I have found this interesting theorem but with no proof:  "$p$ and $p+2$ are twin primes iff $4((p-1)!+1)\equiv -p\mod{(p(p+2)})$"  I have tried to prove it with classical N.T. theorems but I just get nowhere. Any advise?  Thanks in advance.
The Wiki page on  says     The pair $(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}$.   This is obviously connected to Wilson's Theorem. Can anybody provide a proof for that, along the lines of thought given here, ?
How to change default action for file types? There is a certain text editor that I would like to set as the default editor for certain file types but if I try selecting it on the “Open With…” menu, that does not work. However, it has put an explorer context menu item in, which does work. How could I make this menu item the default double-click action, for certain file types?  Thank you SSRT and Josh R. Both good answers. It's a pity I can't give you both best answer.  For what it's worth, I found the answer today (twenty-four hours after asking this question). Showing that $\Bbb R[x] / \langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$ question Show that $\Bbb R[x] / \langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$.      Let $\phi$ be the homomorphism from $\Bbb R[x]$ onto $\Bbb C$ given by   $f(x) \rightarrow f(i)$ (that is, evaluate a polynomial in $\Bbb R[x]$   at $i$).  Then $x^2 + 1 \in \operatorname{Ker} \phi$ and is clearly a   polynomial of minimum degree in $\operatorname{Ker} \phi$.  Thus,   $\operatorname{Ker} \phi = \langle x^2 + 1 \rangle$ and $\Bbb R[x] /\langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$.   My question is:  How is $x^2 + 1$ a polynomial of minimum degree in $\operatorname{Ker} \phi$? Natural Sounding Text to Speech? How can I calculate this limit: $\lim\limits_{x\to 0} x\left\lfloor\frac{1}{x}\right\rfloor$? Calculate the following limit. $$ \lim_{x\to 0} x\left\lfloor\frac{1}{x}\right\rfloor $$ Where $\left\lfloor x \right\rfloor$ represents greatest integer function or floor function, i.e greatest integer less than or equal to $x$.  Thanks. Find out BIOS Version from Windows Is it proper usage to replace a comma in a list with a semi-colon in this situation (see details)? While writing a proposal at work today I needed to mention a list of things and within that list were multi-part items. 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