Proof: For any positive integer $n$, there exists a prime whose first digits are the digits of $n$ Conjecture:      For any positive integer $n$, there exists a prime whose first digits   are the digits of $n$.   Proof:  The infinite set of integers starting with the digits of integer $n$ is distributed as $1$ sequence of consecutive integers for each power of $10$. For example, the integers starting with the same digits as $2$ are those between $20$ and $30$, between $200$ and $300$, between $2000$ and $3000$ $…$ Now let's say $n=1245$, and let's look at much larger integers than $n$. For example, all integers $x$ in the following interval starts with digits $1245$: $$ 12450000000000000000000000000 ≤ x &lt;12460000000000000000000000000 $$  A range of this magnitude is proven to contain at least $1$ prime number, so whether $n$ is a single digit or a million digits long, we only need to add the proper number of zeros to reach an interval where having no prime is impossible.  Questions:   Is my proof valid ? Did I miss something ? Is there other existing proofs of that ?   Of course any suggestions/edits to make it more clear or professional are welcome, I'm still a newbie at proof writing!
Proof that there are infinitely many prime numbers starting with a given digit string To prove the following fact: given any sequence of digits in any base, eg 314159265358979323 base 10, there are infinitely many primes that start with these digits,eg when expressed in decimal they start with 314159265358979323. I think using a prime number theorem will be helpful . But I am still not getting a point of how to start the proof.
Need help to understand Maximum Likelihood Estimation for multivariate normal distribution? In maximum likelihood estimation for the parameter, mean $\mu$ of multivariate normal distribution, I end up with the following result:  $\mu_{MLE} = \dfrac{1}{n}\sum_{i=1}^n x_i$  The way I understand the above equation is the following way below, but I am wondering if the above equation means the following equation:  $\mu_{MLE} = \begin{bmatrix}\mu_1\\\mu_2\\\vdots\\\mu_n\end{bmatrix} = \begin{bmatrix}\dfrac{1}{n}\sum_{i=1}^n x_1\\\dfrac{1}{n}\sum_{i=1}^n x_2\\\vdots\\\dfrac{1}{n}\sum_{i=1}^n x_i\end{bmatrix} = \begin{bmatrix}E(X_1)\\E(X_2)\\\vdots\\E(X_n)\end{bmatrix} = \begin{bmatrix}\bar{X_1} \\ \bar{X_2} \\ \vdots \\ \bar{X_n}\end{bmatrix}$   And also Maximum Likelhood Estimation for $\Sigma$, I have ended up with the following equation:  $\Sigma_{MLE} = \dfrac{1}{n}\sum_{i=1}^n(x_i-\mu)(x_i-\mu)^T = \begin{bmatrix}Var(\sigma_{11}) &amp; Cov(\sigma_{12}) &amp; \dots &amp; Cov(\sigma_{1n}) \\ Cov(\sigma_{21}) &amp; Var(\sigma_{22}) &amp; \dots &amp; Cov(\sigma_{2n}) \\ {}&amp; \vdots&amp;  \\ Cov(\sigma_{n1}) &amp; Cov(\sigma_{n2}) &amp; \dots &amp; Var(\sigma_{nn})  \end{bmatrix}$  Somehow, I got the equation following some algebraic steps, but I can't see how this part, $\dfrac{1}{n}\sum_{i=1}^n(x_i-\mu)(x_i-\mu)^T$, works and produces the covariance matrix. Hope to hear some explanations. Can a bird, previously at earth potential, get electrocuted by landing on a powerline at high-enough voltage due to the initial "equalization charge"? At the voltage levels of typical overhead transmission lines in the US, a bird can land on one and be just fine (as long as it doesn't do something like spread its wings and touch a tree or something else at lower electric potential).  However, what about a hypothetical powerline at much higher voltage (as in tens of megavolts). Could landing on such a powerline fatally-shock the bird even though it does not complete a circuit for sustained current? (Assume that the distance is long enough that electrical arc'ing is impossible.)  NOTE: My understanding of what happens when a bird flies from an earth object to a powerline (please correct me if I'm wrong) is that - upon contacting the wire - its electric potential changes from earth-potential to the powerline's potential. In order for this to happen, there is an initial transfer of electrical energy (i.e. flow of charge i.e. current) from the powerline to the bird which "equalizes" their electric potential, which happens nearly instantaneously. If this is correct, then my question can be restated more generally as "Can an 'equalization charge' such as this result in a fatal shock, if the potential difference that it's equalizing is high enough?" How can i get my community Base URL in apex I am using  Salesforce Community and I have network Id of the community but I don't know how can I get Base URL of my community. Any one know how can I get my community base URL using its id in apex as a internal user.  Is there any other way in which salesforce internal user can get the base URL of all his community in the apex also i don't find any object in which Community Base URL is stored. I checked network sObject but it only store community path prefix not the complete URL so how can i get the base URL for my community? How do I change extension of multiple files recursively from the command line? $\Bbb Z^\ast$ What is this notation? What does $\Bbb Z^\ast$ mean? I would think some subset of the integers but I cannot find a definition. Thank you. Should the first word after a colon be capitalized? Should the first word after a colon be capitalized? Which of the following is correct?   For example: This. For example: this. How is voltage always able to increase, if electrons move at the speed of light? Why are the laws of thermodynamics "supreme among the laws of Nature"? Eddington wrote     The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.   and Einstein wrote     [classical thermodynamics] is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability of its basic concepts.   Why did they say that? Is it a very deep insight they had, or is it something one can be convinced of quite easily? Or even, is it trivial? Why does printf() not print anything before sleep()? I am considering buying a Dell XPS 15 laptop. The laptop has a Nvidia 420m card which should work with Linux but I keep reading that the Nvidia Optimus technology isn't supported on Linux. I am not really interested in switching from Nvidia to Intel to save power but need to know that the Nvidia card will in fact work if I installed Ubuntu.  If anyone has experience using a Nvidia card with Optimus technology or even better the exact laptop in question (Dell XPS 15 with 1GB NVIDIA® GeForce®  GT 420M) it would be great.  A major problem holding people back from adopting Linux is this sort of hardware issue. I am a long term Ubuntu user and supporter who can't afford to make a mistake with a purchase like this. I don't want to spend £500+ to find I have no graphics acceleration because Windows 7 is not an option. Why isn't Higgs coupling considered a fifth fundamental force? When I first learned about the four fundamental forces of nature, I assumed that they were just the only four kind of interactions there were. But after learning a little field theory, there are many other kinds of couplings, even in the standard model. So why isn't the Yukawa Higgs coupling considered a fifth fundamental force? To be a fundamental force, does there need to be a gauge boson mediating it? At one point, I decided to make friends with the low-lying spectrum of QCD. By this I do not mean the symmetry numbers (the "quark content"), but the actual dynamics, some insight.  The pions are the sloshing of the up-down condensate, and the other pseudoscalars by extending to strangeness. Their couplings are by soft-particle theorems. The eta-prime is their frustrated friend, weighed down by the instanton fluid. The rho and omega are the gauge fields for flavor SU(2), and A1(1260) gauges the axial SU(2), and they have KaluzaKlein-like echoes at higher energies, these can decay into the appropriate "charged" hadrons with couplings that depend on the flavor symmetry multiplet. The proton and the neutron are the topological defects. That accounts for everything up to 1300 but a few scalars and the b1.  There are scalars starting at around 1300 MeV which are probably some combination of glue-condensate sloshing around and quark-condensate sloshing around, some kind of sound in the vacuum glue. Their mass is large, their lifetime is not that big, they have sharp decay properties.  On the other hand, there is nothing in AdS/QCD which should correspond to the sigma/f0(600), or (what seems to be) its strange counterpart f0(980). While looking around, I found this discussion: . The literature that it pointed to suggests that the sigma is a very unstable bound state of pions (or, if you like, tetraquarks).   gives strong evidence for an actual pole;  gives a more cursory review. The location of the pole is far away from the real axis, the width is larger than the mass by 20% or so, and the mass is about 400MeV. The authors though are confident that it is real because they tell me that the interpolation the interactions of pions is safe in this region because their goldstone properties dominate the interactions. I want to believe it, but how can you be sure?  I know this particle was controversial. I want to understand what kind of picture this is giving. The dispersion subtraction process is hard for me to visualize in terms of effective fields, and the result is saying that there is an unstable bound state.  Is there a physical picture of the sigma which is more field theoretical, perhaps even just an effective potential for pions? Did anyone who convinced himself of the reality of the sigma have a way of understanding the bound state properties? Is there an analog unstable bound state for other goldstone bosons? Any insight would be welcome. When I create a clan in the multiplayer menu, it says clan created but I cannot add or invite people to my clan. I go to create a new one assuming the request was canceled or not entered correctly, it says clan name already exists as if someone else created it.   Is this a bug? Did I do it wrong? Or am I missing something?   I'm on the PS3. Will this also happen on my Xbox 360 Version? Is "Five dollars is all I have" correct? I want to emphasize that I only have 5 dollars, so can I use the sentence:     Five dollars is all I have.   Or must it be:      Five dollars are all I have.   Please give the explanation as well. Did Darth Sidious and Vader ever discover how to cheat death? In Episode 3 Revenge of the Sith Palpatine/Sidious strings Anakin along to the dark side with the promise that together the two of them will be able to re-discover Darth Plagueis' secret to cheating death.      "To cheat death is a power only one has achieved, but if we work together, I know we can discover the secret."   From the end of Revenge of the Sith to their deaths in Return of the Jedi did Darth Sidious and Darth Vader ever manage to re-discover this secret? Maxing out gear with maxed out smithing, enchanting and alchemy Can a hyphen be used without anything on the right side? An  is defined as $p = q\times d$ (for two point charges $\pm q$ separated by a distance $d$).    What is the physical meaning of this quantity?  Why does the direction of the electric dipole moment go from the negative charge to the positive charge? Another way of saying this, why electric dipole moment is a vector quantity? Janrain has , one of the many OpenID providers. MyOpenID will not be available starting February 1, 2014. (An   hinted at the same but did not set a date for end of service.)   This decision was made by Janrain, and Stack Exchange did not have any part in it. 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Here's how to add a new login to your account. You must be logged into your account when you begin. (This information is also available .)  Open up your user profile page (by clicking on your login name at the top of the screen) and click the "my logins" link above the bio box on the right hand side.   You should see a list of all of your logins.  To add a new login, click "add more logins..." at the bottom of the box. Choose the new login from the list and follow the steps to confirm.    If you choose, you can remove your myOpenID login by clicking the same "my logins" link, then clicking "remove" next to myOpenID.  If you are adding a Stack Exchange OpenID, you will receive a confirmation email. You must click the link in the email to confirm your email address before the Stack Exchange OpenID will show up under "my logins". Please make sure that you are already logged into your account when you click the confirmation link; otherwise, you will create a new account that needs to be merged with your old one.    If you encounter any trouble adding logins, or accidentally create a separate account that you would like merged, please fill out our . Combining two covariance matrices