I think I know that there were times in the past when it was convenient to look at a number $1$ as a prime number, and, as far as I can remember, even then it was dependent on who we ask is it prime or not?  This question is primarily here because in some of the questions I posted in the last few days I wanted that we for the purpose of the question consider $1$ as prime, or because of some similar reason.  So, let us see why, in my opinion, we should take $1$ as prime, and why we should not.  First, the definition of prime numbers as seen on Wikipedia:     A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.   If we interpret here this greater as $n \in \mathbb N \setminus \{1\}$, in other words, as strictly greater, then clearly $1$ cannot be prime because it is, by definition, not taken into consideration.  If we change the definition and remove part of the sentence, the part greater than 1, then we have this definition:     A prime number (or a prime) is a natural number that has no positive divisors other than 1 and itself.   Then we could say that $1$ is prime, because he has no other positive divisors other than 1 and itself, and he has two divisors which coincide and are both equal to $1$ (1 and itself).  But do we stumble upon some problem if we declare $1$ as prime?  Yes we do, but all these problems (that I know of) are not so serious.  First, we should rephrase fundamental theorem of arithmetic which says that:     Every positive integer $n&gt;1$ can be represented in exactly one way as a product of prime powers.   If we would take $1$ as prime then the fundamental theorem of arithmetic could be stated as:     Every positive integer $n&gt;1$ can be represented in exactly one way as a product of prime powers in such a way that in the factorization of $n$ powers of number $1$ are absent.   This is because we have $1^a=1^b$ for $a \neq b$ so if we want unique factorization then powers of $1$ must be absent from the factorization.  But do we gain something positive if we declare $1$ as prime number?  I think we do.  For example, the Goldbach conjecture states:     Every even integer greater than 2 can be expressed as the sum of two primes.   If we would declare $1$ as prime we could shorten the Goldbach conjecture:     Every even natural number can be expressed as the sum of two primes.   Of course, this is because $1+1=2$ so now even two has a representation as a sum of two primes.  But in this case we would gain even more. Now, when $1$ is taken as a prime number, then we have slightly greater chance that every even natural number can be expressed as a sum of two primes because now we can consider sums of the form $2n=(2n-1)+1$ and we cannot consider them when we do not take $1$ as a prime.  It seems (but I could be wrong), that it is the matter of agreement between mathematicians should we or should we not take $1$ as a prime number.  So, I have a few questions:     1)  Is there any serious problem or inconsistency inside of mathematics if we take $1$ as a prime number (other than just re-writing the theorems/lemmas/conjectures/definitions in a different way)?      2)  Should we or should we not take $1$ as a prime number?      3)  Can you give some other examples of conjectures for which it would be better to take $1$ as a prime?      4)  What are your thoughts on this?
Why is $1$ not considered a prime number?  Or, why is the definition of prime numbers given for integers greater than $1$?
When compiling a beamer presentation and using the following \author command  \author{Name \\ \texttt{my.email@domain.com}}   I get the following hyperref warning in my logfile  Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref)                removing `\\' on input line 15.   Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref)                removing `\new@ifnextchar' on input line 15.   I understand that this has to do with hyperref setting the PDF metadata, where the linebreak does’t make much sense and should be removed.  Trying to set  \hypersetup{pdfauthor={Name}}   does’t change the situation, seems like hyperref is still looking at the author command.  How do I get the desired display of author name with email and still keep hyperref happy? A girl who transcends time I remember quite a bit, but can't seem to find the title or (female) author.   A girl accompanies her father as he captains a mission to travel at near-light speed in order to experience the effects of time dilation and return to Earth 100 years into its future. She leaves behind her boyfriend (Tal or Tel or something), who promises he'll find a way to stay with her despite the time travel that will render him, well, dead before her return. She's accompanied by a close friend who is deaf but lip-reads and speaks so well as to occasionally fool others into believing she is hearing. They spend a lot of time in the ship's garden. I believe he ends up either getting on a different ship to survive into the future via time dilation, or else having himself frozen. Either way, if I recall correctly, Earth isn't doing well at all by the time the mission with the girl returns. There's been a war or some other disaster. There's an extended period of vague, ambiguous, existential crap in which the girl (now a woman) eventually ends up transferring her consciousness into her own daughter, presumably in order to go on living. Everything wraps up somehow and the girl and her boyfriend walk together hand in hand on a desolate planet into the sunset near the end of time.   It was all really rather trashy and silly, I remember, and took itself entirely way too seriously, but I'd like to at least know what it was so I could read it again. Either it'll be worth a laugh, or it'll turn out to be surprisingly good. Install MySQLdb (for python) as non-compressed egg The install instructions are:  $ python setup.py build $ sudo python setup.py install # or su first   This gives me an .egg file. How do I tell the install to dump the files as a normal, uncompressed library?  Thanks! Related:    The helpcenter has . It still links to the  for further details. Besides that, the question ban message .  Now that we have the helpcenter, could the following changes be implemented?   Link to the help center instead of the meta post Remove the link to the meta post from the help center, and instead incorporate its contents into the helpcenter itself This picture is a part of a table i am using, the terms subcases are vertically written but i cant center them vertically also in the next columns terms like A, B, etc are also not vertically centered. I made this table using tabulary package a. My code is  \begin{table}[H]  \centering  \caption{Saturation Table}  \label{tab:5}%  \begin{tabulary}{1\linewidth}{J J J J J J J J J J J J J}  \toprule  &amp; \textbf{Method} &amp; \multicolumn{1}{c}{\textbf{Parameters}} &amp; \textbf{Step 0} &amp; \textbf{Step 1} &amp; \textbf{Step 2} &amp; \textbf{Step 3} &amp; \textbf{Step 4} &amp; \textbf{Step 5} &amp; \textbf{Step 6} &amp; \textbf{Step 7} &amp; \textbf{Step 8} &amp; \textbf{Step 9} \\  \midrule  \multirow{4}[4]{*}{\begin{sideways}\textbf{Subcase A}\end{sideways}} &amp;  \multirow{2}[2]{*}{\textbf{A}} &amp; \textbf{No} &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; \textbf{6} &amp; &amp; &amp; &amp; \\[1mm]  &amp; &amp; \textbf{Maximum Saturation {(\%)}} &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 4 &amp; \textbf{5} &amp; &amp; &amp; &amp; \\  \cmidrule{2-13} &amp; \multirow{2}[2]{*}{\textbf{B}} &amp; \textbf{No} &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 &amp; 7 &amp; \textbf{8} &amp; &amp; \\[1mm]  &amp; &amp; \textbf{Maximum Saturation {(\%)}} &amp; 6 &amp; 6 &amp; 7 &amp; 8 &amp; 9 &amp; 9 &amp; 19 &amp; \textbf{8} &amp; &amp; \\  \bottomrule  \end{tabulary}%  \end{table}% Boolean expression (grammar) parser in c++ Partial SSL in rails "done" vs "have done" What WordPress API function lists active/inactive plugins? XTerm only display part of some unicodes and Urxvt displays nothing In my case, (U)XTerm does display some unicodes, but only partially. For exmaple: for letters are: 𝒫φϕ𝒟. Here is a screenshot of UXTerm and Urxvt respectively   It seems that the problem is related to the letter space or letter length. By the way, everything is fine with gnome-terminal.  My current .Xresources file is: Xft.dpi:       120 Xft.antialias: true Xft.hinting:   true Xft.rgba:      rgb Xft.autohint:  false Xft.hintstyle: hintslight Xft.lcdfilter: lcddefault  XTerm*faceName:          Fixed XTerm*faceSize:          14 XTerm*reverseVideo:      false XTerm*selectToClipboard: true  UXTerm.vt100*faceName:          Fixed UXTerm.vt100*faceSize:          14 UXTerm.vt100*reverseVideo:      false UXTerm*selectToClipboard: true  URxvt.font:            xft:Fixed:pixelsize=18  URxvt.depth:                      32 URxvt.background:                 #222D31 URxvt*scrollBar:                  false URxvt*mouseWheelScrollPage:       false URxvt*cursorBlink:                true URxvt*background:                 black URxvt*foreground:                 grey URxvt*saveLines:                  5000  How to make XTerm and Urxvt to display those characters correctly? Is it possible? How can we extract specific files from a large tar.gz file? I found  but, when I tried the mentioned command there, I got the error:  $ tar --extract --file={test.tar.gz} {extract11} tar: {test.tar.gz}: Cannot open: No such file or directory tar: Error is not recoverable: exiting now   How do I then extract a file from tar.gz? Ubuntu 16.04 sftp and vscode ssh not working What gifts would be welcome for a Swedish host family? How do I insert a backslash inside a mathematical statement? I'm trying to write a set subtraction operation in an AMS formatted template.  In the end, what I want should look like A\B.  But I'm doing this in math-mode $A\B$ and this gives me errors.  I've also tried using the \diagdown symbol, but it doesn't work either.  Is there another way to do this? Shimano Deore front derailleur occasionally sticking I have a question about my front derailleur sticking occasionally.  I've had this bike for a 1.7 years and have ridden it through 2 winters.  It has only been after this last winter that my front derailleur has been sticking on occasion where it's like grit has jammed in somewhere (which I'm sure it is the case).  When I'm on the fly I just knock it with my heel and it will downshift and I can push the shifter hard enough to go back up when I need to.  The first time I decided to overhaul both the front and back ones.  Took them apart, cleaned/degreased, regreased the two back cogs, and used WD-40 professional silicone on the rest of the parts as to not attract more dirt.  Then after about a month it got stuck again, and I couldn't see anything that could be jamming it (while it was still stiff as heck).  Didn't take it apart this time just used a tooth brush and silicone again and it's fine at the moment.  I'm assuming it may be that since that the roads have been grittier/dirtier from the winter with some recent rain and slush that it may just be normal.  I haven't been on the trails yet this year so I know it's not that.  Is this normal, has some kind of bur that I can't see come up and is catching grit?  EDIT:  I should also note that this is a hardtail mtb that I use for trails and winter commuting (a lot of salt and grit on the roads). Why are the number of false positives independent of sample size, if we use p-values to compare two independent datasets? If I run the R code below, it will generate two independent vectors then test them to see if they are related in some way (i.e. p-value &lt; 0.05).  If I repeat this 1,000 times, then 50 of these (5%) will have a false positive, with a p-value &lt; 0.05. This is a type 1 error.  If we increase sampleSize to 1000, or even 100,000, the results are the same (5% false positives).  I am struggling to understand this, as I would have thought that if we collected enough samples, the chance of a false positive would drop towards 0 (just like correlation).  So, I guess my question is: "How is it possible that the number of false positives, based on p-values generated from comparing two independent datasets, be independent of sample size?".  # R code to demonstrate that with a large dataset, we can still  # get significant p-values, purely by chance.  # Change this to 1000, and we still get the same number of false positives (50, or 5%). sampleSize = 20  cat("Sample size:",sampleSize,"\n")  set.seed(1010093) n=1000 pValues &lt;- rep(NA,n) for(i in 1:n){     y &lt;- rnorm(sampleSize)     x &lt;- rnorm(sampleSize)     pValues[i] &lt;- summary(lm(y ~ x))$coeff[2,4] } # Controls false positive rate fp = sum(pValues &lt; 0.05) cat("Out of ",n," tests, ",fp," had a p-value &lt; 0.05, purely by by chance.\n",sep="")  ----output---- Running "true positives - none.R" ... Sample size: 1000  Out of 1000 tests, 52 had a p-value &lt; 0.05, purely by by chance. Let $X_1$ and $X_2$ be two independent exponential random variables with the PDFs $f_{X_1}(x_1)=\lambda_1 \exp(-\lambda_1 x_1)$ and $f_{X_2}(x_2)=\lambda_2 \exp(-\lambda_2 x_2)$.   Let $X= \frac{X_1X_2}{X_1+X_2+a}$ be a r.v. for which I want to derive the CDF, where $a$ is a positive constant.  In other words, I need to calculate $\mathbb{P}\{X &lt;x \}$.  Any ideas or hints?  Edit (attempt):  Using a similar approach to that used , we have $X_1 \in [0, \infty)$ and  $X_2 \in [0, \frac{x(x_1+a)}{x_1-x})$. The latter interval results from: $\frac{x_1x_2}{x_1+x_2+a} &lt; x$, thus $x_2(x_1-x)&lt;x(x_1+a)$ which implies $x_2 &lt; \frac{x(x_1+a)}{x_1-x}$; but I am not sure if this is correct since $x_1-x$ can sometimes be negative (?). Based on the above, we can write: \begin{align}  \mathbb{P} \{ \frac{X_1X_2}{X_1+X_2+a} &lt; x \}  &amp; = \int_{x_1=0}^\infty \int_{x_2=0}^{ \frac{x(x_1+a)}{x_1-x}} \lambda_1 \exp(-\lambda_1 x_1) \lambda_2 \exp(-\lambda_2 x_2)  dx_2 dx_1  \\ &amp;= \int_{x_1=0}^\infty \left(1-\exp(-\lambda_2\frac{x(x_1+a)}{x_1-x}) \right) \lambda_1 \exp(-\lambda_1 x_1) dx_1  \\ &amp; = 1-\lambda_1 \int_{x_1=x}^\infty \exp(-\lambda_2\frac{x(x_1+a)}{x_1-x} ) \exp(-\lambda_1 x_1)   dx_1 \end{align} In the last equality: for the integral of $\lambda_1 \exp(-\lambda_1 x_1) dx_1$, I suppose that $x_1 \in [0,\infty)$, whereas for the second term I suppose that $x_1 \ge x$ (since $x_2$ should be positive) (?).  Is my approach correct ? MPPT charging a li-ion battery I just designed this MPPT solar charge controller which outputs 4.2 V regulated via MPPT, it uses the SPV1040 IC. This will be used on a satellite with 4 side panels, each with 8 solar cells in parallel outputting 2.6 V at approximately 122 Ma. Each panel will have this exact circuit and all the outputs will be connected in parallel to a 3.7 V lithium ion camera battery.  When active only 1 solar panel will be active and the other MPPT circuits will be offline or providing a very low amount of current. How should I connect the 4.2v to the battery, just directly or using some sort of controller to stop charging at 4.2v? Keep in mind the battery will always have a load and it will charge for around 50m in a 90m cycle due to being eclipsed. I would obviously like the battery to last as long as possible so I am not sure what are the implications of constantly having a battery connected to 4.2 V during 50m. Thanks.   The schematic was made based off a simulation made using the ST Edesignsuite directly from the provider.  Excuse my previous question about this, wasn't properly worded. Blurry Text On Model After Render How did the writers not notice that the mailbox of Leela's parents says "Turanga," and not "Leela"?  Turanga Leela is Leela's proper name, with Leela being her surname, so shouldn't it follow that her folks would be the same?