smallest number larger than x with sum of digits y
smallest number for given sum of digits
This is the proof from the book:     Theorem. There are infinitely many primes of the form $4n+3$.    Lemma. If $a$ and $b$ are integers, both of the form $4n + 1$, then the product $ab$ is also in this form.  Proof of Theorem: Let assume that there are only a finite number of primes of the form $4n + 3$, say  $$p_0, p_1, p_2, \ldots, p_r.$$ Let $$Q = 4p_1p_2p_3\cdots p_r + 3.$$ Then there is at least one prime in the factorization of $Q$ of the form $4n + 3$. Otherwise, all of these primes would be of the form $4n + 1$, and by the Lemma above, this would imply that $Q$ would also be of this form, which is a contradiction. However, none of the prime $p_0, p_1,\ldots, p_n$ divides $Q$. The prime $3$ does not divide $Q$, for if $3|Q$  then $$3|(Q-3) = 4p_1p_2p_3\cdots p_r,$$ which is a contradiction. Likewise, none of the primes $p_j$ can divides $Q$, because $p_j | Q$ implies $p_j | ( Q - 4p_1p_2\cdots p_r ) = 3$, which is absurd. Hence, there are infinitely many primes of the form $4n +3$.   END  From "however, none of the prime ...." to the end, I totally lost!    My questions:   Is the author assuming $Q$ is prime or is not? Why none of the primes $p_0, p_1,\ldots, p_r$ divide $Q$? Based on what argument?     Can anyone share me a better proof?     Thanks. Is there a minimum damage rule? In D&amp;D3e, if you successfully hit a target you always did at least 1 point of damage (before applying Damage Resistance), even if you had a Strength penalty.  I can't find any mention of that rule in the 5e PHB.  Does the rule no longer apply, or did I just overlook it? Density of random variables Let $X$ and $Y$ be independent random variables with respective densities: $f_X (x) = xe^{-\frac{x^2}{ 2}} 1_{]0,+\infty[} (x)$ and $ f_Y (y) =  \frac{1}{\pi \sqrt{1-y^2}} 1_{]-1,1[} (y)$  Let $U:=XY$ and $V:=X\sqrt{1-Y^2}$  I try to check whether U and V are independent and which law they follow.  What I did:  After some calculations, I found the density of $(U,V)$:  $ f_{(U,V)}(u,v)=\frac{1}{\pi } e^{-\frac{u^2}{ 2}}1_{\mathbb{R}}e^{-\frac{v^2}{ 2}}1_{\mathbb{R}_+^* }$ Which mean that $(U,V)$ follows a $ \mathcal{N}(0,1)$ law (because $u^2+v^2=x^2$) and that  U and V are independent because $f_{(U,V)}$ can be factored as the products of densities of $U$ and $V$  My question:  There is a factor $\frac{1}{\pi}$ and I don't know to which density it will belong: to $U$ density or to $V$ density?  Many thanks! I have a Mac that runs Snow Leopard and I intend to upgrade it to Mountain Lion and sell it on ebay. However, I only have one hard-drive, and I'm sure I can't use something in my OS to delete my OS and all user data.  Is there something for this that's built into the firmware that I can do on boot, or otherwise?   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Does the fact that the input points come from the same underlying function make them somehow correlated or posses commonalities ? and if not why do most methods (ex: Non-parametric splines, Gaussian process) use the $y_i$'s to get information and learn ${y_\star}$ ? Isn't the fact that if samples are independent then they can't learn new information from each other ? How do I space quads evenly? Is there a way in blender or an addon that allows me to select a bunch of faces (or verts) and evenly distributes (or relaxes) them into quads clean quads.  Without screwing up the vert ID/order.  Edit: I would like to have the shapes of my quads to turn into perfect squares while maintaining the volume of the object and the vertex order. Divergent infinite series $n!e^n/n^n$ - simpler proof of divergence? $$ \sum_{n=1}^\infty \frac{n!e^n}{n^n} $$ Where $e$ is Euler's number.  Recently on calculus class we were covering convergence tests and my group got stuck with this infinite series. 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The blank line beneath the row with the avatar of the ignoree is quite annoying.    I use Google Chrome on OS X.  There was a  tagged  but it still looks weird. Why aren't Christoffel symbols tensors? - asked from a fibre bundle perspective Definite article before name of languages Should I use the definite article before name of languages?   (the?) Persian language (the?) German language (the?) Arabic language (the?) Russian language (the?) English language   Some language name may also refer to other things, so you need language word after their name. 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Why are one-way plane tickets more expensive than return tickets? Best word wrap algorithm? Word wrap is one of the must-have features in a modern text editor.  How word wrap be handled? What is the best algorithm for word-wrap?  If text is several million lines, how can I make word-wrap very fast?  Why do I need the solution? Because my projects must draw text with various zoom level and simultaneously beautiful appearance.  The running environment is Windows Mobile devices. The maximum 600&nbsp;MHz speed with very small memory size.  How should I handle line information? Let's assume original data has three lines.  THIS IS LINE 1. THIS IS LINE 2. THIS IS LINE 3.   Afterwards, the break text will be shown like this:  THIS IS LINE 1. THIS IS LINE 2. THIS IS LINE 3.   Should I allocate three lines more? Or any other suggestions? ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­ Repeated squaring method How do I use the repeated squaring method to calcualte 2^176 (mod 177)? 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