We all know that $$15=3 \times 5$$  And $$15 =(-3) \times(-5)$$  Since $3 \neq -3$ and $5 \neq -5$ , we have two different prime factorizations !  Is this wrong ?   If this is wrong , then there are no negative primes !
Do we have negative prime numbers?   $..., -7, -5, -3, -2, ...$
The question is :   Show that $I=\langle 2+2 i\rangle$ is not a prime ideal of $\mathbb Z[i]$. Also find the number of elements in $\mathbb Z[i]/I$ and its characteristic.  My try:  I started with elements $2, 1+i \in \mathbb Z[i]$. Clearly, the product is $0$ in the coset representation.  The part where i am struck at is the number of elements and the characteristic.  Well, $2+2i+\langle 2+2i\rangle=0+\langle 2+2i\rangle$  Can we directly write $i^2=-1$ in the coset representation? How to understand intuitively the Stolz-Cesaro Theorem for sequences? I have to give a presentation on the theorem in Real Analysis with a fellow student. While I've looked over the  and verified that, yes, step B does indeed follow logically from step A, etc. and have internalized the proof to the extent that I can replicate it myself on paper, I still feel that I have made little progress as to why this theorem works the way it does, i.e. what is the intuition behind the theorem, such that it should make sense that it follows the way it does. Therefore I ask: what kind of intuition is there about this theorem? It would also be helpful to understand how the theorem can be used effectively in analysis.  I have seen it referred to as a sort of L'Hopital's rule for sequences, which certainly seems to make sense, but I similarly have little intuitive understanding of how that rule works, either.  I am asking this question not only for my personal understanding, but also for the sake of being able to present it in an illuminating manner, such that the rest of the class can also come away with the same sort of intuition of how the theorem works and how it is useful. Any help would be greatly appreciated.  EDIT: I should point out that the formulation of the theorem we are being tasked to prove is the following:  Let $\lbrace a_n \rbrace$, $\lbrace b_n \rbrace$ be sequences, ${b_n}$ strictly increasing and unbounded. Then if $\lim\limits_{n \to \infty} \frac {a_{n+1} - a_n}{b_{n+1} - b_n} = l$ for some $l \in \mathbb R$, then $\lim\limits_{n \to \infty} \frac{a_n}{b_n} = l$ also. Disable the "You've gone full screen" notification in Chrome What does env x='() { :;}; command' bash do and why is it insecure? There is apparently a vulnerability (CVE-2014-6271) in bash:   I am trying to figure out what is happening, but I'm not entirely sure I understand it. How can the echo be executed as it is in single quotes?  $ env x='() { :;}; echo vulnerable' bash -c "echo this is a test" vulnerable this is a test     EDIT 1: A patched system looks like this:  $ env x='() { :;}; echo vulnerable' bash -c "echo this is a test" bash: warning: x: ignoring function definition attempt bash: error importing function definition for `x' this is a test   EDIT 2: There is a related vulnerability / patch:  which uses a slightly different test:  $ env 'x=() { :;}; echo vulnerable' 'BASH_FUNC_x()=() { :;}; echo vulnerable' bash -c "echo test"   unpatched output:  vulnerable bash: BASH_FUNC_x(): line 0: syntax error near unexpected token `)' bash: BASH_FUNC_x(): line 0: `BASH_FUNC_x() () { :;}; echo vulnerable' bash: error importing function definition for `BASH_FUNC_x' test   partially (early version) patched output:  bash: warning: x: ignoring function definition attempt bash: error importing function definition for `x' bash: error importing function definition for `BASH_FUNC_x()' test   patched output up to and including CVE-2014-7169:  bash: warning: x: ignoring function definition attempt bash: error importing function definition for `BASH_FUNC_x' test   EDIT 3: story continues with: How to imitate commercial fried chicken? I'm trying to develop my own "secret" fried chicken recipe, something that is similar to Popeye's or Church's. I'm a LONG ways from getting there. Here's what I did in my last experiment:   Chicken was marinated in buttermilk with Cajun spices and Tabasco sauce for 24 hours I used a deep fryer and canola oil. 175C. Prior to "dressing" the chicken, I let the excess buttermilk brine drip off the chicken. I then seasoned it with Cajun seasoning I made a batter using an egg, self rising flour, milk, baking powder I also tried using just plain flour for breading   Results:   Using the batter, the chicken came out looking burned and too smooth (didn't have that crunchy look). It burned too much that I couldn't finish cooking the chicken. Using the flour, the chicken came out with a thin crust, somewhat crispy, but also too burnt. Dipping the chicken in the batter AND then flour gave a bulky &amp; crunchy look. However, again, the result looked too burnt.   Modifications:  I then changed the the breading. I used bread flour and corn starch. I dipped the chicken in the batter again, then used the new breading. The bread flour was a bit lighter. And the corn starch was even more lighter. But they were still burn looking (just lighter shades). I also dropped the temperature to 160C.  Results:  Burnt chicken whose crust wasn't really sticking to the skin/meat.  Questions:  Any suggestions on what I can do to improve? I'm guessing something in the batter is causing the burnt coat. What flour should I use? What can I do to make the coat stick to the skin/meat? Any other suggestions on what to do? Where to find my Favorites in Google Chrome? Why should edits be at least six non-space characters? Earlier today I got this message:  Oops! Your edit couldn't be submitted because:   Edits must be at least 6 non-space characters; is there something else to improve in this post?   However, the purpose of my edit was to change the formatting for a message to show code.  There may have been other grammatical mistakes that could have been, but in general it is possible that there would not be.  What's the correct protocol to handle a situation like this? C# getting its own class name If every prime ideal is maximal, what can we say about the ring? Suppose $R$ is a ring and every prime ideal of $R$ is also a maximal ideal of $R$. Then what can we say about the ring $R$? Series involving log $\sum_{n=1}^{\infty} \left( n\log \left(\frac{2n+1}{2n-1}\right)-1\right) = \frac{1-\log 2}{2}$ wireless network adapter supported on ubuntu? I always see pictures of the solar system where our sun is in the middle and the planets surround the sun. All these planets move on orbits on the same layer. Why? Facebook message notification showing blank message Since I updated the system to 10.11.5, the Facebook message notification is not showing message text.   Is this a bug in this software upgrade of OSX? Here in the Figure R5=50K. This resistor value is not in the equations to obtain output voltage, and in many tutorials non-inverting input is directly grounded without a resistor. What might be the reason to use it here? How to convert QString to std::string? Least rank of a matrix Let $A$ be a $7\times 7$ matrix such that it has rank $3$ and $a$ be $7\times 1$ column vector.  Then least possible rank of $A+(a a^T)$ is? ($a^T$ is transpose of the vector)  Intuitively I think it's $2$ as $a a^T$ has rank $1$, and the most it could affect is one row of $A$. In the last episode of season 7, The Doctor goes into his timeline to save Clara. It ends when they see his war regeneration and Clara faints. Did I miss a special episode between that one and the 50th anniversary special? Because they never explained how they escaped his timeline. In the Lord of the Rings it is said     "In Eregion long ago many Elven-rings were made, magic rings as you   call them, and they were, course, of various kinds: some more potent   and some less. The lesser rings were only essays in the craft before   it was full-grown, and to the Elven-smiths they were but trifles — yet   still to my mind dangerous for mortals. But the Great Rings, the Rings   of Power, they were perilous.”   [LotR I 2 (60)]   There is a clear distinction here between the 20 Rings of Power in the Ring Rhyme (the 7,9,3,and 1) and these other rings. What happened to the "lesser rings"? What is the adjectival form for the word "integrity?" I'm looking for the adjectival form of &quot;integrity.&quot; Instead of &quot;Be a person of integrity,&quot; I'd like to say something like &quot;Be [one word I'm looking for]&quot; I did a  for this, but I also wanted to know what stackexchange folks would like to say. "Reflexive only" verbs English can use a lot of verbs in a reflexive context. Even ones that usually are used intransitively.     I laugh myself silly.   However, it seems like there are very few – perhaps no – verbs that ONLY work in a reflexive context.  Are there any?