If $n,m\in\mathbb{N}$ and $n$ is odd, prove that $\gcd(2^n-1,2^m+1)=1$.
Showing $\gcd(2^m-1,2^n+1)=1$
Given two increasing continuous functions $f,g$ prove that $(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx$ Given two monotonically increasing continuous functions $f,g$ prove that  $$(b-a) \int^b_a f(x)g(x) dx &gt; \int^b_a f(x) dx \int^b_a g(x) dx,\; b&gt;a$$  what I have tried:  Let $$h(x) = (x-a)\int^x_a f(t)g(t) dt - \int^x_a f(t)dt\int^x_a g(t)dt $$ and trying to figure out that $h'&gt;0$ using MVT to reform $h'(x)$ as   $$h'(x) = (x-a)(-f(x)g(\xi_1) - g(x)f(\xi_2) + f(x)g(x) + f(\xi_3)g(\xi_3))$$ where $\xi_{1,2,3} \in (a,x)$ (That method does not seem to work) Set shortcuts to change keyboard layout in Windows 10? Is there any way to set keyboard shortcuts to quickly change to a different language/keyboard layout? The old shortcuts still seem to work (when set back in Windows 7), but I can't seem to find anything to change it in Windows 10. Number of equivalence relations on a set If a set has $n$ elements then what are maximum number of equivalence classes and equivalence relations possible on it? Rotate whole animation by 90 degrees I make my animation and now I realized that its pointed in wrong direction I used -y (native blender) as direction but I need to be pointed at x axis so what would be the easiest way to rotate whole animation, to switch -y with x :) How to use z-index in svg elements? Soft Thresholding CUDA implementation Where should I go to ask a question about guns? Are C-style macro names subject to the same naming rules as identifiers?  After a compiler upgrade, it is now emitting this warning for a legacy application:  warning #3649-D: white space is required between the macro name "CHAR_" and its replacement text   #define         CHAR_&amp;        38   This line of code is defining an ASCII value constant for an ampersand.  #define   DOL_SN        36 #define   PERCENT       37 #define   CHAR_&amp;        38 #define   RT_SING       39 #define   LF_PAR        40   I assume that this definition (not actually referenced by any code, as far as I can tell) is buggy and should be changed to something like "CHAR_AMPERSAND"? Does the verb &quot;provide&quot; always have to be used with &quot;with&quot;? For example,  Can you provide me with some good examples? Can you provide me some good examples? Can you provide some good examples?  I suppose it's a transitive verb, isn't it? Moreover, is the following type of usage correct?  You should provide food for your dog before you go on vacation.  If you could give me some good examples regarding &quot;provide&quot; with different types of usage, I'd appreciate that. How the `\alignat` works in ams package I have read the related question(), which had detailed answer but still don't know what the meaning of the \alignat's argument. It is said in  that      This environment takes one argument, the number of “equation   columns”: count the maximum number of &amp;s in any row, add 1 and and   divide by 2.   What is the meaning of adding one in &amp; number and divide by two? I presumed the argument is given by the user, not automatically generated by the system? How can i limit the character length in excerpt? Possible Duplicate:          I have a question after reading this post (). I like this function(Search Term Without Plugin) very much but the character length is too long. What php code should i add to make the excerpt shorter? Would appreciate if someone can suggest it. Thank you! Why use hex? Hey! I was looking at this code at   I noticed that in some situations they used hex numbers, like in line 134:  for (j = 1; val &amp;&amp; j &lt;= 0x80; j &lt;&lt;= 1, q++)   Now why would they use the 0x80? I am not that good with hex but I found an online hex to decimal and it gave me 128 for 0x80.  Also before line 134, on line 114 they have this:  small_n = (n &amp; 0xffff0000) == 0;   The hex to decimal gave me 4294901760 for that hex number. So here in this line they are making a bit AND and comparing the result to 0??  Why not just use the number? Can anyone please explain and please do give examples of other situations.  Also I have seen large lines of code where it's just hex numbers and never really understood why :( Paging Lucene's search results I am using Lucene to show search results in a web application.I am also custom paging for showing the same. Search results could vary from 5000 to 10000 or more. Can someone please tell me the best strategy for paging and caching the search results? Will hardware changes affect system? Story where vampires are created by drop absorbed on neck? I have been trying to identify a book that I read maybe 25-30 years ago (1987-1992), in which there are vampire or vampire-like creatures, but instead of being created in the traditional way (e.g. bite on the neck), something drops a drop of a fluid on the back of their neck, which is then absorbed into the body and transforms them. I have a vague notion that some sort of tree is what creates these drops. The drops may have had a shiny metallic appearance.  I've tried to find this book before and always came up short. I'm starting to think the whole thing was a dream and I never read it at all. But I really feel like it was a book or at least a short story. Is it possible that $a_n>0$ and $\sum a_n$ converges then $na_n \to 0$? (without assuming $a_n$ is decreasing) When I cut a layer using marquee tool, I get blur edges to cut side. How do I get sharp edges after cut? I am using rectangular marquee tool to cut a portion of the layer in photoshop cs5, but after cutting, the edges are not sharp, I get blur edges.      how can I get sharp edges? I have this strange problem with conky. It doesn't show properly on the desktop:    I've tried everything from editing conky widget file to reinstalling conky and conky manager (v.2.4). Who knows how to fix it? We often get questions on bicycles.stackexchange about fitting wider or narrower tires to bicycles. We have a large number of these with answers suiting the specific bike or style of bicycle in question, but I have been unable to find one that provides a generic answer covering off all things that need to be considered. For most bicycles, the answer is the same regardless of the specific style and sizes of tires in question.   This is intended as a canonical question that we can point closed questions to. I got a problem while making a model of a product. I have tried a lot even seen lots of tutorial but did not get the thing that i want. I am using Blender 9.92 and I am just a beginner I got my hands on Blender 7-8 days ago. I've just basic knowledge about I am trying to model a product &quot;Mattress Topper&quot;   This is the exact look of the product, Help me how i can model this product and render realistic images.