We know that there are infinite number of primes so as there are infinite number of primes of the form $4n+3$ where $n\in Z^+$.   A note on Burton's book (Elementary Number Theory) somehow says that it is of high chance to expect that there are also infinite number of primes of the form $4n+1$. However it is not yet proven by the time the book was published.  A quick search on net gives infinitude of primes of the form $3n+1$ and $5n+1$.  The question is: The problem: Are there infinitely many number of primes of the form $4n+1$ still an open problem or it was already proven? If so, where can I find the proof? Thanks a lot.
Question: Are there infinitely many primes of the form $4n+3$ and $4n+1$?  My attempt: Suppose the contrary that there exist finitely many primes of the form $4n+3$, say $k+1$ of them: $3,p_1,p_2,....,p_k$  Consider $N = 4p_1p_2p_3...p_k+3$, $N$ cannot be a prime of this form. So suppose that $N$=$q_1...q_r$, where $q_i∈P$  Claim: At least one of the $q_i$'s is of the form $4n+3$:  Proof for my claim: $N$ is odd $\Rightarrow q_1,...,q_r$ are odd $\Rightarrow q_i \equiv 1\ (\text{mod }4)$ or $q_i ≡ 3\ (\text{mod }4)$  If all $q_1,...q_r$ are of the form $4n+1$, then $(4n+1)(4m+1)=16nm+4n+4m+1 = 4(\cdots) +1$  Therefore, $N=q_1...q_r = 4m+1$. But $N=4p_1..p_k+3$, i.e. $N≡3\ (\text{mod }4)$, $N$ is congruent to $1\ \text{mod }4$ which is a contradiction.  Therefore, at least one of $q_i \equiv 3\ (\text{mod }4)$. Suppose $q_j\equiv 3\ (\text{mod }4)$  $\Rightarrow$ $q_j=p_i$ for some $1\leq i \leq k$ or $q_j =3$  If $q_j=p_i≠3$ then $q_j$ | $N = 4p_1...p_k + 3 \Rightarrow q_j=3$ Contradiction!  If $q_j=3$ ($\neq p_i$, $1\leq i \leq k$) then $q_j | N = 4 p_1...p_k + 3 \Rightarrow q_j=p_t$ for some $1 \leq i \leq k$ Contradiction!  In fact, there must be also infinitely many primes of the form $4n+1$ (according to my search), but the above method does not work for its proof. I could not understand why it does not work. Could you please show me?  Regards
how to add a custom lock for the recovery and/or bootloader? Unlocking the bootloader and/or the recovery is very popular discussion across many devices. But doing so exposes a very obvious and inevitable issue with security!  An unlocked recovery means that no matter how secure you make your android lock screen, someone can easily reset phone to restore the phone to a fully functional (and lockless/insecure) state and the worst part, they don't have to erase your data(meaning they can access all your personal files and folders)!  It is almost hilarious! In fact if the phone supports insertion of external SD cards, you can even flash your own software into the phone and maybe bypass the lock screen without wiping any data.  For example: One possible loophole is using the 'Tasker' app which has a feature that can temporarily disable the keyguard. And the best part is that Tasker can be flashed using the recovery!  So is there any method by which I could add a security measure that prevents complete access to my recovery and/or boot loader that will make my phone less insecure?!  Expected answer: Method by which a custom pin/password must be entered before entering the bootloader/recovery. So that my bootloader/recovery is still technically unlocked but simply requires authentication to use (this also ensures that conventional unlocking methods won't work as the phone is not really locked!). This also means that one does not have to lose root access to ensure security! !! Do not use the future tense after unless   [✓]  I won't go unless you go   [✘] [I won't go] unless you will go. [✘]   Why not? Even if the above is prescriptive, what may be some reasons? In my separate question , I quote: unless = if not. If not can precede future tense; so why can't unless?  Footnote:  motivated this. Does it make sense to say, "The speed of light varies?"  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But calculate them separately so I can get the total area of each type in column 1. Why does adding a curve modifier move my object to the center of the curve? I'm using a curve modifier and an array modifier. My object is at one end of the curve but as soon as I choose the curve for the curve modifier it moves to the center. How can I fix the problem so that I can have the array going from one end of the curve to the other? I'm looking at buying a second-hand Mac, probably one that does not include an OS installation disk.  How can I scan it to make sure that it has not had security threats installed on it?  Is there a CD or DVD image I can boot it up with to perform this scan offline?  Edit: prospective machine would be a 2011- or 2012-vintage MacBook Pro.  Also note that I don't have reliable access to high-speed internet, so solutions that involve downloading 5GB over the Internet are not ideal.  For others, though,  may be helpful. how to auto start or enable service on next boot up or start up? in ubuntu 16.04 lts how to start or enable service on the next boot up or start up? in ubuntu 16.04 lts  without doing it manually enable or start services again everytime i start up or boot up the ubuntu 16.04 lts  thank you I am curious about the proof of these two integrations. It would be great if you could point me in the right direction : $ \int{e^{ax}\cos({bx})dx}= \dfrac{e^{ax}}{a^{2}+b^{2}}[a\cos(bx)+b\sin(bx)] + C $ &amp; $ \int{e^{ax}\sin({bx})dx}= \dfrac{e^{ax}}{a^{2}+b^{2}}[a\sin(bx)-b\cos(bx)] + C $ What can I use to typeset MATLAB code in my document? This question led to a new package:      Does anybody have a quick and easy way to typeset some relatively long MATLAB code in the appendix? I looked up a few methods, and the mcode package emerged as a possibility, but it seems to be unmaintained...  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I'm working on creating an illustration that I will need in many, different guises in differential geometry/manifold theory etc. Following on from a previous (answered) question of mine (, I want to show how two coordinate lines in the plane originate from two curves on a surface via the chosen map:  \begin{tikzpicture}     \begin{axis}[             name=mfd,             declare function={                     f(\x,\y)=10-(\x^2+\y^2);             },                   declare function={                     c_x(\t)=(cos(\t)+(sin(5*\t)/10))/3+1;             },                   declare function={                     c_y(\t)=(sin(\t))/2-1;             },                   declare function={                     c_z(\t)=f(c_x(\t),c_y(\t));             },           ]                    \addplot3[surf,domain=-2:2,domain y=-2:2,]{f(x,y)};             \addplot3[black,opacity=1.0,variable=t,domain=0:360,dashed,thin] ({c_x(t)},{c_y(t)},{c_z(t)});             \addplot3[black,opacity=1.0,variable=t,domain=-2:2,thin] (t,-1.2,{f(t,-1.2)});             \addplot3[black,opacity=1.0,variable=t,domain=-2:2,thin] (0.8,t,{f(t,-1.2)});             \addplot3[black,opacity=1.0,only marks,mark=text,text mark=$\cdot$] (1,-1,{f(1,-1)}) coordinate (a);     \end{axis}     \begin{axis}[             at={($(mfd.north east)+(1cm,-2cm)$)},             anchor=north west,             declare function={                     c_x(\t)=(cos(\t)+(sin(5*\t)/10))/3+1;             },                   declare function={                     c_y(\t)=(sin(\t))/2-1;             }            ]                    \addplot[variable=t,domain=0:360]({c_x(t)},{c_y(t)});             \addplot[black,opacity=1.0,only marks,mark=text,text mark=$\cdot$] (1,-1) coordinate (b);             \addplot[variable=t,domain=0.6:1.4](t,-1.2);             \addplot[variable=t,domain=-1.5:-0.5](0.8,t);     \end{axis}     \draw [-stealth, shorten &lt;= 3pt, shorten &gt;= 3pt] (a) to[bend left] (b); \end{tikzpicture}   However, I have a couple of problems:   The curves representing the coordinate axes pulled back from the plane via the coordinate map insist on being closed off with a straight line - is there a way to avoid that? I want to draw all the black curves with a thin, dashed line, but I probably don't understand how to set those options. What am I doing wrong?   I apologise if these questions are rather elementary, but I find the relevant manuals difficult to search through; they seem to be meant for reading cover to cover rather than reference. Now that the , I've been seeing tag-only suggested edits from 1500-rep users.  For example, one had the summary of 'burninating [tagThatWasRemoved]'.  It's very tempting to reject these as Too Minor, but I know that these users used to have a Retag privilege (and are obviously trying to participate in a burnination.)  What should we do for tag-only edits from high(er) rep users? Why there is a delay between when a key is pressed and when it is accepted? This is one of the wackiest and weirdest problems I have ever encountered. My keyboard was working absolutely fine untill yesterday on Ubuntu 13.04. However, when I boot to Ubuntu today, I suddenly find that all the keys work, but I have to press and keep on holding them for about a second for them to work.   This behavior is true for all the keys except for Num Lock, Caps Lock and Scroll Lock keys, the indicator for them on my keyboard lights up as soon as I press those keys. The keyboard works fine at the login screen where I put my password.  The keyboard also works normally if I use the Guest session instead of my normal user account. I have not done any key-remappings and using the standard English (US) keyboard layout.  I have fiddled around with the settings for Keyboard in System Settings, but to no avail. I have Windows 7 as dual-boot and the keyboard works perfectly well on it.   Why am I facing such a behavior and how to normalize this?  PS: Using Logitech Classic Keyboard K100 (USB Model). How to prove that center of $GL_n(\mathbb R)$ is scalar matrices?  Again what can you say about the center of set of all upper triangular matrices and center of set of all diagonal matrices. I'm trying to give the best experience to users of a very huge system (like a CRM). It has a lot of forms and much of these have a lot of fields. My doubt is about if I should use a single column form (which raises too much vertical scrolling), or a two columns form (which needs less vertical scrolling and uses more horizontal white space) and a last one with four columns (which has few scrolling and uses almost whole white space to try to keep most of the fields above the fold).  It's important to say that our clients hate scrolling, and they always ask why we don't use the horizontal white space when we present a one column form).  So I'm little bit confused about it, because a lot of UX studies and good practices recommend to do not use/avoid multicolumn forms but the clients most of times ask multi column forms.  Our clients are very used to desktop programs and not everybody is very familiar with web based systems.  Below follows some possibilities I would like to share to illustrate an example of a complex form in the system with the three possibilities I mentioned:    In the example above I'm using the one column form (most recommended in UX studies), but with a very long vertical scroll.    In the second example I tried to reduce vertical space by using two columns.    This is the client's favorite one, 4 columns using white space and trying to keep everything above the fold.   The lengths of fields are based on the average of characters amount for each field.   I would like to know your opinion about the best approach to this case. Why does a bezier curve twist when rotated? If I create a bezier curve and rotate it in edit mode it does not accordingly rotate the tilt of the handles which creates an undesirable twisting along the curve.  Here is an example using a bezier circle with a square used for a bevel object:   Here it is after being rotated 90° about the y-axis:   How do I prevent/fix this? Is it possible to dual boot two OS's at the same time? UTF-8 all the way through