Prime numbers are natural numbers not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

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For what numbers $n$ is $\sqrt{n}$ irrational?

I would say it has something to do with the numbers that can be expressed as a factor of different prime numbers, but when I get to $8$, that can be changed to $2^3$, which goes against this. Is there ...
7
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1answer
66 views

How was the 506-digit prime number 999…9998999…999 found?

I was surprised to encounter a claim made on the internet that the following number is prime: ...
23
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4answers
2k views

-1 as the only negative prime

I was recently thinking about prime numbers, and at the time I didn't know that they had to be greater than 1. This got me thinking about negative prime numbers though, and I soon realized that, for ...
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0answers
25 views

Is there an arbitrarily large set of naturals so that the sum of each two has exactly $n$ prime divisors? What about an infinite set? [on hold]

Is there an arbitrarily large set of naturals so that the sum of each $2$ has exactly $n$ prime divisors where $n$ is fixed? What about an infinite set? For $n=1$ this is clearly false, what ...
1
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2answers
60 views

How can Mersenne Prime rule be valid if $2047$ isn't prime?

The rule of Mersenne Prime says that $2^p - 1$ is prime if $p$ is prime. $2^{11} - 1 = 2047$ satisfies the condition, but it's not a prime as it can be divided by two prime numbers $23$ and $89$. ...
0
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2answers
24 views

Euler's proof of Infinitude of Primes: If a prime divides an integer, why would it have to divide 1?

Here's the Euler's proof of Infinitude of Primes in Rosen's Discrete Mathematics: We will prove this theorem using a proof by contradiction. We assume that there are only finitely many primes, ...
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0answers
19 views

Prime division algebra level 5

Let $P$ be the number of integers $n$  for which $n^4-52n^2+595$ is prime, and let $D$ be the number of distinct primes that can be represented in this form. Find $P+D$.
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16 views

Finite amount of consecutive smooth numbers

is there a short proof of the fact that there is a finite amount of consecutive smooth numbers (meaning Given a finite set B, there is a finite amount of pairs $n,n+1$ so that both can be expressed as ...
0
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0answers
16 views

What are some easy to prove results on the density of primes?

Bertrand's postulate states that for any integer $n>3$, there's always a prime $p$ between $n$ and $2n-2$. That result sets a reasonable 'lower bound' on how often we can expect primes to show up, ...
1
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1answer
31 views

How does the fact that Fermat primes are relatively prime imply there are infinite primes?

I was just reading a book called Proofs from the Book. It presented the proof given by George Polya to prove that two Fermat primes (numbers of the form $2^{2^n} + 1$) are always relatively prime, ...
1
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2answers
38 views

Prove that there exists infinitely many primes of Digital root $2,5$ or $8$

I am highly interested in properties of digital root. Digital Root: Digital root of a number is a digit obtained by adding digits of number till a single digit is obtained. It's clear that Digital ...
2
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2answers
60 views

How can I prove that only there continuous odd prime are $3,5,7$?

How can I prove that the only prime number $p$, such that $ p,p+2,p+4$ are primes is 3?
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0answers
16 views

Does there exist a $k$ such that for all $n \ge 3$, $\text{gpf}(\lfloor n^{(\log{n})^k} \rfloor) \gt n$?

Does there exist a $k \in \mathbb{R}$ such that for all $n \in \mathbb{N}, n \ge 3$, $\text{gpf}(\lfloor n^{(\log{n})^k} \rfloor) \gt n$, where $\text{gpf}(x)$ is the greatest prime factor of $x$? I ...
1
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0answers
61 views

A method of writing all primes

I've recently noticed a method of describing primes. As an example: $13=5*11-2*3*7$. This pattern must follow these rules: $x-y$ such that $x*y$ is the product of all previous primes (allowing powers ...
11
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1answer
85 views

Is there a polynomial such that $F(p)$ is always divisible by a prime greater than $p$?

Is there an integer-valued polynomial $F$ such that for all prime $p$, $F(p)$ is divisible by a prime greater than $p$? For example, $n^2+1$ doesn't work, since $7^2+1 = 2 \cdot 5^2$. I can see that ...
2
votes
1answer
28 views

Complexity of generating a prime larger than $N$

Is it provably difficult to generate a prime larger than a prescribed $N$? For instance, if I want a prime of $1000$ digits, is there a way to do that deterministically, i.e., without resorting to AKS ...
11
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0answers
66 views

Consecutive prime numerators of harmonic numbers?

Let $$\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\frac{a}{b}$$ and let $a$ and $b$ are coprime, $h_{n}=a$. $h_{n}$ is prime for ...
0
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0answers
30 views

Find k-th element of the sequence

Please, help me with effective algorithm to: Find k-th element of the sequence {n | (6n-1), (6n+1), (12n+5) are primes} Find k-th element of the sequence {n | (6n-1), (6n+5), (12n-7) are primes}
2
votes
1answer
26 views

Sum of reciprocals of n-digit primes

I have observed, by calculation, that the sum of the reciprocals of all the n-digit prime numbers is approximately 1/n, and that this becomes increasingly accurate as n increases. Is there a simple ...
6
votes
1answer
103 views

Is $k+p$ prime infinitely many times?

I have the following conjecture: Let $k\in\mathbb{N}$ be even. Now $k+p$ is prime for infinitely many primes $p$. I couldn't find anything on this topic, but I'm sure this has been thought of ...
0
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1answer
41 views

What is the simplest way to show that ${(p-1)! \over (k)!(p-k)!}$ is an integer?

In the proof of $p$ | $\binom{p}{k}$ (p divides $\binom{p}{k}$) where $p$ is prime, what is the simplest way to show that $${(p-1)! \over (k)!(p-k)!}$$ is an integer?
0
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1answer
18 views

The form of solutions of $p*k-q*j=r,$ for $(p,q)=1$.

I would like to find the form of solutions of $p*k-q*j=r,$ for $(p,q)=1$ for any fixed $r < pq$ and $k,j \in \mathbb{N}$. I tried to look at the divisibility of $p=cq+b.$ But I didn't have any ...
4
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1answer
41 views

Aren't Legendre's conjecture and Andrica's conjecture same?

If Legendre's conjecture is true, couldn't we easily obtain $\sqrt{p_{n+1}}-\sqrt{p_{n}}<1$ where $p_{n}$ is the $n$th prime? $$p_{n+1}<(\lfloor \sqrt{p_{n}} \rfloor + 1)^{2}<( \sqrt{p_{n}}+ ...
0
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1answer
39 views

Proving the primality of these large numbers?

In 2007, Vautier claimed that the largest known consecutive pair of prime numbers (at the time) was $2003663613\cdot2^{195000}-1$ and $2003663613\cdot2^{195000}+1$. I was wondering how Vautier found ...
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2answers
79 views

Using binomial theorem to prove $a | b^n \Rightarrow a | b$. ( | is divides, a prime, n integer > 1)

I tried expanding $(b-a+a)^n=$[$(b-a)+a$]$^n$ but it just seemed to further complicate the problem. I also tried to prove the contrapositive but that doesn't seem to lead to anywhere to. Is there any ...
2
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1answer
71 views

How does one prove that $2\uparrow\uparrow16+1$ is composite?

Just to be clear, close observation will show that this is not the Fermat numbers. I was reading some things (link) when I came across the footnote on page 21, which states the following: ...
10
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3answers
379 views

BMO2 2016 Number Theory Problem

Suppose that $p$ is a prime number and that there are different positive integers $u$ and $v$ such that $p^2$ is the mean of $u^2$ and $v^2$. Prove that $2p−u−v$ is a square or twice a square. Can ...
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1answer
38 views

estimation for n-th prime

The famous theorem of Hadamard and Vallee-Poussin https://en.wikipedia.org/wiki/Prime_number_theorem implies that $p_n\sim n\ln n$, so $C_1 n\ln n \le p_n \le C_2 n\ln n$ holds for all $n\ge 2$ with ...
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2answers
37 views

On analogy between $\Bbb Z$ and $\Bbb F_q[x]$

There are objects and operations analogous between $\Bbb Z$ and $\Bbb F_q[x]$. For example primes in $\Bbb Z$ and irreducibles in $\Bbb F_q[x]$ are analogous and so is multiplication operation. ...
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2answers
36 views

If $p$ is a prime other than 2 or 5, prove that $p$ must be one of the forms $10k + 1$, $10k + 3$, $10k + 7$, or $10k + 9$

If $p$ is a prime other than 2 or 5, prove that $p$ must be one of the forms $10k + 1$, $10k + 3$, $10k + 7$, or $10k + 9$ -The section we are covering is on the division algorithm, although I am ...
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0answers
13 views

How the sieving part of quadratic sieve actually works?

I am trying to implement quadratic sieve algorithm as it's described in wiki. I understand most of it, except the part of the sieving example. In the example they use $N = 15347$ with base prime ...
1
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1answer
62 views

Are there prime gaps of every size?

Is it true that for every even natural number $k$ there exists some $n \in \mathbb{N}$ such that $g_n = p_{n+1} - p_n = k$? I don't know how to approach the problem at all, and in fact I don't even ...
0
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1answer
64 views

Suppose that $n$ is a factor of $(n-1)!+1$. Prove that $n$ is prime [duplicate]

This is in an Algebra and Combinatorics module and I don't know how to prove this. The full question is, Let $n$ be a natural number greater than $1$. Suppose that $n$ is a factor of $(n-1)!+1$. ...
2
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1answer
39 views

Prove the “Chebyshev's theorem”

I know the Chebyshev's theorem for primes that is : Theres a p between n,2n if n>1 Can you prove it easily? Actually im just 13 years old and I couldn't found an answer that I can understand it ...
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0answers
89 views

Is the Riemann Hypothesis incorrect? [closed]

See the attached image I would like to know your opinion about if the zeros shown in this picture can be considered as the zeros mentioned by Riemann in his Z function. I think yes and that his ...
1
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1answer
30 views

$C(n)+P(n)+S(n)$ always composites?

Let $C(n)$ be the concatenation of first $n$ primes, let $P(n)$ be the product of first $n$ primes, and let $S(n)$ be the sum of the first $n$ primes. It is not surprising that $C(n) - P(n) - S(n)$ is ...
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0answers
28 views

Modular and prime prove [duplicate]

Suppose that $p$ is prime and $p=n^2 +5$ for some natural number $n$, prove that the final digit of $p$ is equal to $1$ or $9$ which is $p=1(mod10)$ or $p=9(mod10)$ What I have to tried so far: in ...
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1answer
40 views

Find all $a$, so $q$ prime number which ,$q\times n= aaaaaaa$ [duplicate]

I need your helping to find all the $a$ numbers,which follow the next rules: there is prime number $ 2\lt n\in \mathbb N$ and $ 5\neq q\in \mathbb N$ so that the digits of $n\times q$ are only $a$. ...
0
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0answers
29 views

On primes between consecutive $n-th$ powers. [duplicate]

The Opperman conjecture, is the statement that for every integer $x\geq 2$, there always exists a prime btween $x^2$ and $(x+1)^2$. How about for every integer $n\geq 3$, is there always a prime ...
0
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1answer
26 views

What is an upper bound for number of prime powers and semi primes in the interval $[n^2+1,n^2+n]?$

What is an upper bound for number of prime powers in the interval $[n^2+1,n^2+n]?$ What is an upper bound for number of square free semi primes in this interval$?$
0
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2answers
65 views

Prove that there is prime number and natural so $n\times q$ digits are $1$.

I need your helping to prove that there is a prime number $ 2\lt n\in \mathbb N$ and $ 5\neq q\in \mathbb N$ so that the digits of $n\times q$ are only $1$. for example:if $n=3$ then $3\times 37=111$ ...
2
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2answers
76 views

What is the probability of being prime?

Given some arbitrarily large $a$, what is the probability that this number is a prime number? My attempt involves seeing that for $a$ to be prime, then it must not have a factor $N$ in the following ...
4
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2answers
53 views

What integers are coprime to the first $x$ prime numbers?

I have noticed that there is a very specific pattern to numbers that are coprime to $2$, it is simply all of the odd numbers. More specifically, it is in the following pattern, where $n$ is an ...
2
votes
3answers
69 views

Some hints for “If a prime $p = n^2+5$, then $p\equiv 1\mod 10$ or $p\equiv 9\mod 10$”

I tried to prove this question by first considering the possible last digit of $p$ when $p=n^2+5$, but that reasoning got me nowhere. Then I tried to prove it by contrapositive, and however I just ...
2
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1answer
41 views

Density of numbers with exactly $n$ distinct prime factors in $\mathbb{N}$

It is quite well known that the density of the primes in $\mathbb{N}$ is $0$, that is, $$\lim_{n\to\infty}\frac{|\{p\mid p\leq n, p \text{ prime}\}|}{|\mathbb{N}_{\leq n}|}=0$$ It is less well-known, ...
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0answers
13 views

Truncating Legendre's Formula

Let $p_n$ denote the $n^{th}$ prime. Legendre's Formula, $\phi(x,a)$, counts the number of integers less than or equal to $x$ that are not divisible by the first $a$ primes. Define therefore ...
3
votes
1answer
90 views

Is $\sum_{p}\frac{1}{p!}=\frac{1}{2!}+\frac{1}{3!}+\frac{1}{5!}+…$ irrational?

Is there known way to determine whether the infinite sum below is rational or not? $$\sum_{p}\frac{1}{p!}=\frac{1}{2!}+\frac{1}{3!}+\frac{1}{5!}+...$$
3
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1answer
70 views

Asymptotic relation for the following series?

Questions Is the asymptotic relationship correct? How do I determine $c_1$ and $\kappa$? As, $|s| \to 0$ $$ \sum_{r=1}^\infty s^r \ln(r) \sim c_1 \sqrt{s} + (\kappa - 1 + \frac{\ln(2 \pi)}{2} ...
2
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0answers
52 views

infinitude of primes with the form $n^2+1$ [closed]

Is there any progress in proving the infinitude of prime numbers of the form $n^2+1$ ?
-5
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2answers
135 views

Can I belive that : $e^{e^{e^{e^{\cdots}}}}$ is $\infty$? [closed]

Definetly this number : $e^{e^{e^{e^{\cdots}}}}$ is not an integer this implies that is not prime number or perfect number , now i would like to know really what is the nature of this number ...