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17h
comment Function that is the sum of all of its derivatives
Does it remain to determine whether there are some more solutions where the convergence of the series is (point-wise but) not uniform, then?
2d
comment How can I show that this function is discontinuous at the point $x=1$?
OK, but I think it would be more appropriate as a comment since it does not directly help him answer the question.
2d
comment How can I show that this function is discontinuous at the point $x=1$?
He wanted an $\epsilon$ $\delta$ proof.
Feb
1
comment Formula for the simple sequence 1, 2, 2, 3, 3, 4, 4, 5, 5, …
And in some programming languages the expression $\left\lfloor \frac{n}{2} \right\rfloor$ is written as a single infix operator, for example n/2 (where integer division with discarding the remainder is implied) or n\2.
Feb
1
comment Formula for the simple sequence 1, 2, 2, 3, 3, 4, 4, 5, 5, …
From the question, $n\in\mathbb{N}$. Any function $\mathbb{N} \to \mathbb{N}$ is continuous. The two formulas represent the same continuous function on $\mathbb{N}$ (as required when they are both correct). But the second expression generalizes to $\mathbb{R} \to \mathbb{R}$ or $\mathbb{C} \to \mathbb{C}$ and this extension is still continuous (and even analytic).
Jan
28
comment Is 641 the Smallest Factor of any Composite Fermat Number?
In fact, Lucas (in 1878) strengthened the result to $q\equiv1\pmod{2^{n+2}}$. See MathWorld (Weisstein). Other sources (Caldwell) say Euler already knew this, but I think it was Lucas.
Jan
28
comment Is 641 the Smallest Factor of any Composite Fermat Number?
Also see A023394 where all terms below $10^{19}$ are given.
Jan
28
comment What is the smallest prime factor of the number $14^{14^{14}}+13\ $?
At least it would be possible to raise the search limit well above $6\cdot 10^9$ by patiently waiting for some (good) software.
Jan
28
comment Sum of $1-\frac{2^2}{5}+\frac{3^2}{5^2}-\frac{4^2}{5^3}+…$
@Ananya Naturally. My comment related to the fact that the above answer apparently "cheated" in one step and asked Wolfram. Which is fine enough if you know how you could find the same answer, in principle, without Wolfram, and you only want skip the "boring" details. However, some would regard an answer with no appeal to Wolfram as better.
Jan
28
comment What is the smallest prime factor of the number $14^{14^{14}}+13\ $?
How did you come up with this question? Is it a problem you are supposed to solve (assigned to you by a teacher etc.)? Otherwise, why do you think anyone knows a prime divisor?
Jan
28
comment Sum of $1-\frac{2^2}{5}+\frac{3^2}{5^2}-\frac{4^2}{5^3}+…$
In fact, if you are OK with using Wolfram, it can solve the entire problem with no work by you.
Jan
22
comment Distributive Law and how it works
It is hard to answer "why". We can prove by exhibiting a counter-example that the alternative distributivity you propose does not hold. For example $1 + (2 * 3) = 1 + 6 = 7$ while $(1 + 2) * (1 + 3) = 3 * 4 = 12$, and we know $7 \ne 12$ in $\mathbb{R}$.
Jan
22
comment Optimal escape route out of a half-space in $\mathbb{R}^3$
I have optimized it already. The $\sqrt{\frac73}$ goes from $(0,0,0)$ to $A=(-1,\frac{1}{\sqrt{3}},1)$ on the "roof". then I follow the path like in your English link in height $z=+1$. Then I go vertically down to $z=-1$ (length $2$ vertically down). Finally trace the same kind of curve at $z=-1$, but in "reverse". I never return to the $z$ axis (the line $x=y=0$).
Jan
22
comment Optimal escape route out of a half-space in $\mathbb{R}^3$
I think the curve I mention is $\sqrt{\frac73}+(\frac{1}{\sqrt{3}}+\frac{7\pi}{6}+1) \cdot 2 + 2 = \frac{2+\sqrt{7}}{\sqrt{3}}+\frac{7\pi}{3}+4 \approx 14.01$ in length.
Jan
22
comment Optimal escape route out of a half-space in $\mathbb{R}^3$
What if you made a right "cylinder" whose base face looked like the optimal curve in 2D (with $z=-1$), and repeated that curve as a "roof" at $z=+1$? Maybe it is not better than visiting all eight vertices of a cube?
Jan
22
comment Optimal escape route out of a half-space in $\mathbb{R}^3$
What bounds do you know already?
Jan
15
comment Notation of the second derivative - Where does the d go?
I guess the same could occur with a delta, for example with the formula $U=\frac12 k \Delta x^2$ one might interpret it as $U=\frac12 k (\Delta x)^2$ (the elastic potential energy in a Hooke spring).
Jan
14
comment Is there a known method for finding extremely huge squarefree numbers?
@Peter By the way, a new world-record prime number has just been identified; it will be announced this Tuesday at 14:00 UTC.
Jan
13
comment Applications of Perfect Numbers
Also, a semiperfect number (i.e. an abundant one which is not weird) is good enough to give away all of the items.
Jan
3
comment When would a graphing calculator make an error while plotting a function?
Upvoted for the link illustrating the issue.