_3$...
Aug
1
comment Does there exist a cubic polynomial $f(x)$ such that $f(x)\equiv0 \pmod p $ has no integer solutions if $p\equiv 3\pmod 4$?
No, the question doesn't require the polynomial to have a root mod all p=4k+1 — and since 1/3<1/2 I don't see an immediate contradiction with Chebotarev's theorem.
Aug
1
comment Does there exist a cubic polynomial $f(x)$ such that $f(x)\equiv0 \pmod p $ has no integer solutions if $p\equiv 3\pmod 4$?
Ah, I see. But IMHO the question asks about a cubic polynomial that (like $x^2+1$) doesn't have zeroes mod all primes of the form $4k+3$.
Aug
1
comment Does there exist a cubic polynomial $f(x)$ such that $f(x)\equiv0 \pmod p $ has no integer solutions if $p\equiv 3\pmod 4$?
@gammatester p=3 is not 1 mod 3
Jul
18
comment Intuition for the definition of the Gamma function?
(Somewhat) related: How to come up with the gamma function? & Understanding the Gamma function
Jul
15
comment Sum involving binomial coefficients
Well, have you tried guessing general answer from first few values?
Jul
15
comment How to compute homotopy classes of maps on the 2-torus?
(I wish I knew a reference where all this is explained more clearly...)
Jul
15
comment How to compute homotopy classes of maps on the 2-torus?
@Qiaochu ...i.e. $\pi_2(X)/\langle t-t^a,t-t^b\mid t\in\pi_2\rangle$ is actually $H^2(\mathbb T^2;\pi_2(X))$, where $\pi_2(X)$ is the local system s.t. generators of $\pi_1(\mathbb T)$ act as $t\mapsto t^a$ and $t\mapsto t^b$.
Jul
15
comment How to compute homotopy classes of maps on the 2-torus?
@Qiaochu The answer for $D^2$ is different because the coboundary map is different. For $[S,X]$ (and let's consider the case $\pi_1(X)=0$ for simplicity) we're talking about $H^2(S;\pi_2(X))$.
Jul
15
comment How to find the length of diagonal of a rhombus
(...and now we have answers to two different questions in one place...)
Jul
15
comment How to find the length of diagonal of a rhombus
I don't know why someone decided to overwrite an old question with a different one — rolled back.
Jul
15
comment How to compute homotopy classes of maps on the 2-torus?
@Qiaochu Let's see. In the same spirit for $D^2$ we have 'a' trivial element of $\pi_1$ and an element of $\pi_2$ up to some equivalence relation; and in the part 'Let's see if this element of $\pi_2$ is well-defined' we have $s'=s+t$ — i.e. equivalence relation identifies all elements of $\pi_2$. That's the correct answer.
Jul
12
comment Probability of their's constituting a triangle?
...and see also mathoverflow.net/questions/2014
Jul
12
comment Probability of their's constituting a triangle?
...or of math.stackexchange.com/questions/26424
Jul
9
comment Prove that $\sum_{k=0}^{m}\binom{m}{k}\binom{n+k}{m}=\sum_{k=0}^{m}\binom{n}{k}\binom{m}{k}2^k$
related: wiki: Delannoy numbers
Jul
8
comment Elementary, direct proof of when $5$ is a quadratic residue mod $p$
related: Special Cases of Quadratic Reciprocity and Counting Fixed Points
Jun
24
comment Contractible Subspace and Homotopy Equivalence
E.g. $X=S^1$, $A=S^1-pt$.
Jun
20
comment Relation between $K$-theories
Related: Relationship between topological and Quillen's K-theory
Jun
11
comment Help me ID this weird $\pi$ formula
Terms are clearly related to $\binom{1/2}n=(-4)^n\binom{2n}n/(1-2n)$ and $\pi/4$ is of course $(1/2)!^2$. So the identity looks similar to Vandermonde's convolution $\sum_k\binom nk^2=\binom{2n}n$ for $n=1/2$ or smth like that.
Jun
10
comment Example of something easier to count with $q$-analog?
(Re: 'justifying its relation to the gamma function... is more complicated'): the definition of q-gamma is completely analogous to a definition of ordinary gamma-function (the lesser-known-but-easier-to-comprehend one).