Consider the polynomial $$p(x) = a_0 + a_1x + a_2x^2 + · · · + a_nx^n$$ where $a_0, a_1, . . . , a_n ∈ \Bbb Z$. Show that if $p(x_i) = 7$ for 4 distinct integers $x_0, x_1, x_2, x_3$, then $p(z) \neq 14$ for any $z ∈ \Bbb Z$.    How do I start this question?
Given a polynomial $f(x)$ with integral coefficients and $f(x) = 37$ has 5 distinct integral roots, find the number of integral roots of $f(x) = 41$?  My Approach: Say $f(x) = (x-r_1)(x-r_2)(x-r_3)(x-r_4)(x-r_5)g(x) + 37$, where $r_i$ are the distinct integers.   Now for $f(x) = 41$ we have $(x-r_1)(x-r_2)(x-r_3)(x-r_4)(x-r_5)g(x) = 4$, so the factors can be $\pm 1, \pm2$ or $\pm 4$. Given $r_i$ are distinct at most two of them will give $\pm1$, then there can be both of $\pm 2$ or one of $\pm 4$. This is where I get lost, since even if I use all of $\pm1, \pm2$, I will be still be left with one $x-r_i$ factor. What about that? Does it matter that 37 and 41 are primes, or is it just a coincidence?  Thanks in advance.
How find this limit $\lim\limits_{x\to 0^{+}}\frac{\sin{(\tan{x})}-\tan{(\sin{x})}}{x^7}$ Let $\int_{- \infty}^{\infty} f(x) dx =1$. Then show that $ \int_{- \infty}^{\infty} \frac{1}{1+ f(x)} dx = \infty.$ I am a software engineer but try to keep up with maths as I really enjoyed the subject at school. I just saw a great TED talk:   The talk states that the current largest known prime is $2^{57885161}-1$  This got me thinking - how does one actually prove whether a number this large is prime or not?  Is it just done by brute force via a computer algorithm or are there any well known mathematical techniques for doing so?  I would be really grateful for any pointers as I am eager to learn about these techniques (if they exist). How to Disable Copy Paste (Browser) How to test that Apex Batch finish() sends an email? I have a Batch class that sends an Email from its finish method and want to test that. a But....I cannot test it. The the above test fails as there was no email invocation at all. BUT: I see it in the Limits view of my Dev Console that indeed one email invocation happens. Why?!  The batch class:  public MyBatch implements Database.Batchable&lt;SObject&gt; {      ...      public void finish(Database.BatchableContext context) {         sendEmail();     } }   The test method:  @isTest private static void sendsEmailOnFinish() {      Test.startTest();        MyBatch b = new MyBatch();     Database.executeBatch(b);     Test.stopTest();       // Verify     System.assertEquals(1, Limits.getEmailInvocations()); }   I thought this might be connected to using Test.startTest() and stopTest() but even moving the assertEquals before the Test.stopTest() doesn't make my test run. change citation style ( [1-4] to [1]-[4], and [5, 8-9] to [5], [8]-[9] ) with natbib? I have spent a whole day trying to solve the following problem. Please help if you know how to do this.  I am using natbib (\usepackage[sort&amp;compress,square,comma]{natbib}), and my current citation style looks like [1-4], [5, 8-9].  What is the easiest way to change the citation style to [1]-[4] and [5],[8]-[9]? It seems that containing lava in solid glass, stone and iron blocks may not be sufficient to prevent wool from burning, even if the lava is completely encased.         After some delay, it seems that fire will jump onto the wool above the iron.   Are solid blocks more effective than airspace, or not? If so, Does it matter whether the corner block is filled in? What are the exact rules on distance that I need, and how can I prevent flammable blocks from being eliminated by nearby lava? Bonus: Does a flowing lava block behave differently than a lava source block when it comes to fire creation? TikZ or cryptocode package for cryptographic protocol diagrams How to remove the lines which appear on file B from another file A? A Regex that will never be matched by anything Related:  (not a dupe, that's a feature request and was closed as a dupe of another one)        (old) post does indeed say that Link-only answers can be flagged as NAA, but there are a lot of posts like  which define Not An Answer is a way that it doesn't apply to link-only answers.  Can we get some up-to-date consensus on this? That link only answers are not allowed is . Can you explain more?  I do not know when we can use them in the sentence.  Can we use them instead of each other?     They have a beautiful home.   They have a beautiful house. Take the following C# class:  c1 {  event EventHandler someEvent; }   If there are a lot of subscriptions to c1's someEvent event and I want to clear them all, what is the best way to achieve this? Also consider that subscriptions to this event could be/are lambdas/anonymous delegates.  Currently my solution is to add a ResetSubscriptions() method to c1 that sets someEvent to null. I don't know if this has any unseen consequences. Recently, I have answered a few questions that were upvoted by factors more than the accepted answer, but not accepted. , at the time of my writing, has an accepted answer that the asker said was not what he wanted. I don't know why. In the future, when people view this, people might only look at the accepted answer and move on. Would it be possible to have a "community accepted" feature? Maybe only the people at the top can vote to accept it or something. How much rep, if any, should go to that person? Order and degree of a differential equation Suppose $f$ is a continuous single-variable function, prove that:  $$2\int_a^b \int_a^x f(x)f(y) \, dy \, dx = \left[ \int_a^b f(x) \, dx \right]^2$$  This question was just on my Calculus III final exam and I couldn't figure it out, could someone provide some insight, please? What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$? /A problem from the 2012 MIT Integration Bee is $$ \int_0^1\frac{x^7-1}{\log(x)}\mathrm dx $$ The answer is $\log(8)$. Wolfram Alpha gives an indefinite form in terms of the logarithmic integral function, but times out doing the computation. Is there a way to do it by hand? bi-directional sync with rsync Generalizing $\,r(n^2) = r(n)^2,\,$ for $\,r(n) := $ reverse the digits of $n$ I'm assuming this theorem was found by someone else before, but I found this relationship between square numbers of 3 digits or less. The theorem is this: If you reverse the digits in a square number, then the result will also be a square number. Take the square 961. It is 31 squared, and if you reverse the digits you will get 169, which is also a square number. Plus, 31 and 13(the roots of these reversed squares) are also reverses of eachother. The problem is this breaks with 4 or more digits. If I take the square 1024 and reverse the digits, I get 4201, which is not a square. How can I expand this theorem to fit 4 or more digits? I need to graph a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? I can't find it anywhere.