Let $n$ and $d$ denote integers. We say that $d$ is a divisor of $n$ if $n = cd$ for some integer $c$. An integer $n$ is called a prime if $n &gt; 1$ and if the only positive divisors of $n$ are $1$ and $n$. Prove, by induction, that every integer $n &gt; 1$ is either a prime or a product of primes.  My try: First, that there's nothing to prove because a number is always a prime or not, so do not what to think. Step:  $P(n): n$ is either a prime or a product of primes. If n=2 then 2 is prime.  $P(n): True$ I want to see $P(n) \rightarrow P(n+1)$ If $n$ is a prime then  $2$ is a divisor of $p+1$, then is a product of primes. If $n$ is a product of primes... I can't say anything about $n+1$.  Some help... please.
I'm trying to get through Spivak's Calculus on my own and even though I kinda understand induction I'm not so sure that's the case when it comes to complete induction. So I tried to do a starred problem which involves using it. But I'm not sure that my proof is valid, can someone check this for me?  So the problem is that I have to prove that     For every natural $n&gt;1$, if $n$ is not prime we can write it as a product of primes.   I'm starting with a base case. For $n=2$, $n$ is prime, so the assumption holds.  Let's assume that for some $n &gt; 1$ and also all numbers $p &lt;= n$ the assumption holds.  Then $n+1 = cd$. If $n+1$ is not prime, then $c &lt; n+1$ and $d &lt; n+1$, but as they are natural numbers, we can also write that $c &lt;= n$ and $d &lt;= n$. But we assumed that if $p &lt;= n$, and $p$ is not prime, we can write it as a product of primes. So if $c$ or $d$ are not primes, we can write them as a product of primes. That means that $n+1$ can be written as a product of primes if it's not prime, which completes the proof.
Methods to work around the problem of missing data in machine learning Virtually any database we want to make predictions using machine learning algorithms will find missing values ​​for some of the characteristics.  There are several approaches to address this problem, to exclude lines that have missing values ​​until they fill with the mean values ​​of the characteristics.  I would like to use for a somewhat more robust approach, which would basically run a regression (or another method) where the dependent variable (Y) would be each of the columns that have missing values ​​but only with the rows of the table that contain all the data , and predict the missing values ​​with this method, complete the table by the table and move to the next 'column' with missing values ​​and repeat the method until everything is filled.  But that gives me some doubts.  Why any column start? I believe that the one with the smallest missing values ​​until the one with the most  Is there any threshold of missing values ​​that is not worth trying to complete it? (for example, if this characteristic only has 10% of the values ​​filled would not it be more interesting to exclude it)  Is there any kind of implementation in traditional packages or other methods that are robust to missings? Why is the last digit of $n^5$ equal to the last digit of $n$? Profile changes in Q&A are not pushed to Area 51 Object of class mysqli_result could not be converted to string I have designed a circuit where I want to give a pulse of very short duration using a switch. How can I implement it in spice. memoryless property of exponential distributions with random variables It is true that $P(X&gt;t+s|X&gt;t)=P(X&gt;s)$ for certain values $t$ and $s$.  However, how can I show that this still holds if:   $T$ is a continuous random variable. That is $P(X&gt;T+s|X&gt;T)=P(X&gt;s)$ Both $T$ and $S$ are continuous RVs: $P(X&gt;T+S|X&gt;T)=P(X&gt;S)$   All the random variables are independent. change terminal title I want to make a shell alias that starts a certain program and also changes the title of the terminal to the name of the program.  How can I do that?  In case it matters, I am running the terminal on Ubuntu and connecting to a Debian server via ssh. Limiting Length of Input in JTextField is not working Let $R$ be a commutative ring.  The nilradical $\text{nil}(R)$ is the set of all nilpotent elements, and it is the intersection of all the prime ideals of $R$.  Is the following true in the polynomial ring? $$\text{nil}(R[X])=(\text{nil}(R))[X]$$  Any polynomial in the right hand side has all its coefficients nilpotent in $R$, hence also in $R[X]$, and therefore it is nilpotent in $R[X]$ as the sum of nilpotent elements.  The other inclusion is not clear to me.  Is there a counterexample? I'm trying to figure out a good workflow for developing scripts and add-ons in Blender. Here's what I'm currently doing as I try my luck with a PyNode add-on:   Open blender Create a text editor panel Open my script (external text file) Edit my script in a nice external editor Reload the script Save the .blend file Run the script   Now that I can see what my code does, this Blender instance has been modified, so I can't (make changes to my script and) run my script again. So I do this:   Close blender Open blender with the .blend from step 6 and repeat steps 4-7, over and over and over..   Is there a better workflow for this sort of thing? Write $\frac 11 + \frac12 + ...\frac1{(p-1)} =$ $\frac ab$.Such that $(a,b)=1$. Show that $p^2 \mid a$ $\text{if}$ $p\geq 5$.  I was trying to apply something with prime modulus, but I am unsure how exactly to go about it. How to set up passwordless SSH access for root user I need to configure a machine so software installation can be automated remotely via SSH. Following , I was able to setup SSH keys so my user can access the machine without a password, but I still need to manually enter my password when I use sudo, which obviously an automated process shouldn't have to do.  Although my /etc/ssh/sshd_config has PermitRootLogin yes, I can't seem to be able to log in as root, presumably because it's not a "real" account with a separate password.  How do I configure SSH keys, so a process can remotely log in as root on Ubuntu? What is the spell save DC and spellcasting modifier for a Rogue with Magic Initiate? As a rogue I'm planning to take Magic Initiate with warlock class. The Magic Initiate feat says:     [...] Your spellcasting ability for these spells depends on the class you   chose: Charisma for warlock [...]    cantrip 1: Booming blade or Green flame blade cantrip 2: Minor illusion 1st-level spell: Hex    My stats bonuses are: STR : -3 | DEX +6 | INT 0 | WIS +1 | CHA +4 | prof : 4  Questions:   Green flame blade :      [...] The second creature takes fire damage equal to your spellcasting   modifier. [...]   What is my spellcasting modifier? 0 from INT as rogue or +4 from CHA as warlock?   Minor illusion :      [...] If a creature uses its action to examine the sound or image, the   creature can determine that it is an illusion with a successful   Intelligence (Investigation) check against your spell save DC. [...]   What is my spell save DC? 12 (8 + 0 from INT + 4 from prof) or 16 (8 + 4 from CHA + 4 from prof)? I'm looking for the antonym of -cide (as in suicide, pesticide, etc.). Essentially the ending to words that would mean "to create or embrace". Why is asymmetric cryptography bad for huge data? I've been told that asymmetric cryptography requires that the message to be encrypted be smaller than its key length. Why is this?  I know about hybrid encryption, which . But I still want to know why public-key cryptography needs the data to be shorter than the key length. How to find the period of the sum of two trigonometric functions "Missing return statement" within if / for / while Single quote within double quotes and the Bash reference manual In section 3.1.2.3  titled Double Quotes, the Bash manual says:      Enclosing characters in double quotes (‘"’) preserves the literal   value of all characters within the quotes, with the exception of ‘$’,   ‘`’, ‘\’, and, when history expansion is enabled, ‘!’.   At the moment I am concerned with the single quote(').  It's special meaning, described in the preceding section, section 3.1.2.2 is:     Enclosing characters in single quotes (') preserves the literal   value of each character within the quotes. A single quote may not   occur between single quotes, even when preceded by a backslash.   Combining the two expositions,    echo "'$a'"   where variable a is not defined (hence $a = null string), should print $a on the screen, as '', having it's special meaning inside, would shield $ from the special interpretation. Instead, it prints ''. Why so? Andrew Price released a nice tutorial on how to nicely put 3D objects on top of a real image:   I have seen images, such as this:  which have 3D elements over images and have actual reflections from the object showing up on the wet asphalt.  How can this effect be achieved using render passes (or any other way)?  Here is another example (from ): I am having difficulty in how to calculate the Pearson's $\chi^2$ statistic for a contingency table with entries: (3, 0) and (0, 3) (row-wise).  I was studying Agresti's categorical data analysis and found this table. He calculated it as 6.  Through what calculation did he reach this answer?