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2h
answered how to add supremums
13h
comment How to prove taht a product of two complete residue system is not a complete residue system?
Hint/strategy: first show that if it were possible, then the product $\sigma(i)i$ would have to have the same gcd with $n$ as $\sigma(i)$ and $i$ do. (If it got any higher, then some gcd would occur too many times.) Then if $n=pm$ for a prime $p$ and $m>1$, the subset $\{m,2m,\dots,(p-1)m\}$ must be multiplied by a permutation of itself, and the proof you have for the prime case will work again.
13h
answered Calculating sum of all permutations
13h
comment Riemann and Ihara's $\zeta$ Function Variable Question
Why is it okay that $x^5$ and $5^x$ are different? Because there's no reason to think they should be the same.
1d
comment Bounds on the numbe of groups of degree n
I seem to remember that the number of groups is largest when $n$ is a power of $2$.
1d
comment Bounds on the numbe of groups of degree n
@flawr: That's not true, actually. For example, there's only one group of order 15. And the number of ways to fill an $n\times n$ table with entries from a list of $n$ elements isn't $n^3$; it's $n^{n^2}$.
1d
answered Erasing numbers from circle and writing down sum
1d
comment If $(m,n)\in\mathbb Z_+^2$ satisfies $3m^2+m = 4n^2+n$ then $(m-n)$ is a perfect square.
To find this solution in the first place, write $m=n+k$; it's easier to see the right way to rewrite the equation then.
1d
answered Given $n$ linear functionals $f_k(x_1,\dotsc,x_n) = \sum_{j=1}^n (k-j)x_j$, what is the dimension of the subspace they annihilate?
2d
revised Why this approach to differentiate $\log_{10}(x+1)^x$ does not work?
added 7 characters in body; edited title
2d
comment Order of elements in finite fields
Not as an integer, no, but modulo 11251, the calculation is indeed easy. en.wikipedia.org/wiki/Modular_exponentiation
2d
answered Order of elements in finite fields
2d
comment Symmetry and the zeta function
To the content of your question: the current definition of $\zeta$ has its symmetry centered at $s=1$, as you point out; this perhaps isn't as tidy as centering it at $s=0$, but I don't think it qualifies as "neglecting" that symmetry. And the traditional definition has certain tidiness that would be lost with a shift of variables - the primary one being that $\zeta(s) = \sum_{n=1}^\infty 1/n^s$ is the initial reason why the function was introduced.
2d
comment Symmetry and the zeta function
I think the people who wrote the books asserting that "his" naturally implied "his or her" were probably all male :) Seriously though, thank you for the change - there are plenty of forces pushing women away from mathematics, so every little bit to change that helps.
2d
comment Symmetry and the zeta function
"... a mathematician can have at his or her disposal".
2d
comment Normal subgroups in groups of odd order
When you say a proof that applied to all cases, what do you mean? $S(p)$ is false for non-Fermat primes, as you pointed out.
Dec
18
answered Normal subgroups in groups of odd order
Dec
18
comment Primitive Root mod 26 and 25?
A primitive root modulo $q$ is, by definition, an element of (multiplicative) order $\phi(q)$. What is $\phi(26)$, and what integers have order $\phi(26)$ modulo $26$? (Hint: there are four residue classes of primitive roots modulo $26$, and $7$ is one of them.) Same with $25$: what is $\phi(25)$...?
Dec
18
comment Linear composition
Yes, the relevant four vectors are linearly independent over any field, in this case.
Dec
17
comment Linear composition
$\Bbb Z_4^2$? Are you sure you're not in a vector space over a field, say $\Bbb R^4$?