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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All prime co-prime
The number $30$ has a curious property:
All numbers co-prime to it, which are between $1$ and $30$ (non-inclusive) are all prime numbers!
I tried searching(limited search, of course) for numbers ...
2
votes
0answers
63 views
Prime numbers; how to do this?
You can write every number as the product of some prime numbers, for example $33 = 11 \cdot 3$. However, how can you do this when you're dealing with a prime number? If you write $29 = 29 \cdot 1$ you ...
6
votes
2answers
235 views
How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis if at all?
How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis? What are the connections between both conjectures if any?
1
vote
1answer
26 views
Power analogies to primes
As we all know, a natural number $n$ is prime if and only if there do not exist natural numbers $x, y$ exclusively between $1$ and $n$ such that $xy = n$.
Is there any generally recognized analogy ...
3
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1answer
88 views
+50
Only 3 $n$ where $q=\left\lfloor (2 p_n+p_{n+2}) (p_{n}+p_{n+1}+p_{n+2})\over p_{n}\right\rfloor,\;\text{isPrime}(q)$?
Consider: $$q=\left\lfloor (2 p_n+p_{n+2}) (p_{n}+p_{n+1}+p_{n+2})\over p_{n}\right\rfloor,\;\text{isPrime}(q)$$
where $p_n$ denotes the $n$th prime.
Other than: $$n=6\quad\text{or}\quad ...
2
votes
0answers
44 views
finding out linear decomposition of $x$ into $k$ prime numbers
Some $k$ prime numbers $n_1, n_2, ..., n_k$ are given. Then some natural number $x$ is provided.
Then we want to figure natural numbers (including zero) $m_1, m_2, ..., m_k$ so that $n_1m_1 + n_2m_2 ...
1
vote
1answer
580 views
Conjecture on cycle length and primes
That thanks for Peter Košinár's answer,I change the conjecture a lot.
Suppose $a$ is natural number, and $b = A179382((a+1)/2)$ named " cycle length of $a$", if $b = (a-1)/(2^c)$ for some $c > ...
11
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0answers
143 views
What is the smallest integer $n$>1 such that $n^{5000}+n^{2013}+1$ is prime?
Which is the smallest integer $n>1$, such that $$n^{5000}+n^{2013}+1$$ is prime ?
Since $x^{5000}+x^{2013}+1$ is irreducible over $\mathbb{Q}$ and has value $1$ for $x=0$,
there should be ...
3
votes
3answers
65 views
Right triangles with integer sides
Most of you know these triples:
$3: 4 :5$
$5: 12 :13$
$8: 15 :17$
$7: 24 :25$
$9: 40 :41$
More generally we can construct such triangles such as
$$2x:x^2-1:x^2+1$$
My question is why one of ...
4
votes
3answers
76 views
How do you prove that the mean of the co-primes of a number is half the number?
Say $n = 6$, The set of co-primes is $\{1, 5\}$, $\text{mean} = 3$
For $n = 9$, the set of co-primes is $\{1, 2, 4, 5, 7, 8 \}, \text{mean} = 4.5$
Question: Prove that the mean of co-primes of ...
9
votes
1answer
230 views
Prime with digits reversed is prime?
Well, just another idea came up into my mind and i have no idea how to solve it :D
Is there infinitely many prime numbers, which are not repunits and their inverse is also prime? (For example, inverse ...
2
votes
0answers
72 views
Prime numbers problem - discrete math
Show that natural numbers of the form $n^2+1$ are not divisible by primes of the form $p=4k-1$.
I can't really find a place to start.
Thank you very much in advance,
Yaron.
12
votes
3answers
301 views
$\sum\limits_{\text{prime }p} 2^{-p}$ is an irrational number
I need help to prove the following result.
$\displaystyle\sum_{\text{prime }p} 2^{-p}$ is an irrational number.
4
votes
1answer
56 views
What's the asymptotic distribution of $p^n$ (powers of primes)?
We know by the prime number theorem that
$\lim_{n\to\infty}\frac{\pi(n)}{n\,/\ln n} = 1$
An even better approximation is
$\lim_{n\to\infty}\frac{\pi(n)}{\int_2^n\frac{1}{\ln t}\mathrm{d}t} = 1$.
Is ...
4
votes
0answers
81 views
The divergence of the series of reciprocals of primes (proof check):
I just wanted to check my attempt at a proof for the divergence of:
$$\sum_{n=1}^{\infty} \frac{1}{p_n} \tag{ $\star$ }$$
We begin with assuming that $(\star)$ converges. If $(\star)$ ...
2
votes
2answers
52 views
On the Pell-like $px^2-qy^2 = 1$ for prime $p,q$
Given any prime of form $p_n = u^2+nv^2$ for non-zero integers $u,v$. Consider,
\begin{aligned}
&p_2x^2-2y^2 = 1\\
&p_3x^2-3y^2 = 1\\
&p_7x^2-7y^2 = 1\\
&p_{11}x^2-11y^2 = 1\\
...
2
votes
3answers
38 views
Does there exist a k such that the kth prime is balanced in order k-1?
A balanced prime of order n is a prime number that is equal to the arithmetic mean of the nearest n primes above and below.
For example, 5 is a balanced prime in order 1 because it is the average of ...
7
votes
1answer
227 views
interpolating the primorial $p_{n}\#$
The primorial $p_{n}\#$ is given by the product $p_n\# = \prod_{k=1}^n p_k$ (where $p_{k}$ is the $k$th prime) -- is there a natural (a la the gamma function $\Gamma(z)$) way of interpolating it for ...
6
votes
2answers
144 views
Do we know if there are more primes with even leading digits or odd leading digits?
I was just wondering, out of curiosity, do we know if there are more primes with even leading digits or odd leading digits? For example, primes with even leading digits would be $23$ or $29$ and ...
3
votes
2answers
344 views
How to find number of prime numbers between two integers
I have two integers, $x$ and $y$ so that $x \lt y$. How many prime numbers are there between $x$ and $y$ (exclusive). Is there a formula or algorithm to compute?
6
votes
4answers
803 views
The least prime greater than 2000
I'm a bit curious as to how "real" mathematicians would solve this problem.
"Find the least prime number greater than 2000."
Of course, I can always go brute force:
...
1
vote
2answers
62 views
If $2n+1$ and $4n+3$ are prime, then $2n-1$ and $4n+1$ are not when $n>2$
How do you prove that, for $n>2$, if $2n+1$ and $4n+3$ are prime numbers, then $2n-1$ and $4n+1$ are composite numbers?
4
votes
1answer
63 views
$p^{3}+m^{2}$ is square of a number.
Well i thought it is a nice problem so i will post it here.
1) Prove that for every natural numbers $m$, There is at most two primes $p$ where $p^{3}+m^{2}$ is the square of a number.
2) Find all ...
10
votes
2answers
201 views
$x^2+x+1$ is the cube of a prime.
Please help me find all natural numbers $x$ so that $x^2+x+1$ is the cube of a prime number.(Used in here)
2
votes
1answer
58 views
Is this elementary number theory proof correct?
Let $A(n)$ be the number of primes less than $n$, divided by $n$ (so for example, $A(n) \leq 1$, as there cannot be more primes less than $n$ as there are integers less than $n$). Suppose that $n$ is ...
3
votes
4answers
43 views
Proving $\frac{1}{2}(5x+4),\;2 < x,,\;\text{isPrime}(n)\Rightarrow n = 10k+7$
How is it possible to establish proof for the following statement?
$$n = \frac{1}{2}(5x+4),\;2<x,\;\text{isPrime}(n)\;\Rightarrow\;n=10k+7$$
Where $n,x,k$ are $\text{integers}$.
To be more ...
5
votes
2answers
106 views
Do there exist $29$ consecutive integers so that every of them has exactly $2$ distinct prime factors?
Do there exist $29$ consecutive integers, denote $a,a+1,\cdots,a+28$, so that every of them has exactly $2$ distinct prime factors?
For example, $25$ has only one distinct prime factor, and $30$ ...
5
votes
3answers
217 views
What would be the immediate implications of a formula for prime numbers?
What would be the immediate implications for Math (or sciences as a general) if someone developed a formula capable of generating every prime number progressively and perfectly, also able to prove (or ...
6
votes
0answers
294 views
Partial Solution to the Twin Primes Conjecture — What does it imply? [closed]
But now, as the Mathematician Zhang Yitang from University of New Hampshire in Durham has shown, there is a kind of weak version of the twin prime conjecture. He didn’t prove that a distance of 2 ...
3
votes
3answers
292 views
Is it true that all senary numbers ending in 1 and 5 are primes?
I was reading the Wikipedia article on senary numbers (base 6), which states that:
all primes, when expressed in
base-six, other than 2 and 3 have 1 or
5 as the final digit
Unless I am ...
3
votes
1answer
74 views
Why is $n=\frac{2p^2+2pq+2pr+q^2+2qr+r^2}{p+q+r}$, where $n$ is $\text{prime}$, of form such that $p\pm a,p\pm b,$ are $ \text{prime}, 1<a<b<n$
Why is $n= \left\lfloor \frac{2p^2+2pq+2pr+q^2+2qr+r^2}{p+q+r} \right\rfloor$, where $n$ is $\text{prime}$, of form such that $p\pm a,p\pm b,$ are $ \text{prime}, 1<a<b<n$?
Consider this:
...
2
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0answers
122 views
primes of the form $p=8k+1, 8k+3$ can be expressed as $p=a^2+2b^2$
I have trouble showing that primes of the form $p=8k+1, 8k+3$ can be expressed as $p=a^2+2b^2$.
Thanks in advance.
10
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3answers
472 views
Cardinality of the set of prime numbers
It was proved by Euclid that there are infinitely many primes. But what is the cardinality of the set of prime numbers ?
Cantor showed that the sets $\mathbb{Q}$ and $\mathbb{Z}$ have the same ...
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0answers
53 views
Creation of Prime numbers [closed]
Suppose I have a set of 4 algebraic equations (none exceeding the second order) which create ALL Prime numbers.
A breakthrough or not?
Constantine
0
votes
1answer
45 views
Why is $\{n=4r+1,r = {n-1\over 2}\}\subset \mathbb{P}$ true under these conditions?
Let $p=p_k$, $q=p_{k+1}$ and $r=p_{k+2}$, where $p_m$ denotes the $m$th prime.
I conjecture that whenever $n$ is prime, where $n$ is defined as follows:
$$n = 1+\left(\left\lfloor{p\over ...
1
vote
2answers
42 views
Proving x and y is divisible by p (prime).
If p is a prime number and x and y are integers, how do I prove "if xy and x+y are both divisible by p, then x and y is divisible by p"?
I started like this..
1) p divides xy, so p divides x or p ...
0
votes
0answers
25 views
Tight bounds on the prime counting function
What are the best bounds for $\pi(x)$ i.e. the number of primes less than or equal to $x$ ?
From Wikipedia I saw that:
$$\frac{x}{\ln x}\left(1 + \frac{1}{\ln x}\right) < \pi(x) < \frac{x}{\ln ...
5
votes
7answers
574 views
Prime number generator, how to make
Can anybody point me an algorithm to generate prime numbers, I know of a few ones (Mersenne, Euclides, etc.) but they fail to generate much primes...
The objective is:
given a first prime, ...
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0answers
38 views
Revised: Primes of form $p \equiv m \in S \mod x \ $
Refer to this question for background.
I was speculating if there was an elegant way to define sequences
A007645,A002313,A045357,A045407,A042986,A045331,
A045425,A045374,A045400,A045350,A042988;
...
4
votes
3answers
45 views
Infinitely many primes of the form $4n+3$
I've found at least 3 other posts$^*$ regarding this theorem, but the posts don't address the issues that I have.
Below is a proof that for infinitely many primes of the form $4n+3$, there's a few ...
1
vote
1answer
53 views
Regarding definition of cuban primes
While considering the relationship between $6n-1$ (OEIS A002476) and generalized cuban primes(OEIS A007645) I came across something I thought was interesting:
Seems like the description of ...
3
votes
2answers
48 views
effective version of Mertens Theorem for the Euler product
I'm referring to the theorem given here, which is
$$\displaystyle\lim_{n\to \infty} \:\: \left(\frac1{\ln(n)} \cdot \left(\displaystyle\prod_{p\leq n} \frac1{1-\frac1p}\right)\right) \;\;\; = \;\;\; ...
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0answers
30 views
Could a determinstic primality test specialized to this form of prime exist?
Is it possible there could be an "efficient" deterministic primality test for prime numbers of the form
$$(2^n + 1)^2 - 2$$
or
$$(2^n - 1)^2 - 2$$
in the same vein as the Lucas-Lehmer test for ...
12
votes
5answers
584 views
How many prime numbers are known?
Wikipedia says that the largest known prime number is $2^{43,112,609}-1$ and it has 12,978,189 digits. I keep running into this question/answer over and over, but I haven't been able to find how many ...
3
votes
0answers
68 views
Prime norm ideals that are also principal
Landau's prime number theorem tells us asymptotic formula for counting the number of prime ideals of a number field K, with norm at most X.
I am interested in the the prime ideals with a prime norm. ...
3
votes
1answer
64 views
On the primality of integers of the form $p^2+k$
I am not able to find an answer to the following question:
For which positive even integers $k$ is the integer
$$p^2+k$$ prime, where $p$ is a prime number $\gt5$?
14
votes
3answers
326 views
An Elementary Number theory Problem
A mathematician friend gave me this question (partly as a joke) a few months ago and it has puzzled me for a long time:-
Do there exist infinitely many pairs of primes $(p,q)$ such that ...
1
vote
1answer
55 views
Does $a \mid bc$ imply $\frac{a}{(a,b)} \mid c$?
If $a \mid bc$, then does $\frac{a}{(a,b)} \mid c$? I doubt anybody here is industrious enough to show this via a diagram, but who knows.
1
vote
2answers
43 views
Classify the odd primes $q$ such that a NEGATIVE number is a quadratic residue $\mod{q}$
Suppose we are given $y < -1$. I wish to classify all primes $q$ such that $y$ is a quadratic residue $\pmod{q}$, i.e. such that there exists a number $x$ satisfying $$y \equiv x^2 \pmod{q}.$$
How ...
2
votes
1answer
49 views
Finding a prime $p$ to solve a quadratic congruence $\pmod{p}$
I have a congruence of the form $$ax^2+bx \equiv -1 \pmod{p},$$
where $p$ is an odd prime and $a,b \in \mathbb{Z}$. Given $a$ and $b$, is there a general method to finding $p$ such that the above ...
