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.

learn more… | top users | synonyms

1
vote
0answers
58 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 ...
1
vote
1answer
268 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 ...
2
votes
3answers
119 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
votes
1answer
153 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. ...
7
votes
3answers
599 views

Proof of Wolstenholme's theorem

According to the theorem, if $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots+\frac{1}{p-1} =\frac{r}{q}$$ then we have to prove that $r\equiv0 \pmod{p^2}$. (Given $p>3$, otherwise ...
2
votes
1answer
90 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 ...
3
votes
2answers
601 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
202 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 ...
-1
votes
1answer
112 views

The asymptotic of the first Chebyshev function, using the Prime Number Theorem [closed]

Using the prime number theorem, show that: $\vartheta (x) \sim x$ Where $\vartheta (x) := \sum_{p \le x} \log p$ Any help on this would be great, thanks in advance.
1
vote
1answer
676 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 ...
1
vote
1answer
90 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 $p = 8933$ ...
3
votes
2answers
96 views

Ratio of logarithmic primes

Any help is appreciated in proving/disproving the following inequality $$ \frac{\ln{p_{n+1}}}{\ln{p_{n}}} < \frac{n+1}{n} $$
7
votes
5answers
2k 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 ...
-1
votes
3answers
535 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 $p$.
1
vote
0answers
100 views

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 ...
5
votes
1answer
162 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.
4
votes
2answers
2k 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?
5
votes
3answers
962 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?
5
votes
2answers
337 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 $n$ ...
1
vote
1answer
111 views

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
0
votes
0answers
61 views

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 $f(n)$?
0
votes
1answer
310 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 ...
6
votes
1answer
100 views

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.
4
votes
2answers
3k 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 ...
0
votes
0answers
193 views

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, ...
1
vote
2answers
38 views

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 ...
2
votes
2answers
508 views

Show that $n!+1$ has a prime factor $\;>n$; showthat there are infinitely many 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 ...
7
votes
1answer
164 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 ...
1
vote
2answers
117 views

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
4
votes
3answers
131 views

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 ...
2
votes
2answers
236 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 ...
1
vote
1answer
120 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 ...
3
votes
1answer
99 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} ...
0
votes
1answer
115 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 ...
4
votes
1answer
150 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)$ ...
11
votes
4answers
1k views

Is it true that the book 'Calculate Primes' has found the pattern? [closed]

I read about a book called 'Calculate Primes' by James McCanney. It claims to have cracked the pattern for generating families of primes, and also the ability to factorize large numbers. ...
3
votes
1answer
216 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 ...
2
votes
3answers
411 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 ...
5
votes
0answers
153 views

Primes of the form $\frac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and ...
8
votes
2answers
495 views

Primes of form $x^2+x\pm k$

Let $\pi(n) = $ number of primes $ \leq n.$ Let $x_i <n,~~ i = 1,2,3,...$ such that $x_i^2+x_i \pm k $ is prime, in which $k \ll n$ is an odd integer. Let $\pi_k(n)$ be the number of such primes ...
2
votes
1answer
72 views

What error bound would an epsilon closer to the Riemann hypothesis give?

$s=1$ line gives: $$\psi(x) = x(1+o(1))$$ classical zero free region gives: $$\psi(x) = x + O(x e^{-c \sqrt{\log x}})$$ for some positive constant $\delta$ RH gives: $$\psi(x) = x + ...
2
votes
1answer
128 views

Show that Fermat number $F_n$ and its index $n$ are coprime.

I want to show that $\gcd(F_n,n)=1$, where $F_n=2^{2^n}+1$. How to prove this? I can show that that $\gcd(F_n, F_m)=1$ for any natural $n$ and $m$, and that $F_{n+1}=(F_n)^2-2F_n+2=F_0\dots ...
7
votes
0answers
179 views

Prime numbers, analysis of polylogarithms

Can any interesting results concering prime numbers be obtained using the analytic properties of the polylogarithm, similar to how analytic methods are used on the zeta function to obtain results ...
5
votes
1answer
144 views

Sequence involving primes of form $n^2 + n+1$

Looking at prime numbers $p_i $ of the form $n^2+n+1$ and the derived expression $$1 - \prod_{i=1}^{j}\frac{(p_i-1)}{p_i}$$ it seems (I do not claim it and do not see why it should be true) that ...
3
votes
1answer
95 views

Question involving prime numbers, *brothers* numbers.

I thought about the following problem, probably it already appears in mathematical literature. Definition 1: Operator $\unrhd$, is binary operation, defined for natural numbers as follows: To every ...
3
votes
2answers
427 views

Proof of lack of pure prime producing polynomials.

I recently encountered this following proposition: For every polynomial, there is some positive integer for which it is composite. What is the most elementary proof of this?
3
votes
2answers
611 views

Prove that if $a$, $b$, and $n$ are positive integers such that $a^n|b^n$ then $a|b$

This is how I did it, but not sure if it is a correct proof. Assume that $a^n | b^n$. Then $(a^n, b^n) = a^n$. So, $$b^n = a^n(p_1p_2p_3...p_k)^n$$ $$b^n = (ap_1p_2p_3...p_k)^n$$ $$b = ...
2
votes
1answer
313 views

How does sieve that Chen used to prove Chen's theorem work?

In the Number Theory for Computing, Song Y. Yan states that Chen used "complicated arguments based on sieve method", when proving what is now called Chen's theorem. How does this sieve work? Does it ...
0
votes
2answers
90 views

Congruences and Primes

Show that if $p$ is an odd prime, with $p = 3 \pmod{4}$, then $$ (\mathbb{Z}_{p}^{*})^4 = (\mathbb{Z}_{p}^{*})^2 $$ More generally, show that if $n$ is an odd positive integer, where $p = 3 ...
5
votes
6answers
2k views

Prove that if $n$ is a composite and $p \gt \sqrt[3]n$, then $n/p$ is a prime.

Also, $p$ is the least prime factor of $n$. I'm trying to do this by way of contradiction. Since $n$ is a composite, $n = pq$, for some $q \in \Bbb Z$. So, we have $p | n$, $q|n$ and $q = \frac ...