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Jul
17
awarded  Curious
Jul
16
comment Prove that for any give sequence of digits, there is a perfect square starting with that sequence
From the educational perspective "so the solutions provided should include these tools (a bit of constraint to the problem)." ... I am sure there are other ways to prove it.
Jul
16
answered Prove that for any give sequence of digits, there is a perfect square starting with that sequence
Jul
16
asked Prove that for any give sequence of digits, there is a perfect square starting with that sequence
May
23
comment Given two concentric circles with radiuses r < R, can we estimate the number of chords in between the circles?
Another way, probably, to tackle the problem (and this is a brainstorming): $$n_{of chords}\sim \frac{\pi }{\sqrt{1-\alpha ^{2}}}=\pi \cdot \sqrt{1+\alpha ^{2}+\alpha ^{4}+...}< \pi \cdot \sqrt{n}\cdot \sqrt{1+\frac{1}{n}\sum_{i=n}^{\infty }\alpha ^{2\cdot i}}$$
May
23
comment Given two concentric circles with radiuses r < R, can we estimate the number of chords in between the circles?
Yes, sort of. For this particular case when $r=\sqrt{p_{n}}$ and $R=\sqrt{p_{n+1}}$, knowing that $\sqrt{p_{n+1}}<n$ from some n and $\sqrt{p_{n+1} - p_{n}} \geq \sqrt{2}$, $n_{of chords} < \frac{\pi }{\sqrt{2}}\cdot n$.
May
23
comment Given two concentric circles with radiuses r < R, can we estimate the number of chords in between the circles?
Yes, it goes to infinity, but how "quickly"?
May
23
comment Given two concentric circles with radiuses r < R, can we estimate the number of chords in between the circles?
yes, that's the ultimate goal ;)
May
22
comment Given two concentric circles with radiuses r < R, can we estimate the number of chords in between the circles?
And probably "Is there a better one?" is a wrong formulation, should have written "Is there a different one?".
May
22
comment Given two concentric circles with radiuses r < R, can we estimate the number of chords in between the circles?
Ok, let me reformulate my previous message. 1. I knew of that asymptotic expression (Remark 1). 2. Unfortunately, it doesn't help. 3. Is there a better one? "Intuition result", had a different context, I shouldn't have mentioned it, probably.
May
22
comment Given two concentric circles with radiuses r < R, can we estimate the number of chords in between the circles?
So far, I have got to $\sqrt{p_{n+1}} - \sqrt{p_{n}} < \frac{\sqrt{p_{n+1}}}{2}\cdot \ln\frac{\sqrt{p_{n+1}}}{\sqrt{p_{n}}} < C\cdot \frac{\sqrt{p_{n+1}}}{n_{of chords}^2}$
May
22
comment Given two concentric circles with radiuses r < R, can we estimate the number of chords in between the circles?
Well, that's one of it, mentioned in "Remark 1". Considering that $\sin \angle AOB = \sqrt{1-\frac{r^{2}}{R^{2}}}\sim \angle AOB$ when $\frac{r}{R}\rightarrow 1$ this leads to the close to the intuition result $n_{of chords}\sim \frac{\pi }{\angle AOB}$. However, this doesn't help with estimating $\frac{n_{of chords}}{R}$, when $r\rightarrow \infty $ ... for more details, I am trying to "asses" Andrica's conjecture and see if I get any luck with estimating $n_{of chords}$.
May
22
comment Given two concentric circles with radiuses r < R, can we estimate the number of chords in between the circles?
True. I was thinking of something in form of an asymptotic function $f(r,R)$. Can't find anything with google and I don't believe this problem was never touched.
May
22
asked Given two concentric circles with radiuses r < R, can we estimate the number of chords in between the circles?
Jan
22
comment Maximum likelihood and fisher information of uniform and binomial
Probably, it has no sense to consider $N$ math.stackexchange.com/questions/396982/…
Jan
22
comment Maximum likelihood and fisher information of uniform and binomial
No, it's not $\max z_{i}$. Having the samples $z_{i}$ you will have to estimate $N$ or $p$ parameters using MLE (typically only $p$), as it is very well explained here projectrhea.org/rhea/index.php/… Or, for another explanation of the same topic, here onlinecourses.science.psu.edu/stat504/node/28
Jan
10
accepted Ratio of limits
Jan
10
comment Let $U$ and $V$ be any two open sets with $U\cap V=\emptyset$ and $F\subset U\cup V$. Show that there is an integer $n$ with $F_n\subset U\cup V$.
Think about $K_{n} = F_{n} \setminus \left ( U \cup V \right )$
Jan
2
comment Maths without irrational numbers
Computers, for example, have no idea about irrationals and this doesn't stop us from using computers to approximate irrationals, well at least those computable (e.g. non computable one en.wikipedia.org/wiki/Chaitin's_constant)
Dec
28
answered Prove or disprove these statements on prime numbers