Are there infinite primes of the form n!+1 ? I searched the internet but couldn't find the answer.
121077
For some numbers $n!+1$ is prime, but all such numbers are not prime. For example, $5!+1 = 11\times 11$. The question is this:  Are there infinitely many primes of the form $n!+1$?
57235
Let $p_n$ be the $n$th prime number in its natural order. We want to show that  $$p_{n+1} \leq p_1\dotsc p_n + 1 \tag{1}$$  I know $p_1\dotsc p_n + 1 &gt; 1$, so it has at least one prime divisor, but the text says that $p_{n+1}$ cannot exceed the smallest of these divisors, but I cannot understand why, especially when I imagine $p_{n+1}$ being very large. Although some examples confirm that this statement is true, I cannot make a connection as to why the $n+1$ prime number cannot exceed the smallest prime divisor. Once this has been shown to be true, $(1)$ immediately follows.
87687
Find all natural numbers $n$ and $m$ for which $$n^m=(n-1)!+1.$$  By the Wilson theorem, $n$ is prime. Obviously, $m &lt; n$.  For example, some solutions $(n, m)$ are $(2, 1)$, $(3, 1)$ and $(5, 2)$.
112475
I am trying to prove that ($\forall \ n\in\mathbb{N}$) there exists a prime number $q$ such that $n &lt; q \le 1 + n!$  I have made a graph with $n=0$ through $n=10$ and found solutions to all of them looking for a pattern and I see that $n!$ gets enormous fast and it becomes quite obvious that there is a prime number in between them.    I have considered trying to prove by contradiction that $q$ does not exist on that interval, but I don't know where to go from that statement.  Could anybody help me figure it out?  I have been staring at it for hours and I can't figure out where to go.  Thank you.
10396
My coach said that for all positive integers $n$, $n^4+4^n$ is never a prime number.  So we memorized this for future use in math competition.  But I don't understand why is it?
89037
Let $n$ be an integer such that for any integer $d$, if $d | n$ then $d + 1 | n + 1$.   Show that $n$ is a prime number or equal to $1$  My workout...  Suppose $n = xy$ for some postive integers $x$ and $y. x|n$, and hence we assume without loss of generality $x + 1|n + 1.$   Then $x|xy + 1$. However, $xy + x$ is divisible by $x$.  This implies $x − 1$ is divisible by $x$, which is an impossibility if $x$ is not equal to $1$.  Therefore whenever $n$ is decomposed into two factors, one must turn out to be $1$. Hence $n$ must be a prime.  I am looking for any other method which solves this quickly and more quickly...
26619
I was thinking of how to prove this -  Let $p_{k}$ be the $k$th prime.Then to show that there exists infinite number of $k$ such that $p_{k+1}-p_{k} &gt; 2$.  I was thinking of Twin Prime Conjecture stating that there are an infinite number of primes differing by 2, but how can it help with infinite number consecutive primes differing greater than 2?  I was thinking of going through the method of contradiction but could not proceed
67698
I have shown that the infinite product $$\prod_{p \in \mathcal{P}}\frac{p^2+1}{p^2-1}$$ is equal to $\frac{5}{2}$ (pretty remarkable!). I have checked this numerically with Wolfram Alpha for up to $500000$ primes and it seems true.  I was wondering if this result is recorded anywhere?  Also if true, does this mean that there aren't infinitely many primes of form $p^2+1$?
84012
I need help with this question: There are infinitely many prime numbers $p$ for which $p + 2$ and $p + 4$ are also primes.  Where should I start? what proof techniques will be useful?  Any input will be very useful to me.
104546
My question is about number theory about the prime numbers.  Here is the question:  Prove that there is only one prime trio of the form $p$,$p+2$,$p+4$.  I mean,we can only find the primes 3,5,7  That satisfies the condition.
65396
I need something explained or corrected:  In my number theory class we proved that there are an infinite number of primes using Euler's Phi Totient. It went something like this:  Let $M = p_1 p_2 \dots p_n$ be the product of all primes. Consider $1 &lt; A \le M$:  Some prime must divide $A$, call it $q$. Since $q$ must be one of the primes, $q$ must divide $M.$  So the $gcd(A,M) &gt; 1.$ Thus $\phi(M) = 1$ ...?   Which is not even and contradicts the theorem that $\phi(N)$ is even for $N&gt;2.$ Therefore there exists an infinite number of primes.  I get confused by the statement "Thus $\phi(M) = 1$"....  Did i possibly copy this proof down wrong? or am I missing something? Thank you in advance.   Edit: by consider a such that...i meant consider an integer '$a$' I replaced it with $A$ to hopefully make it more clear.   Edit2: I'm sorry I am not familiar with using the equation editor. This is not a homework assignment, just studying for my exam. I just want to be able to understand this or clearify it.
111147
Prove that there is no arithmetic progression that consists only of primes.  A question that I've been set; I'm guessing it makes use of primes being written in the form 4k+1 and 4k+3? Not sure where to start.  Thanks.
51712
I was going through the fundamental theorem in Number Theory where any non zero integer n can be represented as a product of distinct primes. A related problem with this theorem is to prove that for every such number, there exists a prime $p$ such that $p&lt; \sqrt n$. I was wondering if there is any mathematical proof that no prime $p$ exists for the number $n$ such that $p&gt; \sqrt n$.
104818
Theorem: There are infinitely many primes congruent to $3 \mod 4$. Proof: Assume that $p_1 = 3, . . . , p_n$ are primes of the form $p_j ≡ 3 \mod 4$. We will construct a new one by looking at $N = 4p_1 · · · p_n − 1$ (putting $N = 4p_1 · · · p_n+3$ would also work).  There is more to this proof but the rest of it I understand, what I don't get is where it says &quot;$N = 4p_1 · · · p_n − 1$ (putting $N = 4p_1 · · · p_n+3$ would also work)&quot;. I do not understand why we are writing $4p_1 · · · p_n − 1$. Is there a more clear way to prove this theorem?
6229
Where pn is the nth prime number, does the infinite product $$\prod_{n=1}^{\infty}\left(1-\frac{1}{p_n}\right)$$ converge to a nonzero value? (Any help would be much appreciated!)
81964
Why is $1$ not considered a prime number?  Or, why is the definition of prime numbers given for integers greater than $1$?
112393
Prove that if $p$ is a prime of the form $3n+1$, $p$ is also of the form $6m+1$.  I guess one could use the division algorithm for this; but, what's the best method?