the only solution of prime $q|(p^2-1)$ with $p<q$ is $p=2$ and $q=3$. I am trying to prove a lemma that the only solution of $q|(p^2-1)$ with $p&lt;q$ is $p=2$ and $q=3$. i.e if $p&lt;q$ are primes and $q|(p^2-1)$, then $p=2$ and $q=3$. Since $q|(p^2-1)$, then $q|(p+1)$ or $q|(p-1)$. Since $p&lt;q$, we have only $q|(p+1)$. But I am not sure if $q|(p^2-1)$ implies $q|(p+1)$ or $q|(p-1)$, since $p&lt;q,$ we have $q|(p+1)$. But I got stuck here.
If a prime divides a product then it must divide a factor of the product [Euclid's Lemma] We have $q, p$ prime and $tq = (p-1)(p+1)$ for some positive integer $t$. The book I'm reading says that because $q$ is prime, it divides $p-1$ or $p+1$.   Question: Why exactly does it matter that $q$ is prime here?
Series as an integral (sophomore's dream) I need help with this exercise.  I need to prove $$\int_{0}^{1}x^{-x}\ dx=\sum_{n=1}^{\infty}n^{-n}$$  I think I should use some convergence theorem, but I'm stuck.  Thanks a lot! Proper capitalization of commonly used acronyms and initialisms Say $X_1, X_2, \ldots, X_n$ are independent and identically distributed uniform random variables on the interval $(0,1)$.  What is the product distribution of two of such random variables, e.g., $Z_2 = X_1 \cdot X_2$?  What if there are 3; $Z_3 = X_1 \cdot X_2 \cdot X_3$?  What if there are $n$ of such uniform variables? $Z_n = X_1 \cdot X_2 \cdot \ldots \cdot X_n$? Can we balance the whitespace above/below headings? In January, some additional whitespace  below h1, h2 and h3 headings,  more whitespace below the headers than above them.  Could we have the whitespace above the headings be more than the whitespace below them?  I find some posts a bit hard to read, like the current rendering on the left:   &nbsp; &nbsp; &nbsp; Speak Slower or Speak Slowlier? AFAIK the correct grammar for "speak slow" is "speak slowly" (slowly being an adverb). Please correct me if I am mistaken.  But in daily life I have not heard anyone saying "Speak slowlier".  I think I heard folks saying "Speak slower" but I also think it is incorrect.  Which is the best or correct form to convey the message?  Could that be the longish "Speak more slowly?" So, for fun, I am taking some of the data of calls from the call center I work at and trying to do some hypothesis testing on them, specifically the number of calls received in a week, and using a Poisson distribution to fit it. Due to the subject matter of my job, there are two types of weeks, let's call one of them on-weeks where I hypothesize there are more calls, and off-weeks where I hypothesize there are fewer.  I have a theory that the $\lambda$ from on-weeks (let's call it $\lambda_1$) is larger than that of the one from off-weeks (let's call it $\lambda_2$)  So the hypothesis I want to test is $H_0: \lambda_1 &gt; \lambda_2, H_1: \lambda_1 \leq \lambda_2 $  I know how to test for one parameter (say  $H_0: \lambda_1 &gt; 1, H_1: \lambda_1 \leq 1 $) but not so sure how to go about doing 2 given a data set. Let's say I take two week's worth of data from each one $X_1 = 2$ and $X_2 = 3$ for the off-week and $Y_1 = 2$ and $Y_2=6$ for the on-week. Can someone help walk me though this simpler version such that I can apply it to a larger data set? Any help is appreciated, thank you. I'm fairly new to bicycling; apologies if this is a standard question but I couldn't figure out how to phrase it to search for it elsewhere.  I ride a KHS Urban-X with Kenda Kwest tires (just what it came with). I've gotten barely any use out of it since owning it, and it's been resting comfortably in the living room through the winter, so I decided today was a great day to get back on.   So I unscrewed the back tire cap, and began to hand-pump air into the tire. All seemed to go well, and the tire held pressure fine, but when I removed the tire pump, the pin appeared to remain pushed down slightly, and air flowed back out of the tire. I looked, and this is what I saw:    Several narrow gashes that appear as if the rubbed was ripped, cut, or separated somehow.  Questions   Is this a common occurrence? Again, I've ridden on this bike very little and thus filled up the tires even less. We have no animals and no bugs around that I know of that would gnaw or chew on such material. Is this fixable, or is my tube completely shot? It would seem to me that the latter is the case. Is there anything I can do to prevent this in the future? I use a hand pump but am gentle; would a stand-up pump be a better choice?  Do bike tires come with a standard warranty for cases like this, or do you know of any brands that do? If I have to purchase new tires, I'm going to purchase Is there any kind of "off-season" maintenance that I should be performing on my bike that could prevent things like this?   Thanks in advance for any help you can give! I have 13.04 installed from a DVD that I got on eBay but the application launcher is missing. I've tried CCSM but it won't work! How to prove polynomials with degree $n$ does not form a vector space? This is one of my linear algebra problems:     Prove that polynomials of degree $n$ does not (The professor made these words bold intentionally) form a vector space.   From what I read, the set of polynomials of degree $n$ should be a vector space, because:   There is an "One" and a "Zero" in this set; We can find inverse for addition and multiplication from this set; It follows all the axioms of addition. It follows all the axioms of scalar multiplication.   Then can someone give me some hints to prove it does not form a vector space? What is NDSolve`FEM`*? Numbering rows within groups in a data frame Is Borden's true identity real, or another ruse? Towards the end of , it's revealed that Borden accomplishes his Transported Man trick by using his twin brother, Fallon, as a double. It then becomes clear that he has been doing this in his personal life as well as on stage - each brother leads 'half a life', sometimes appearing as Borden and sometimes as Fallon. This is how he completes the illusion: one Borden goes into the first door, the other Borden emerges from the second door a moment later. Angier doesn't know how Borden does the trick, and doesn't believe Borden uses a double despite being told as much by Cutter. He follows what he believes to be a clue in Borden's notebook and travels to Colorado to meet Nikola Tesla, who Angier believes can build a machine that can help him perform the trick. Angier believes that Tesla built a similar machine for Borden. However, the final pages of the notebook reveal that Tesla was merely a red herring, designed to get Angier out of Borden's way for a long time. Tesla, it turns out, had never built such a machine before. At this point, Tesla agrees to build Angier's machine anyway. Instead of building a machine that transports matter, he builds one that duplicates it. Angier uses this machine in his act, but in order to avoid having to deal with multiple clones (the &quot;prestige materials&quot;), one of them must die as part of the act. This all makes sense, so far, but I find it hard to believe that Borden's use of the word 'Tesla' as the key to his cipher is random. It's too coincidental that he sends Angier on a wild goose chase, but Angier returns with a working machine. A film that is otherwise so tightly plotted shouldn't rely on coincidence to move the narrative forward, so there must be something else going on. Is it possible that Borden cloned himself using Tesla's machine? Can internal forces do work? How a Gmail username can be taken but not recognised simultaneously? Being at  I enter a desired username (without @gmail.com which is supposed), and I get error:     "That username is taken. Try another."   But when I try to login into Google with that xxx@gmail.com at  it fails with:     "Sorry, Google doesn't recognise that email."   How that can be?  P.S.: I want that username still. I want to be able to calculate the confidence interval from the estimated coefficient and respective standard errors.  I have a linear regression model which can be summarized (in R):  summary(fit1)  Call: lm(formula = bwt ~ height + weight + parity, data = data)  Residuals:     Min      1Q  Median      3Q     Max  -66.913 -10.624   0.991  10.979  55.621   Coefficients:             Estimate Std. Error t value Pr(&gt;|t|)     (Intercept) 31.19217   13.56879   2.299   0.0217 *   height       1.24964    0.23083   5.414 7.48e-08 *** weight       0.06781    0.02823   2.402   0.0164 *   parity1     -1.83309    1.19838  -1.530   0.1264     --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  Residual standard error: 17.9 on 1170 degrees of freedom Multiple R-squared:  0.04898,   Adjusted R-squared:  0.04654  F-statistic: 20.08 on 3 and 1170 DF,  p-value: 1.071e-12   With this model I can calculate the respective confidence intervals:  &gt; confint(fit1)                   2.5 %     97.5 % (Intercept)  4.57029503 57.8140351 height       0.79676227  1.7025207 weight       0.01243198  0.1231932 parity1     -4.18429933  0.5181151   I would expect the intervals of the predictor height to be given by  $$ 1.24964 \pm (1.96*0.23083) = [0.7972132,1.702067] $$  where 1.24964 is the estimated value for the coefficient and 0.23083 is the standard error for this coefficient. The numbers are close but not quite the same.  What am I doing wrong? Is  "choose from one of four options" wrong? I need backup in pressing my case that the phrase “choose from one of four options” is grammatically incorrect. Is there some resource that can prove my case, that the incorrect phrase should be replaced with one of the two following ones?        Choose one of four options   Choose from four options The cardinality of a language over a set of variables $V$ with $\#V= κ$ In predicate logic, to define a language, we consider the following $3$ sets:  (1) $V$, the set of variables;  (2) $F$, the sets of function symbols;  (3) $P$, the set of predicate symbols with their arities.   Now I am concidering about determine the cardinality of a language over a set with arbitary cardinality, $κ$(Note that $κ$ can be either finite, countablely infinite or uncountablely infinite ).  May I please ask:  If the cardinality of $V$, $\#V = κ$, what conditions on $V$ , $F$ and $P$ will give us a language over $V$, $F$ and $P$ with cardinality $κ$?  And may I please ask if it is the fact that if $\kappa$ is finite, then to get a language with cardinality $\kappa$ we need $(V+F+P)^{&lt;\omega}=(\kappa+F+P)^{&lt;\omega}=\kappa$, so we can never have a language of cardinality $\kappa$. I think it is something weird here. Am I wrong with something?  I had googled it but I have get nothing from websites, now I am confused. Could someone please help? Thanks so much!  EDIT: Latter I am told that in my course, the cardinality of a language refers to the set of expression instead of this of the set of strings. So now I need to consider further about when a string makes sense.  EDIT':Now I am considering that if we can find out a general way to determine the cardinality of expression, which may related to combinatorics. See this question:  Any help would be appreciate. Prove that the graph is connected I was wondering if someone can help me understand how prove that this graph is connected.  Given a graph with n vertices, prove that if the degree of each vertex is at least $(n − 1)/2$ then the graph is connected.  So far I know that about connected graphs:  An undirected graph is called connected if there is a path between every pair of distinct vertices of the graph  The distance between two vertices in a graph is the length of the shortest path between them.  The diameter of a graph is the distance between the two vertices that are farthest apart. Force newlines with cat wildcard printing I want to use cat with wildcards in bash in order to print multiple small files (every file is one sentence) to standard output. However, the separate file contents are not separated by a newline, which I'd like to for ease of reading.  How can I add a file delimiter of some sort to this command? State of the art in quantum memory Presently, how much information can a quantum computer store, in how many qubits? What restrictions are there and how does it vary across realizations (efficiency of data storage, ease of reading and writing, etc)?