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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What is the smallest positive common difference of a 6-term arithmetic progression consisting entirely of (positive) prime numbers?
What is the smallest positive common difference of a 6-term arithmetic progression consisting entirely of (positive) prime numbers?
are divisibility rules applicable here?
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Proof that there exists an integer $n \geqslant2$ such that $n^2$ divides $2^n + 3^n$
Proof that there exists an integer $n \geqslant 2$ such that $n^2$ divides $2^n + 3^n$. I came up with this problem and I don't have a clue how to start, or even if it is not trivial at all.
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1answer
76 views
Is there a list of all known Sophie Germain prime numbers?
Is there a list of all known Sophie Germain prime numbers available anywhere for download? I found a small list from OEIS and the top 20 biggest of such primes, but I can't find a list that would ...
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2answers
59 views
What is the biggest known safe prime number?
I am looking for the biggest known safe prime number. Can someone provide some reference to what that number is and a proof that it is indeed a safe prime number?
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3answers
126 views
Is there a rule for prime numbers?
I've passed by this article:
http://gauravtiwari.org/2011/12/11/claim-for-a-prime-number-formula/
and this paper:
http://www.m-hikari.com/ams/ams-2012/ams-73-76-2012/kaddouraAMS73-76-2012.pdf
They ...
6
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1answer
56 views
Proving $\lim_{n\to\infty}\left(n-\sum_{k=2}^{n}\frac{1}{\sum_{i=1}^{\infty}\frac{1}{i^k}}\right)=1+\sum_{p\in P}\frac{1}{p\left(p-1\right)}$?
$$\lim_{n\to\infty}\left(n-\sum_{k=2}^{n}\frac{1}{\sum_{i=1}^{\infty}\frac{1}{i^k}}\right)=1+\sum_{p\in P}\frac{1}{p\left(p-1\right)}$$
$P$ is primes.
Interesting question ran across while tutoring. ...
2
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1answer
69 views
Converting Prime Numbers from Base $10$ to Base $4$
It seems like when you convert a prime number from base $10$ to base $4$, the base $4$ number, when read back as a base $10$ number is also prime.
Example:
$13_{10} = 31_{4}$.
$31$ in base $10$ ...
6
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2answers
119 views
Currently, what is the largest publicly known prime number such that all prime numbers less than it are known?
So recently, an absurdly large prime number was found, but a lot of prime numbers less than it are still not known. I am wondering up to where we know all the primes.
I put "currently publicly known" ...
2
votes
5answers
109 views
On prime numbers
let $q$ be a prime
let $p = 2^q -1 $
is p must be prime always for any prime q ?
is this is true always ?
or it is false for some prime q ?
if it is false , give an example to show that there ...
7
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1answer
129 views
Prime number generating function as product expansion
I am interested in prime number generating function.
$$f(x)=1+\sum \limits_{k=1}^\infty p_{k}x^k=1+2x+3x^2+5x^3+7x^4+11x^5+....$$
I would like to find the function as product expansion and to check ...
3
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1answer
77 views
question on prime number
Just came across the following question:
Suppose $p$ is a prime number and $p+1$ is a perfect square. Find the sum of all such prime numbers.
This is simple and there is a unique $p$, namle ...
1
vote
1answer
83 views
Solving Quadratic Congruences in P Mod P
Please help me solve the following:
$$2p^2 - 42p + 221 = 0 \mod p.$$
Just messing around with the numbers I noted the following:
$p = 0 \mod p$,
therefore:
$2p^2 - 42p + 221 = p \mod p$,
...
4
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3answers
39 views
Problems with proof that $p|2^m-2^n$ if $p-1|m-n$
This was a homework assignment that I have already made unsuccesfully. However, no answers were given and I'm still curious. The question is as follows:
"If $p$ is an odd prime number and $m > n$ ...
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0answers
45 views
Galois invariant of Tate twists
let $k$ be the maximal extension of $\mathbb{Q}$ unramified outside a set $T$ of primes in $\mathbb{Z}$.
Take a $p\in T$ and set $G=Gal(k/\mathbb{Q})$. I would like to now if there is a classical ...
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1answer
229 views
Article about primes.(Revised)
I'm trying to write a article about primes, and I'm curious whether I can really involve other topics (like complex numbers) and relate them to observe peculiar properties of primes. Or can I try ...
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3answers
91 views
Prime number divisibility
The following line is in a proof I'm reading, and I don't understand the logic:
Let $\frac{a}{b}$ be an arbitrary element ($a$ and $b$ both integers). Since $p$ is a
prime, and $p$ doesn't ...
0
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1answer
128 views
Ratio of primes $(x^2+x+(5+6m))$ to $(x^2+x+(3+6m))$
What I did:
For a large n and $x\leq n$ I counted the number of primes of the form $x^2+x+(5+6m)$ for $m = 0, 1, 2, 3,..., n/2,$ added the number of primes for each m together and called the sum A.
...
8
votes
3answers
188 views
Proof of Wolstenholme's theorem.?
According to the theorem :
$$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...+\frac{1}{p-1} =\frac{r}{q}$$
And we have to prove that $r= 0 \pmod{p^2}$.
(Given $ p>3$, ...
2
votes
1answer
78 views
Distance function: $d(x,x)$ must equal zero?
Let $p$ be prime and assume $\lVert r\rVert_{p}=p^{-k}$, if $r=p^k(m/n)$, where $m$ and $n$ are relative primes of $p$. Define $$d(x,y)=\lVert x-y\rVert_{p}$$ on $\mathbb{Q}$. Show that $d(x,y)$ is a ...
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2answers
79 views
Questions regarding p-adic expansion and numbers
As opposed to real number expansions which extend to the right as sums
of ever smaller, increasingly negative powers of the base $p$,
$p$-adic numbers may expand to the left forever, a property ...
5
votes
1answer
61 views
finite field to rational fraction
Suppose I have a number $n\in\mathbb F_p$, i.e. an element of the finite field obtained by arithmetic modulo some (odd) prime $p$. I'm looking for a way to find a simple description of $n$ as a ...
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1answer
119 views
Set of numbers pairwise relatively prime
Given a positve integer n, we can find infinitely many positve integers $b$ such that the $n-1$ integers in the set $\{b+1,\,2b+1,\,3b+1,\,...,\,(n-1)b+1\}$ are pairwise relatively prime.
I assume ...
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1answer
27 views
Proof of Generalized Primorial Primes
Let's call the numbers of the form $k\times p\# \mp1$, the Generalized Primorial Primes.
One can find many $k$ for a fixed $p$ such that $k\times p\# \mp1$ be prime. As an example for $p = 8933$ ...
2
votes
1answer
37 views
Ratio of logarithmic primes
Any help is appreciated in proving/disproving the following inequality
$$
\frac{\ln{p_{n+1}}}{\ln{p_{n}}} < \frac{n+1}{n}
$$
5
votes
5answers
211 views
prime divisor of $3n+2$ proof
I have to prove that any number of the form $3n+2$ has a prime factor of the form $3m+2$. Ive started the proof
I tried saying by the division algorithm the prime factor is either the form ...
0
votes
3answers
152 views
When a prime number p divides $ab$ then we have either p divides a or p divides b.Prove that $\sqrt {p} $ is not rational for any prime number p.
When a prime number $p$ divides $ ab $ then we have either $p$ divides $a$ or $p$ divides $b$. Prove that $ \sqrt p $ is not rational for any prime number $p$.
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0answers
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intuitive meaning behind Mertens' theorem
I have just been introduced the topic of distribution of primes, big O notation and aymptotic functions so please correct me if I say something that does not make sense. I am looking to get an ...
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1answer
71 views
What are primes in the form of $2^n+1$ called?
What are primes in the form of $2^n+1$ called? I know that those of form $2^n-1$ are Mersenne primes, but I'm not sure about the other ones.
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2answers
119 views
Where can one find a list of prime numbers?
I am looking for the biggest list of precomputed prime numbers one can find and download. Where should I look?
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2answers
142 views
Is there a list of safe prime numbers?
I am looking for a list of precomputed safe prime numbers. Where can I get such a list?
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243 views
Proof of Prime Maker Conjecture
In my mind the following conjecture is true:
Prime Maker Conjecture
I call a number $n$ factor-resistant to $q$ if $q\not\mid n$. Considering $n$ as a composite number, the idea is to make $n$ ...
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1answer
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3 primes conjecture
let be $ p,q,r $ prime numbers AND 'n' an integer
is then true that we can always look for p,q,r and an integer n so
$$ p^{n}+q=r $$
$ 5+2=7$
$ 2^{3}+3=11 $
$ 3^{4}+2=83 $
abnd so on
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Are the Prime Numbers $O(f(n))$ where $f(n)$ is some polynomial?
Are the prime number, denoted $ p(n) $, $O(f(n))$, for any polynomial $f(n)$?
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1answer
100 views
If n is an odd pseudo prime number, then $M_n = 2^n-1$ is a larger one
I came across this Theorem in "Elementary Number theorem" by David B. Burton :
"If n is an odd pseudo prime number, then $M_n = 2^n-1$ is a larger one."
I am not able to understand why this result ...
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1answer
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Testing for convergence $\sum_{j=1}^{\infty}\frac{1}{\sum_{i=1}^{j}p_i}$
How would we test for convergence the series below?
$$\sum_{j=1}^{\infty}\frac{1}{\sum_{i=1}^{j}p_i}$$
where $p_i$ is the $i$th prime number. I'd be glad to learn an elementary way. Thanks.
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1answer
176 views
Using the Euler totient function for a large number
So I have a test in a couple of hours and I'm having trouble finding information on how to use the Euler totient function for a large number so I'm wondering if someone could give me step-by-step ...
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Iterate over combinations ordered by sum
I have a sorted list of a large number of primes. I want to iterate over combinations of fixed size $n$ in increasing order of their sum. Naturally the standard approach for $n=4$:
$$s_0 = \sum(A, ...
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2answers
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Is there a pattern (or a name and expression for the pattern) of the intervals between all primes?
With the recent interest in Mersenne primes, I got thinking whether there was any mathematical expression for the pattern of intervals (or sequence composed of interval lengths) between ordinary prime ...
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2answers
256 views
Show that $n!+1$ has a prime factor $\;>n$; show $\exists$ infinite number of primes
I don't know how to prove this and it's really bugging me. Thanks to anybody that can help!
Let $n$ be any natural number. Prove that $n! + 1$ contains a prime factor greater than $n$ and use that to ...
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1answer
46 views
Apparent patterns in ratios of consecutive primes
I was plotting the values of $\frac{P(n+1)}{P(n)+2}$, where $P(n)$ is the nth prime number. I noticed very easily that the values seem to belong very nicely to a set of "trajectories". They clearly ...
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Problem over prime numbers
Which is the largest integer $n<1000$ so that $n$, $n+2$ and $n+4$ are primes?
I have tried to solve this problem but have not reached an argument worth
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prove , if $p,q$ be two primes with the property , $q$=$p$+1 then $p$=2 and $q$=3
prove , if $p,q$ are two primes with the property , $q$=$p$+1 then $p$=2 and $q$=3
how can we prove something like that ?
my information in number theory is not big , and i have no idea about the ...
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2answers
126 views
Finding a counterexample to a Prime Factorization Conjecture
Let $\mathbb{Z}_{\geq 2}$ be the set of natural numbers starting at 2:
$$\mathbb{Z}_{\geq 2}= \{2, 3, 4, 5,\ldots\}.$$
An natural number's prime factorization is odd if the total number of primes in ...
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0answers
79 views
Is this prime formula too general?
I managed to develop a working sequence formula for primes but I think it is too general so I wanted to post it here as a question and let the community say if we could get something from it or not.
...
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1answer
62 views
Math expression for an infinite sequence of primes
At the beginning I would like to ask if there are infinite prime numbers of the form:
$$\prod_{i=1}^{n} p_i + 1$$
where $p_i$ is the $i$-th prime number; but after a google search I found that they ...
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1answer
76 views
For which prime $p$ is $x^4 \equiv -1 \pmod{p}$ solvable?
Let $p$ be a prime. I know, due to Euler's criterion, that if $x^2 \equiv -1 \pmod{p}$ is solvable, then $p \equiv 1 \pmod{4}$ simply because I inspect which $p$ that are such that $(-1)^\frac{p-1}{2} ...
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32 views
Proving finite vs infinite representation of $p/q$ in base-$b$?
Reading up on positional notation and converting between different bases, I came across this statement:
For integers p and q with gcd(p, q) = 1, the fraction p/q has a finite representation in base b ...
3
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1answer
52 views
What happened to the Mertens constant in the strong prime twins conjecture ??
To estimate the amount of primes in an interval $\left(2,x\right)$ one might naively sieve by computing $ x \left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)...\left(1-\dfrac{1}{p_i}\right)$ ...
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1answer
117 views
Is this the way to estimate the amount of lucky twins?
To estimate the amount of prime twins between $3$ and $x$ we just take $x \prod_{p}(1-2/p)$ where $p$ runs over the primes between $3$ and $\sqrt x$. Lucky numbers are similar to prime numbers. Does ...
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3answers
104 views
What can primes, except 2, 3, and 5, be congruent to $\pmod {30}$?
After some trials, I found out that a prime $p \gt 5$ is congruent to $q\pmod{30}$, where $q$ is also a prime, and $1 \le q \lt 30 \;$ (i.e. $p \equiv q\pmod{30}.$
Is there a way to write a formal ...
