How do I escape a global variable in PHP?
How can I prevent SQL injection in PHP?
What are the integer solutions to $a^{b^2} = b^a$ with $a, b \ge 2$ I saw this in quora.  What are all the integer solutions to $a^{b^2} = b^a$ with $a, b \ge 2$?  Solutions I have found so far:  $a = 2^4 = 16, b = 2, a^{b^2}  = 2^{4\cdot 4} =2^{16}, b^a = 2^{16} $.  $a = 3^3, b = 3, a^{b^2} = 3^{3\cdot 9} =3^{27}, b^a = 3^{3^3} =3^{27} $.  In the general case, $a$ and $b$ have the same set of prime divisors, so let $a =\prod_P p_i^{a_i}$, $b =\prod_P p_i^{b_i} $ with each $a_i \ge 1, b_i \ge 1$.  $b^a =b^{\prod p_i^{a_i}} =(\prod p_j^{b_j})^{\prod p_i^{a_i}} =\prod p_j^{b_j\prod p_i^{a_i}} $  $a^{b^2} =a^{\prod p_i^{2b_i}} =(\prod p_j^{a_j})^{\prod p_i^{2b_i}} =\prod p_j^{a_j\prod p_i^{2b_i}} $  Therefore, for each $p_j$, $b_j\prod p_i^{a_i} =a_j\prod p_i^{2b_i} $.  I haven't gotten any further than this.  I conjecture that there are no other solutions. For abbreviations - both acronyms and initialisms - where the last word in the abbreviation is the noun, I'm accustomed to adding an 's' with no apostrophe (e.g. ATMs), as described in the answer to this question:   But how do you pluralize abbreviations where the noun is not the last word? E.g. 'Point of Presence' (POP)  I can see how 'POPs' could read 'Point of Presences', when the intention is 'Points of Presence.' There should be infinitely many primes of the form $5+6n$. How do you prove it? The same should be true for $7+6n$. Where can I get a Master Ball? I remember in previous generations that finding a Master Ball would typically come after a major Boss fight. You could have also gotten through some form of lottery (I think this was in Pearl).  If anything, where and when will I be able to find Master Balls in Pokemon X&amp;Y? Complex zeros of the polynomials $\sum_{k=0}^{n} z^k/k!$, inside balls this is a question from a Temple prelim exam, and i'm trapped in it! We have $p_n(z)=\sum_{k=0}^n\frac{z^k}{k!}$ and we have to prove that $\forall r&gt;0 \quad \exists N\in\mathbb{N}$ s.t. $p_n(z)$ has no zeros in $B_r(0)$ for $n&gt;N$.  I tried to use Rouche's theorem observing that $1+\frac{z^n}{n!}$ has the nth roots of $n!$ as complex zeros, which are outside a fixed ball for n sufficiently large, but i'm having trouble proving that $|1+\frac{z^n}{n!}|&gt;|z+\cdots+\frac{z^{n-1}}{(n-1)!}|$ in $|z|=r$ (if this is true). If anyone could help.. Wrong kernel version returned by uname Ubuntu 16.04.1 LT uname -a returns:    Linux decebal-mobile 4.2.0-35-generic #40-Ubuntu SMP Tue Mar 15 22:15:45 UTC 2016 x86_64 x86_64 x86_64 GNU/Linux   but E: Package 'linux-headers-4.2.0-35-generic' has no installation candidate  and my machine says linux headers version is at linux-headers-4.4.0-21-generic.   I have seen a similar problem described in a .  How can I update that? Let $a$, $b$, and $c$ be integers, where a $\ne$ 0. Then $$ $$ (i) if $a$ | $b$ and $a$ | $c$, then $a$ | ($b+c$) $$ $$ (ii) if $a$ | $b$ then $a$|$bc$ for all integers $c$; $$ $$ (iii) if $a$ |$b$ and $b$|$c$, then $a$|$c$.  Prove that if $a$|$b$ and $b$|$c$ then $a$|$c$ using a column proof that has steps in the first column and the reason for the step in the second column.  My book is really vague. I'm not really sure what to do..  First I was thinking something like this: $a\mid b\Rightarrow b=as$ and  $b\mid c\Rightarrow c=bt$ and $a\mid c\Rightarrow c = au$ $$ $$ $b + c + c = b + 2c$  $$ $$ $=as + 2(bt+au)$  Yea then I just got lost. Given an integer $n$, we are asked to investigate about the existence of integer divisors of $n^2+1$ of the form $4k+3$. Can you provide some insights about it? In JavaScript can I make a "click" event fire programmatically for a file input element? I'd like to make a click event fire on an &lt;input type="file"&gt; tag programmatically.  Just calling click() doesn't seem to do anything or at least it doesn't pop up a file selection dialog.  I've been experimenting with capturing events using listeners and redirecting the event, but I haven't been able to get that to actually perform the event like someone clicked on it. Why isn't temperature frame dependent? In (non-relativistic) classical physics, if the temperature of an object is proportional to the average kinetic energy ${1 \over 2} m\overline {v^{2}}$of its particles (or molecules), then shouldn't that temperature depend on the frame of reference - since $\overline {v^{2}}$ will be different in different frames?  (I.e. In the lab frame  $K_l = {1 \over 2} m\overline {v^{2}} $, but in a frame moving with velocity $u$ relative to the lab frame, $K_u =  {1 \over 2} m \overline {(v+u)^{2}}$). How do you copy a record in a SQL table but swap out the unique id of the new row? Solutions to $y^2 = x^3 + k$? The equation $y^2 = x^3 + k$ for $k = (4n-1)^3 - 4m^2$, with $m, n \in \mathbb{N}$ and no prime number that p is congruent to 1 modulo 4 divids m, doesn't have any answer and its proof can be obtained by using quadratic reciprocity law.  Do you know answers of this equation for two or three different values of $k$? In addition, do you know any reference about that? SSD for production database Bolzano-Weierstrass Theorem Proof Abbott (Ax. of Choice) In Abbott's Proof of the Bolzano-Weierstrass Theorem, does Abbott use any form of the Axiom of Choice ?   I think he is since he chooses an $a_{n_k} \in I_k$ where there are multiple such $a_{n_k}$. How prove this nice limit $\lim\limits_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$ Nice problem:  Let $a_{0}=1$ and $$a_{n}=a_{\left\lfloor n/2\right\rfloor}+a_{\left\lfloor n/3 \right\rfloor}+a_{\left\lfloor n/6\right\rfloor}.$$  Show that  $$\lim_{n\to\infty}\dfrac{a_{n}}{n}=\dfrac{12}{\log{432}},$$  where $\lfloor x \rfloor$ is the largest integer not greater than $x$.  It is said this problem was created by Paul Erdős, and I can't find this problem's solution, does anyone have any nice methods? Thank you. $\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$. I've tried, but I can't solve the question. Please help me prove that:  $\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to  $U_n$. How to get data off a completely dead Android phone? Phone worked perfectly. Then one day completely dead, nothing at all. It may have been that "one day" was after some time of laying around, cannot recall.  Model: Android AT&amp;T Samsung Galaxy S3 (SGH-i747 SKU S9255)  Already tried;   Charging phone: no visual output at all, no leds Removing battery while charging: no visual output at all, no leds Removing SIM and SD card: no visual output at all, no leds Attempt to turn on: no visual output at all, no leds   I need to get our family photo's off this phone (stored on internal memory). I care less about the hardware/phone itself.  Thank you for any help you can offer! I know that for every $n\in\mathbb{N}$, $n\ge 1$, there exists $p(x)\in\mathbb{F}_p[x]$ s.t. $\deg p(x)=n$ and $p(x)$ is irreducible over $\mathbb{F}_p$.     I am interested in counting how many such $p(x)$ there exist (that is, given $n\in\mathbb{N}$, $n\ge 1$, how many irreducible polynomials of degree $n$ exist over $\mathbb{F}_p$).   I don't have a counting strategy and I don't expect a closed formula, but maybe we can find something like "there exist $X$ irreducible polynomials of degree $n$ where $X$ is the number of...".  What are your thoughts ? For any triangle with sides $a$, $b$, $c$, show that $a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0$ For any triangle with sides $a$, $b$, $c$, prove the inequality $$a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0 .$$   This is IMO 1983 problem 6.  I tried substituting $a=x+y$, $b=y+z$, $c=z+x$ but well it doesn't help in any sense except wasting 3 pages that lead to nothing (please don't mind the joke). Using $a=2R\sin A$, $b=2R\sin B$, $c=2R\sin C$ also didn't lead to anything for me. Could you give me a hint for finding the proper substitution? In Ubuntu releases prior to Ubuntu 19.10 Eoan Ermine, when I run a command with sudo, that command receives my home directory in the $HOME environment variable. This is the behavior I have long expected and . If I want sudo to reset the $HOME environment variable, so that it refers to the target user's home directory instead of my own, I have to pass the -H option (or -i, though that does more).    ek@Kip:~$ lsb_release -d Description:    Ubuntu 18.04.3 LTS ek@Kip:~$ sudo printenv HOME  # Shows ek's home, not root's. /home/ek ek@Kip:~$ sudo -u as printenv HOME  # Shows ek's home, not as's. /home/ek ek@Kip:~$ sudo -H printenv HOME  # Shows root's home. /root ek@Kip:~$ sudo -Hu as printenv HOME  # Shows as's home. /home/as   When I first upgraded to Ubuntu 19.10, I was surprised to discover that sudo seems to reset $HOME no matter what! I continue to observe this now that 19.10 is released and I've installed updates--both on newly installed systems and one I upgraded to 19.10.  ek@Cord:~$ lsb_release -d Description:    Ubuntu 19.10 ek@Cord:~$ sudo printenv HOME  # Shows root's home, even without -H or -i. /root ek@Cord:~$ sudo -u as printenv HOME  # Shows as's home, even without -H or -i. /home/as ek@Cord:~$ sudo -H printenv HOME  # Also shows root's home. /root ek@Cord:~$ sudo -Hu as printenv HOME  # Also shows as's home. /home/as   I thought this might be due to .  But I checked, and always_set_home does not appear in any Defaults line in my 19.10 /etc/sudoers file.  What makes sudo treat $HOME differently starting in 19.10, and why was this change made? Does this make it safe to use plain sudo in cases where I would previously have used sudo -H?