How come if $\ i\ $ not of the following form, then $12i + 5$ must be prime?
How come if $\ i\ $ not of the following form, then $12i + 5$ must be prime?
Convert NSDate to NSString How do I convert, NSDate to NSString so that only the year in @"yyyy" format is output to the string? How do I record a screen cap video of the iPad? I'm looking for a method to record the video without actually filming the device. Does such a method exist? An inequality for arctan$x$ Why it is true that $|\arctan x-\arctan y|\leq |x-y|$? I have tried to use $\tan x$ but I didn't conclude to something. In Stein's Fourier Analysis i'm having trouble Justifying the following calculations for the Fourier Coefficients in $(1.)$  $(1.)$  Let $f(\theta) = \theta$ for $ -\pi \leq \theta \leq \pi$ in the case when $n \neq 0$:  $$f(n)=\frac{1}{2 \pi}\int_{-\pi}^{\pi}\theta e^{-in \theta} d \theta = \frac{1}{2 \pi}[-\frac{\theta}{in}e^{-in \theta}]_{-\pi}^{\pi} + \frac{1}{2 \pi in} \int_{-\pi}^{\pi}e^{-in\theta}d \theta = \frac{(-1)^{n+1}}{in} $$  Then the Fourier Series of $f$ is given by:  $$ f(n) = \sum_{n \neq 0}  \frac{(-1)^{n+1}}{in}$$ The definition of span I am trying to implement the following K-map with a table    AB C     00 01 11 10       ______________     0| 1  0  1  0     1| 0  1  1  0   I implemented it like this:   \begin{table}[htdp] \begin{center} \begin{tabular}{c|c|c|c|c|}  &amp;00 &amp; 01&amp; 11&amp; 10 \\ \hline 0&amp;1&amp;1&amp;1&amp;1\\ 1&amp;0&amp;0&amp;0&amp;0\\ \end{tabular} \end{center} \label{default} \end{table}   However on the top row I have vertical lines that I'd like to not see. I went with this approach because if i was to use a one row with the information so that I wouldn't have to worry about aligning the headings. So is there a way I can get the vertical lines not to appear in the heading row? Does the proof of Bolzano-Weierstrass theorem require axiom of choice? Proving any number of the form $2^n$ can't be written as the sum of k consecutive numbers. I think i have proved that any number of the form $2^n$ can't be written as the sum of k consecutive positive integers, but i would be grateful if i could have some clarification as to whether this is a correct and valid proof or not.  Let $g=x+(x+1)+(x+2)+(x+3)+...+(x+(k-1))$, hence g is the sum of k consecutive positive integers.  $g=kx+\sum_{d=1}^{k-1}d$  Hence, $g=kx+\frac{k(k-1)}{2}$  Rearranging for x we find, $x=\frac{g}{k}-\frac{(k-1)}{2}$  So, for certain values of $k$, we can be sure that a positive integer value of $x$ can be found, and hence $g$ can be written as the sum of $k$ consecutive positive integers.There are two cases for $k$ to considor:  Case 1)  $k$ is odd. If $x$ exists, it must be the case $k$ | $g$, as for odd values of $k$, $\frac{(k-1)}{2}$ will alwyas be an integer, and hence $x$ will.  Case 2)  $k$ is even. $k$ does not divide $g$, but $\frac{g}{k}$ must be a multiple of $\frac{1}{2}$, and hence equals $\frac{h}{2}$, for some odd $h$.  Letting $j=2^n$, assume that we can write $j$ as the sum of $k$ consecutive positive integers.  If $k$ is odd, then if a $k$ exists, it must divide $j$, by Case 1. But, $j$ has no odd factors as it is a power of two, hence $k$ could not be odd.  If $k$ is even, by Case 2 it mut be the case $k$ does not divide $j$. Also, $\frac{j}{k}=\frac{h}{2}$, for some odd $h$.  But, this implies $2j$ has an odd factor, $h$. but, as $2j$ is a power of $2$, it can't have odd factors. So, $k$ can't be odd.  Hence, $k$ is neither odd or even, and this is a contradiction to our assumption. Hence no $x$ exists, and hence and number of the form $2^n$ can't be written as the sum of $k$ consecutive positive integers.  Any feedback on this proof would be appreciated. INT comparison without WHERE statement Limit of $L^p$ norm when $p\to0$ Let ($\Omega$, $\cal{F}$, $\mu$) be a probability space and $f\in L^1(\Omega)$. Prove that  $$\displaystyle\lim_{p\to 0} \left[ \int_{\Omega}|f|^pd\mu \right]^{\frac{1}{p}}=\exp \left[ \int_{\Omega}\log|f| d\mu  \right],$$  where $\exp[-\infty]=0$. To simplify the problem, we may assume $\log|f|\in L^1(\Omega).$ bash regex with quotes? Charging Li-ion batteries in parallel I wanted some opinion on my set up here, just to make sure I got the basics right.  I have a Li-ion battery charging circuit based on the MCP73113. This is designed to be a single-cell battery charger.   The battery itself (3.7V, 650mAh) comes with its own PCB with Schottky diode and current regulators as protection. EDIT: Not a Schottky diode. Current limiter and a Protection IC.  By design, they work together just fine.  I have more batteries from the same manufacturer and wanted to make higher capacity packs by putting two cells in parallel. The two cells come with their own PCB, but I only kept one of them, as I soldered their leads together. It seems to work and they are charged and discharged just like regular batteries (3.7V, now 1300 mAh).  However, not all the packs I made work smoothly and I get a failure rate of 2 in every 10 packs. They all start out normally, but occasionally when the batteries are not charged for some time, or are used up, they tend to not work anymore. So far, in only 1 pack I found the voltage to be different (both now below 2V, aka dead).  I assume my method is not the right approach to ensure 100% success rate, so I was wondering how can this be solved. All the controllers I looked into was designed for multiple cells in series, not parallel. Am I fundamentally missing something here? How can I remove the Trash icon from the Unity launcher? I would like to not show the trash icon at the bottom of the Unity launcher.  Is this possible without having to recompile anything? :P `testl` eax against eax? I am trying to understand some assembly.  The assembly as follows, I am interested in the testl line:  000319df  8b4508        movl   0x08(%ebp), %eax   000319e2  8b4004        movl   0x04(%eax), %eax   000319e5  85c0          testl  %eax, %eax   000319e7  7407          je     0x000319f0     I am trying to understand that point of testl between %eax and %eax? I think the specifics of what this code isn't important, I am just trying to understand the test with itself - wouldn't the value always be true? Blender 2.8 - How to access material nodes? My Ubuntu is running fsck on every bootup How can I use ranges in a switch case statement using JavaScript? How do I show that $\ln(x) = \lim_{n\to\infty} n (x^{1/n} - 1)$?  I ran into this identity on . I haven't been able to find any proof online and my efforts to get from $\ln(x) := \int_1^x \frac{\mathrm dt}t$ to that limit have been a failure. Active login not recognized on Super User What is the difference between equation and formula? Sometimes equation and formula are used interchangeably, but I was wondering if there is a difference.  For example, suppose we can calculate a car's fuel efficiency as:  mpg = distance traveled in miles / the fuel used in a gallon   Is that an equation or formula?