mercio
Reputation
25,960
122/100 score
 13h comment Solutions of $f : \Bbb Q^*_+ \to \Bbb Q$ such as $f(\tfrac1x)=f(x)$ and $f(x) = (1+\tfrac1x)f(x-1)$ for all $x$ because $p/q \mapsto (p+q)/2$ is a solution and the image of $1$ by this is $1$. Alternatively I could have just said that $f \mapsto f(1)$ is an isomorphism of $\Bbb Q$-vector spaces from the set of solutions to $\Bbb Q$. If you know $1$ (nonzero) solution, you know all of them. 13h comment Solutions of $f : \Bbb Q^*_+ \to \Bbb Q$ such as $f(\tfrac1x)=f(x)$ and $f(x) = (1+\tfrac1x)f(x-1)$ for all $x$ for example, take $x=2/3$. For every $f$, $f(2/3) = f(3/2) = (1+2/3)f(1/2) = (1+2/3)f(2) = (1+2/3)(1+1/2) f(1)$ so we have $y(2/3) = (1+2/3)(1+1/2)$. $y(x)$ is just the product of all the factors corresponding to the transformations that bring $x$ to $1$. 13h comment Solutions of $f : \Bbb Q^*_+ \to \Bbb Q$ such as $f(\tfrac1x)=f(x)$ and $f(x) = (1+\tfrac1x)f(x-1)$ for all $x$ yes but your choice of $y$ depends a priori on some solution $f$. 13h answered Solutions of $f : \Bbb Q^*_+ \to \Bbb Q$ such as $f(\tfrac1x)=f(x)$ and $f(x) = (1+\tfrac1x)f(x-1)$ for all $x$ 1d comment What do you call the subset of all the Gaussian integers having the smallest magnitude given an argument? so it's the set of gaussian integer that have coprime real and imaginary part ? 2d comment Number of $q$-th residues modulo $n$ well I don't have any reference, it's a pretty basic result. 2d comment Is this simple proof that the Frobenius endomorphism of an elliptic curve defined over $\mathbb F_q$ is surjective valid? well then $E(\Bbb F_q)$ is finite and $\phi$ is injective, so ... 2d comment Unique solution of a simple functional equation if $\theta$ is such a function, then adding $2\pi$ to it gives you another solution. 2d answered Measure 11 liters using bottles of 16, 6, and 3 liters 2d answered Is this simple proof that the Frobenius endomorphism of an elliptic curve defined over $\mathbb F_q$ is surjective valid? 2d comment Number of $q$-th residues modulo $n$ um what result ? 2d answered Number of $q$-th residues modulo $n$ 2d comment Finding the size of a Galois Group of a splitting field for a polynomial of degree 6. well your reasoning would prove that the Galois group of any degree $6$ polynomial has order bounded by $48$ which is plainly false. There's no reason for $\overline{\alpha}$ to be in $\Bbb Q(\alpha)$. Also sometimes, polynomials have real roots. Apr22 awarded algebraic-number-theory Apr21 answered How to write $\sqrt{4x^2 - 3}$ in the ring $\mathbb{Q}[x]/(x^3 - x - 1)$? Apr20 comment What causes long sequences of consecutive 'collatz' paths to share the same length? I never claimed this was an answer or else i would have written one. I actually don't know what kind of answer you're expecting here. "special" is a rather vague term, especially when we are dealing with heuristic arguments. Do you want us to fabricate a "heuristic argument" that goes with a formula that gives the right order of magnitude in this particular case ? Apr20 comment What causes long sequences of consecutive 'collatz' paths to share the same length? Also, the trajectories tend to coalesce with a lot more regularity than independant random variables do. As the examples of 100 and 101 show, the sequences for $4k$ and $4k+1$ always have the same length (and you can make up a lot more relationships like this) , so you shouldn't model the length of adjacent numbers with independant random variables. Apr20 comment What causes long sequences of consecutive 'collatz' paths to share the same length? I don't think that approximating the length with a uniform random variable between $1$ and $\log n$ is accurate. For starters we have a lower bound $\log n / \log 2$. Apr19 comment Which one is the ring of integers of $K$; $\mathbb Z[\alpha,\frac{\alpha^2}{2}]$ or $\mathbb Z[\alpha,\frac{\alpha+\alpha^2}{2}]$? that's exactly what i did except that i kept the indeterminate $T$ instead of pointlessly putting $\sqrt T$ (whatver that would mean), then replaced $T^2$ with $T$ to go from "a polynomial that vanishes for $T= \alpha$" to get "a polynomial that vanishes for $T = \alpha^2$" Apr19 revised Which one is the ring of integers of $K$; $\mathbb Z[\alpha,\frac{\alpha^2}{2}]$ or $\mathbb Z[\alpha,\frac{\alpha+\alpha^2}{2}]$? added 4 characters in body