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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Ratio of logarithmic primes

Any help is appreciated in proving/disproving the following inequality $$ \frac{\ln{p_{n+1}}}{\ln{p_{n}}} < \frac{n+1}{n} $$
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5answers
1k 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 ...
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3answers
503 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
98 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
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1answer
160 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
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?
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3answers
851 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
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2answers
329 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
110 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
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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
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1answer
290 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
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1answer
99 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.
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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
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0answers
181 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, ...
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2answers
37 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
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2answers
477 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 ...
7
votes
1answer
149 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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2answers
115 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
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3answers
130 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
224 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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1answer
117 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
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1answer
97 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
112 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
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1answer
146 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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4answers
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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
208 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
377 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
151 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
492 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
69 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
124 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
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0answers
176 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
91 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 ...
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2answers
342 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
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2answers
580 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
292 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 ...
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2answers
88 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 ...
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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 ...
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1answer
60 views

Are there infinitely many primes of the form $n^k+l_0$ for fixed $l_0$ when $(n,k)$ runs through the $\mathbb N\times ({{\mathbb N}\setminus\{1\}}$)?

I do not know if this what I am going to ask is immediate consequence of something known but if not it may have an easy answer which I do not see, so any help would be great. Let us define sequence ...
0
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1answer
966 views

Prove that if $n$ is a composite, then $2^n-1$ is composite. [duplicate]

Not sure if I'm doing this correctly but this is what I've done: Assume that $n$ is composite and suppose $2^n-1$ is a prime for $n \gt 2$. Then, $2^n-1 = 2k$ for some $k \in \Bbb Z $, $\forall n$. ...
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vote
2answers
201 views

Indices - Numbers as a product of prime numbers

I've checked the internet which only provides basic $x^2 \times x^3 = x^5$ information and have concluded that I need resort to a Q & A website. The basics of indices are fine for me, but it's ...
3
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2answers
529 views

When is the number 11111…1 a prime number?

For which $n$ is the sum: $$\sum_{k=0}^{n}10^k$$ a prime number? Are they finite?
2
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1answer
117 views

Primes of form $n^{n+1} - (n+1)^n$

I was playing with some numbers today and saw (with a bit of joy) that $3^4 - 4^3$ is the $(3 + 4)$th prime number, which is sort of neat. Then naturally I asked the question, what kind of number $n$ ...
2
votes
1answer
74 views

How many solutions to prime = $2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$

Let $a,b,c$ be integers, no sign restriction. Let $p$ be a given prime. How to find the number of solutions to $p = 2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$ ? Note, from Heron's ...
3
votes
3answers
169 views

How many solutions to prime = $a^3+b^3+c^3 - 3abc$

Let $a,b,c$ be integers. Let $p$ be a given prime. How to find the number of solutions to $p = a^3+b^3+c^3 - 3abc$ ? Another question is ; let $w$ be a positive integer. Let $f(w)$ be the number of ...
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vote
2answers
93 views

How many solutions to prime = $(d^2-2ad+b^2-2ab+2a^2)(d^2-2cd+2c^2-2bc+b^2)$?

Let $a,b,c,d$ be integers $>-1$. Let $p$ be a given prime. How to find the number of solutions to $p = (d^2-2ad+b^2-2ab+2a^2)(d^2-2cd+2c^2-2bc+b^2)$ ? I assumed that this polynomial above does not ...
4
votes
2answers
214 views

Period of a set of primes

Let $S$ be a set of primes. We call a positive integer $n$ a period of $S$ if $p\in S$ implies $q\in S$ for all primes $q$ with $q\equiv p\mod n$. Show that if $n_1$, $n_2$ are periods of $S$, then ...
0
votes
1answer
71 views

Primes $n=\overline{10101\cdots01}$ with $k$ ones.

Find all primes $n=\overline{10101\cdots01}$ with $k$ ones. The number is in standard base 10.
0
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1answer
26 views

What is the pair $(n,k)$ called where $n$ is an integer and $k$ is the ordered factorization index?

I’m developing a number class (as in Object-Oriented Programming) and am wondering what to call it. At its core, it represents an integer, but in a way in which not all integers are unique. What it ...