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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Why it is important to find largest prime numbers?

It always takes a lot of effort and money to find the next largest prime number. Why is it so important to do this work and what is the application those numbers?
0
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1answer
76 views

To which extent distribution of Riemann non-trivial zeros follow a gauss process?

I am trying to clearer and preciser understand to which extent the distribution of the non-trivial zeros of the Riemann $\zeta$-function follow a Gauss process? Yet, what I figured out from ...
2
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1answer
153 views

There is a real number $\alpha >1$ such that $\Bigl\lfloor2^{2^{{.}^{{.}^{{.}^{2^{\alpha }}}}}}\Bigr\rfloor$ is prime for all $n\geq 1$

Theorem: There exists a real number $\alpha >1$ that if $$\alpha =\alpha _0,\quad 2^{\alpha _0}=\alpha _1,\quad \dots\quad 2^{\alpha _n}=\alpha _{n+1},\quad \dots$$ then for all $n\geq ...
3
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1answer
120 views

A curious fact on partitions of 7 integer and related question.

Let's start writing $7$ partitions, marking them with $n\ A$, where $n$ is a number of terms in partition and $A$ is a set of terms in it. $$\underbrace {7}_{1\ \{7\}} = 7$$ $$\underbrace {6 + ...
0
votes
1answer
84 views

Discrepancy between terms of sum and sum

My question is why the following happens, and whether we can correct (2) below to account for an errant factor of 2. By a slight generalization* of the argument of this problem we have I think that ...
17
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1answer
532 views

How can we prove $\pi =1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\cdots\,$?

I saw the beautiful result that was proved by Euler in Wikipedia but I do not know how it can be proved. $$\pi =1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \frac{1}{6} + ...
1
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1answer
92 views

Series equivalent to $\sum p_k$

Looking at a theorem of Chebyshev, I noticed that $$\sum_{n=0}^{\infty} \sum_{p_k < n} \frac{(\log p_k)^n}{n!} = 2 + 3 + ...+ p_k.$$ Proof. Letting $x = \log p_k$ and writing out the expansion of ...
4
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2answers
187 views

What is the need for classifying numbers like integer, whole number etc?

what are the everyday life examples where we use the classification. I feel all the math behind the scenes(in computers weather etc ) is highly abstracted. I am looking for strong answers to tell the ...
0
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3answers
624 views

Prove that every integer is either prime or composite

In the book I'm reading, the following proof is given for the stated theorem: Let n be any integer that is greater than 1. Consider all pairs of positive integers $r$ and $s$ such that $n = rs$. ...
4
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2answers
145 views

Estimating $\sum_{p_2 \leq x} (\log p_2)^2$

This was an exercise to use the approach here to estimate the sum $\sum_{p_2 \leq x} \log (p_2)^2,$ in which $p_2$ are numbers containing two prime factors (repetitions allowed). $\pi_2(x)$ is the ...
4
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2answers
148 views

Limiting value of $\lim \frac{1}{k}\sum_{n=1}^k \frac{p(n+1)-p(n)}{\log p(n)}$

Empirically it seems $$\lim_{k\to \infty} \frac{1}{k}\sum_{n=1}^k \frac{g(n)}{\log p(n)} = 1\tag{1} $$ in which p(n) is the nth prime and g(n) is the prime gap $p(n+1)-p(n).$ Cramer conjectured ...
12
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5answers
2k views

Why does one counterexample disprove a conjecture?

Can't a conjecture be correct about most solutions except maybe a family of solutions? For example, a few centuries ago it was widely believed that $2^{2^n}+1$ is a prime number for any $n$ . For ...
2
votes
1answer
166 views

Limit of $\sum\frac{1}{p(\pi(n))}$

Let $p(n)$ be the nth prime and $\pi(n)$ the number of primes not exceeding n. I wonder if we can show that $$\tag{1} S = \sum_{n= 2}^k \frac{1}{ p (\pi (n))} \sim \log k. $$ We know by comparison ...
13
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2answers
246 views

Prove that $m^{2013}-m^{20}+m^{13}-2013$ has at least $N$ prime divisors

for positive integer $N>1$,There always exists $m$ such that $$m^{2013}-m^{20}+m^{13}-2013$$ has at least $N$ prime divisors Thank you all, this is good problem, but I don't know how to solve it.
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0answers
103 views

Lower bound for $\pi(x)$

Is there a way to show that $$\frac{x}{\ln x} < \pi(x),$$ for sufficiently large $x$, using only elementary calculus? Apparently it is true for $x \geq 17$ (see this article). However, I am looking ...
7
votes
1answer
206 views

Prime numbers in Collatz sequences

This question/request is twofold. First, if this is a stupid question or if it has been addressed before, please say so (bluntness is optional), and I will crawl back into my cave... My question: is ...
0
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1answer
322 views

Quadratic expression that generate primes

I recently learned that there exist quadratic expression that generate some primes and some of these equations generate more primes than others. In the following video, the person shows the following ...
13
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2answers
301 views

Number of digits until a prime is reached

Begin with a random digit from $1$ to $9$. Add a random digit to the right-hand side from $0$ to $9$ until a prime number is reached. How many digits are necessary in the avarage ? More precisely: ...
2
votes
1answer
939 views

Sum of the first n Prime numbers

Let $P_i$ denote the i-th prime number. Is there any formula for expressing $$S= \sum_{i=1}^m P_i.$$ We know that there are around $\frac{P_m}{\ln(P_m)}$ prime numbers less than or equal to $P_m$. ...
2
votes
2answers
181 views

Some questions about Goldbach's conjecture

I was thinking about the usage of Dirichlet's theorem in proving some facts about the Goldbach's conjecture. I will start with an example. Using Dirichlet's theorem, we know that there are infinitely ...
3
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0answers
87 views

An intuitive interpretation of Montgomery pair corrlation function vs. prime divisibility?

Theorem - If the Riemann hypothesis would be true, and the Montgomery pair correlation conjecture (see linked article page 183-184) true too; let $p \in \Bbb P$ prime, $n \in \Bbb N$ and $$\mathcal ...
2
votes
2answers
81 views

A function that brings back the prime number just before it?

Is there a function that brings the prime number just before it? I.e P(18)=17 P(6)=5 P(28)=23; I know how weird that sounds.
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2answers
145 views

Prime number based calendar: how to calculate intermediate values?

I'm trying to define a new [silly] calendar, because there aren't enough of them yet. My calendar, so far, is specified: epoch (orthodox) is the moment the leading edges of the dinosaur-killing ...
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3answers
451 views

What are the connections between number theory and topology ?? [closed]

What are the connections between number theory and topology ? How does topology relate to number theory ? In particular I wonder about primes and diophantine equations. I do not see how the amount of ...
3
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1answer
263 views

(Not) Surprising Result on Natural Numbers as Sum of $k$-Almost Primes

I started with the following idea: Let $P_k$ be the infinte set of all $k$-almost primes. The counting function for $k$-almost primes less than $x$, is $\displaystyle \pi_k(x)\sim\frac{x}{\log ...
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1answer
73 views

Conjecture on limit of $1-(n^{p-1}\mod p)$

Given $p \in \Bbb P$ prime, $n \in \Bbb N$ and $$\mathcal V_p=1-(n^{p-1}\mod p)$$ let me conjecture that $$\lim_{p\rightarrow \infty}\mathcal V_p = \operatorname{sinc}(2\pi \,n)$$ Question: Is ...
0
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2answers
337 views

Algebraic Representability of Prime Number Generators

Does anyone happen to have at hand a short, elegant proof that demonstrates that there do (or do not) exist one or more algebraically representable prime number generating functions?
2
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0answers
193 views

Prime number finding via polynomials

I try to find approximation polynomial to estimate which number is prime or not. Addtion to this, (If It is possible) To find the closed form of coefficients of the series ($c(n)$) Euler found the ...
2
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1answer
127 views

The relation of $\zeta$-function and $p^k$ for $Re(s) \le 1$?

The von Mangoldt function: $$\Lambda(n) = \begin{cases} \log p &; \mbox{if }n=p^k \mbox{ for some prime } p \mbox{ and integer } k \ge 1, \\ 0 &; \mbox{otherwise.} \end{cases}$$ establishes a ...
7
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3answers
237 views

If $q$ is prime, then $w=(1+q+q^2)/3$ is a composite integer iff $w$ has a prime divisor less than $\sqrt{w}$ and congruent to $1$ modulo $6$

I have a question regarding prime numbers. Specifically, I wonder if the following is true: If $q$ is a prime and $w=(1+q+q^2)/3$ is an integer, then $w$ is composite iff $w$ has a prime divisor ...
7
votes
1answer
750 views

Is the $n$-th prime smaller than $n(\log n + \log\log n-1+\frac{\log\log n}{\log n})$?

Let $p_n$ be the $n$-th prime. Wikipedia gives the following known bounds on $p_n/n$ when $n\geq6$: $$ \log n+\log\log n-1 \leq \frac{p_n}{n} \leq \log n+\log\log n. $$ If I take the first few terms ...
2
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1answer
237 views

Confused about harmonic series and Euler product

So Euler argued that $$1 + \frac{1}{2} + \frac{1}{3} + \frac {1}{4} + \cdots = \frac {2 \cdot 3 \cdot 5 \cdot 7 \cdots} {1 \cdot 2 \cdot 4 \cdot 6 \cdots} $$ which you can rearrange to $$ \left( \frac ...
1
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1answer
34 views

Proving that $\sum_{i=2}^{M}\frac{\pi(x^{1/i})}{i}=O(x^{1/2})+O(Mx^{1/3})$

How do I prove that $$\sum_{i=2}^{M}\frac{\pi(x^{1/i})}{i}=O(x^{1/2})+O(Mx^{1/3}).$$ I tried to use Prime Number theorem for $\pi(x)$ and then approximating the summation by integral, but when I used ...
6
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0answers
145 views

Density of products of a certain set of primes

I have an infinite set S of prime numbers with relative density 0 (that is, $\lim_ns_n/p_n=\infty$ with $S=\{s_1,s_2,\ldots\}$ and $s_1 < s_2< \cdots$). I would like to find the size (in some ...
4
votes
2answers
113 views

Valid Alternative Proof to an Elementary Number Theory question in congruences?

So, I've recently started teaching myself elementary number theory (since it does not require any specific mathematical background and it seems like a good way to keep my brain in shape until my ...
0
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1answer
72 views

Why does a non-zero density function not imply infinitude of what it measures?

Consider the following density function for the twin primes: Numbers $x-2$, $x-4$ are twin primes iff: $x \ne 2,4 \ mod \ 2 $ $x \ne 2,4 \ mod \ 3 $ $x \ne 2,4 \ mod \ 5 $ $x \ne 2,4 \ mod \ 7 $ ...
2
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1answer
521 views

The smallest example of a Carmichael number

A composite integer n is a Carmichael number if the only Fermat witnesses for $n$ are those $a \in \mathbb Z_n^+$ which are not coprime with $n$. The smallest example of such a number is $561 = ...
0
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3answers
85 views

Prime Numbers and Multiples?

Other than prime numbers are all numbers multiple of 2,3,5 and 7 (Other Prime numbers as well). Suppose like if we need 8 it's the combination of 2.2.2, and 15 as 5.3 etc.
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1answer
3k views

use contradiction to prove that the square root of $p$ is irrational

On a practice exam, our teacher provides us with this question and this answer. Let $p$ be a prime number. Use contradiction to prove that $\sqrt{p}$ is irrational. ANSWER: BWOC assume ...
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3answers
59 views

Distance beetwen primes [duplicate]

How to prove that distance between two neighboring primes can be arbitrarily large? Thanks for help, John
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2answers
86 views

Proof that from FTA follows $n>\alpha_{n,k}$

Recall the FTA in canonical form $$n=p_1^{\alpha_{n,1}}p_2^{\alpha_{n,2}} \cdots p_k^{\alpha_{n,k}} = \prod_{i=1}^{k}p_i^{\alpha_{n,i}}$$ where $n,i,k \in\Bbb N$ and $\alpha \in\Bbb N_0$. Two ...
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3answers
595 views

Does the correctness of Riemann's Hypothesis imply a better bound on $\sum \limits_{p<x}p^{-s}$?

This is follow up question on this: How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \mathrm{Re}(s) < 1 $? There it is stated that: $$ \sum_{p\leq x}p^{-s}= \mathrm{li}(x^{1-s}) + ...
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1answer
65 views

2 questions on logarithms of $1\over x$ where $x = 10^k -1$

Let $a_n$ denote the logarithm of the $n$'th sum of $\{{1\over9},{1\over99},{1\over999},...\}$, such that $a_n = ...
2
votes
1answer
73 views

A passage in the newman proof of the prime number theorem.

In the proof of the statement that $\theta(x) \sim x$ based on the fact that $\int_1^\infty { \frac{\theta(x) - x}{x^2}dx } < \infty$ We assume that for some $\lambda > 1$ there are ...
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2answers
580 views

A problem about the largest prime factor of $n^2+1$

Let $f(n)$ be the largest prime factor of $n$. The image of function $g(n)=\sqrt{f(n^2+1)}$ is like this: Question: If we want to draw a horizontal line which bisects the points from $n=1$ to ...
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1answer
2k views

How to find generator in a finite group?what is generator?

Suppose that a group $Z_p=${$1,2,3......(p-1)$} where p is a prime number. How to Determine the generator/generators of this group? what are the possible method of finding it?
3
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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 ...
2
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4answers
2k views

Get numbers that have only 2,3 and 5 as prime factors

I am given an integer N. I have to find first N elements that are divisible by 2,3 or 5, but not by any other prime number. ...
4
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3answers
181 views

Prime number in a polynomial expression

Will be glad for a little hint: let x and n be positive integer such that $1+x+x^2+\dots+x^{n-1}$ is a prime number then show that n is prime
1
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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 ...