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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Paper of Paul Erdös

I'm trying to understand On Arithmetical Properties of Lambert Series by Erdös, but am stuck on the first page. He states: Put $k=\left[(\log n)^{1/10}\right]$ and let $p_1,p_2,\ldots$ be the ...
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1answer
619 views

The Goldbach Conjecture and Hardy-Littlewood Asymptotic

A source I am reading refers to the Goldbach conjecture (that every even number is the sum of two primes), and then immediately follows with the "Hardy-Littlewood conjecture" that $\sum ...
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1answer
46 views

If a 10 digit number is formed using all the digits from 0 to 9 then find the following . [closed]

A) Find the largest such number divisible by 11111 . No matter what I try , I end up with atleast a digit repeating . Since the question says that the no. has all from 0 to 9 , therefore I cant ...
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1answer
46 views

Prove that ($\frac{-2}{p}$)= 1 if and only if p is of the form $8k + 1$ or $8k + 3$

Let p be a prime number. Prove that ($\frac{-2}{p}$)= 1 if and only if p is of the form $8k + 1$ or $8k + 3$, and then from there conclude that there are infinitely many primes of the form $8k + 3$ ...
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1answer
39 views

How to show that this polynomial is a rational polynomial?

In a algebra class, the following polynomial was given: $$f(x) = x^{rs}+11x^{rs-1}+x^{rs-2}+2016x^{rs-3}+rx+s,$$ where $r$ and $s$ are distinct primes (there were several problems stated, but the one ...
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55 views

Prove composite number [closed]

A question for high school students: $a, b, c, d \in \mathbb{Z}$ and $a^2 + b^2 = c^2 + d^2$. Prove $a + b + c + d$ is a composite number (not a prime number).
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How to show this identity is true?

$$\left(\sum_{k=1}^{\infty}\frac{i^{\Omega (k)}}{k^{s}}\right)^{2}=\frac{\zeta (4s)}{\zeta (2s)}\frac{2^{s-1}}{i-2^{s-1}}$$ where $i=\sqrt{-1}$ and $\Omega (k)$ is the number of (not necessarily ...
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45 views

Unclear on how a sieve function is being generalized into a function that uses the möbius function

I am reading through an AMS.org article on prime counting. Let $\Phi(x,b)$ be the number of integers $i$ where $1 \le i \le x$ and $\gcd(i,p_b\#)=1$ where $p_b$ is the $b$th prime and $p_b\#$ is the ...
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1answer
56 views

Counting how many primes exists between square root of a given range .

I am given a range say $l,r$ ( with $1 \leq l,r \leq 10^{14}$). I am also given a cumulative count of prime numbers that exists between $1$ and $10^7$ . For example : For $1$ count=$0$ (as no ...
3
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1answer
44 views

Closed form for $\sum_{p\le n, p\text { prime}}\frac {(-n)^p} {p^n}$

I'm looking for a closed form for the following sum $$\sum_{p\le n, p\text { prime}}\frac {(-n)^p} {p^n}$$ Motivation: This sum is part of a EXP-calculation formula in a game I'm helping develope ...
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30 views

Modular Euler product?

We know the Euler product. $$\zeta (s)=\prod_{p}\frac{p^{s}}{p^{s}-1}$$ I wonder if there is formula or any kind of work for this kind of prime product below? $$\prod_{p\equiv a \ (mod \ ...
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1answer
74 views

Sums of digits of prime numbers: reference request

I wonder if someone could point out to me a paper on the following problem, if it has been considered at all. If not, it would still be nice to have some good references to good papers related to the ...
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1answer
27 views

Proof that there exists a larger prime than prime number P, which is the largest of a finite set of primes?

I am currently working on a problem in which I must prove that there exists a larger number prime number than prime $P$, the largest prime of a finite set. Here are a list of considerations: There ...
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35 views

Lower and upper bounds for the $n$th prime number [duplicate]

Is there anything one could say about in what region a prime number is guaranteed to be in? A general equation for $a$ and $b$, dependent on $n$, in $a<p_n<b$?
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1answer
38 views

Show that $(2^p-1,2^{q-1}-1)=2^{(p,q-1)}-1$ [duplicate]

Let $p$ and $q$ are two prime numbers. Also, let us assume $q|(2^p-1)$. Then show that $(2^p-1,2^{q-1}-1)=2^{(p,q-1)}-1$. Note- $(p,q)$ denotes HCF of $p$ and $q$.
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1answer
50 views

Question on convergence of sum(prime(n)/prime(n+1)/n^2,n=1…infinity)

According to WolframAlpha partial sums for http://www.wolframalpha.com/input/?i=sum%28prime%28n%29%2Fprime%28n%2B1%29%2Fn%5E2%2Cn%3D1...infinity%29&h=1 (I actually used the Maple notation for ...
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3answers
1k views

Does the sum of reciprocals of primes converge?

Is this series known to converge, and if so, what does it converge to (if known)? Where $p_n$ is prime number $n$, and $p_1 = 2$, $$\sum\limits_{n=1}^\infty \frac{1}{p_n}$$
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4answers
269 views

Convergence of series involving the prime numbers

Given the series $A=\sum\limits_{n=1}^{+\infty}\frac{p_n}{p_{n+1}}$ and $B=\sum\limits_{n=1}^{+\infty}\frac{p_{2n}}{p_{2n+1}}$, where $p_n$ is the sequence where the nth number are the nth prime ...
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2answers
130 views

Does this series of primes converge? [duplicate]

Denote the prime numbers $2,3,5,7,\ldots$ as $p_1,p_2,\ldots$. Determine whether the infinite series $\dfrac{p_1}{p_2}+\dfrac{p_3}{p_4}+\cdots = \dfrac{2}{3}+\dfrac{5}{7}+\cdots$ converges. I was ...
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30 views

If $\alpha\approx\sqrt p$ and $\beta\approx\log p$, is $\alpha\beta^{-1}\bmod p\approx p$ with probability $1-o(1)$?

Given $p$ a prime and a random $\alpha\in\Bbb Z_p$ with $\alpha\approx\sqrt{p}$ suppose we pick a random $\beta\approx\log p$ then what is the probability that remainder $\alpha\beta^{-1}\bmod p$ is ...
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1answer
246 views

simple proof of a theorem that is weaker than chen's theorem?

I want to see a simple proof of a theorem that is weaker than chen's theorem. Thus let $m,n$ be positive integers. An m-almost prime is a squarefree integer that is the product of at most $m$ primes. ...
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1answer
30 views

Is there some function which return probability to select prime number from $n$ first Fibonacci numbers.

So my question is: is there function return probability to select prime number from $n$ first Fibonacci numbers. So maybe it realize with $\pi(n)$ function?
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3answers
160 views

Can exist an even number greater than $36$ with more even divisors than $36$, all of them being a prime$-1$?

I did a little test today looking for all the numbers such as their even divisors are exactly all of them a prime number minus 1, to verify possible properties of them. These are the first terms, it ...
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2answers
122 views

A curious pattern on primes congruent to $1$ mod $4$?

It is known that every prime $p$ that satisfies the title congruence can be expressed in the form $a^{2} + b^{2}$ for some integers $a,b$, and unique factorisation in $Z[i]$ ensures exactly one such ...
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2answers
38 views

Sum and product of greatest prime factors

Consider this functions below. $$f(n)=\sum_{k=2}^{n}gpf(k)$$ $$g(n)=\prod_{2}^{n}gpf(k)$$ where $gpf$ is the greatest prime factor function.(For example, $gpf(30)=5$) Is it possible to find an ...
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2answers
67 views

If $30$ divides $p_1^4 + p_2^4 + \ldots + p_{31}^4$. Prove that $p_1=2$, $p_2=3$ and $p_3=5$.

Let $p_1<p_2<\cdots<p_{31}$ be prime numbers such that $30$ divides $p_1^4 + p_2^4 + \cdots + p_{31}^4$. Prove that $p_1=2$, $p_2=3$ and $p_3=5$. No clue how to start..Hints are welcomed.
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1answer
26 views

Proving that there exists a certain set of equations

This may be a dumb question, but it bothers for quite a while. Lets say, we have a certain equation, like $ab-a$ where $a, b$ are primes. Then we generate a sequence for every $a$ and $b$ which looks ...
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210 views

Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?

I have made the following observation: for those n even numbers that do not have primitive roots modulo n ,$Pr(n)$, the set $M(n)$ of those $k$ having a maximum multiplicative order $ord_n(k)$ ...
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Trying to understand how Lehmer's method represents a simplification of Meissel's method for counting primes

My question stems from a wikipedia article on prime counting. The details on Meissel's method can be found in the wikipedia article. As I understand, Meissel proposed two formulas which I asked ...
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3answers
89 views

On the asymptotic growth of the products of prime numbers

Something must be known about the asymptotic growth of the products of prime numbers. Let $p_n$ be the sequence of prime numbers and define $$P_k=\prod_{n=1}^k p_n$$ I'm looking for a sequence $n_k$ ...
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2answers
77 views

A mathematical way to say that the given number is a prime.

I was doing a test on number theory when I encountered this problem- What is the value of $n$ for which $n^2+1$ is a prime? a.$50$ b.$60$ c.$40$ d.$100$ But I am not able to answer ...
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1answer
47 views

On the number of Goldbach partitions

http://members.chello.nl/k.ijntema/partitions.html?text1=8&area1=You+entered%3A+6%0D%0ANumber+of+Goldbach+partitions+%3D+1%0D%0A%0D%0AGoldbach+partitions%3A%0D%0A3+%2B+3+++%0D%0AEnd Here you can ...
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1answer
159 views

A conjecture about prime numbers based on $\sigma_1(n)$ and the Highly Abundant Numbers

I am trying to find the smallest expression $E(n)$, whose distances between the value of the expression and the next prime closer to the expression, $\mathcal{N}(E(n))$, and from the expression to the ...
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Unclear on why Meissel's approach to counting primes works

I am reading through the Wikipedia article on prime counting. The following is presented to describe Meissel's approach: Let $p_1, p_2, \dots, p_n$ be the first $n$ primes. Let $\Phi(m,n)$ be the ...
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3answers
73 views

Prove that every odd prime divides a number of the form $l^2+m^2+1$ $(l,m\in \mathbb {Z})$

I understand this proof http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Ng.pdf (Lemma 2.2) until the point "and hence of $-1 - m^2\mod p$ ". Why is this true, and how does the final line ...
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1answer
68 views

Is there an upper bound for $\pi (n)-\pi (n/2)$?

Is there a nice upper bound for $\pi (n)-\pi (n/2)$ where $\pi$ is the prime counting function?
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3answers
97 views

The greatest common divisor of multiple numbers

What is the cardinality of the following set $\{{\bf x}=(x_1,\ldots,x_d): \text{each } x_i\in \{ 1,\dots,n \},\text{ and } \gcd({\bf x})=1\}$, where $\gcd({\bf x})$ is the greatest common divisor of ...
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2answers
30 views

Is there list of composite Mersenne numbers with their factorization?

Here is a list of known Mersenne primes. http://mathworld.wolfram.com/news/2009-06-07/mersenne-47/ I'm looking for a list of composite Mersenne numbers(when $p$ is prime $2^p-1$ isn't) with their ...
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3answers
48 views

$n = 2^p - 1$ Prove that if n is prime then p must also be prime.

My first thought was to try a contradiction; So given n is prime assume p is not prime i.e $p = p_{1}^{\alpha1} .... p_{r}^{\alpha r}$. But i didnt know where to go from there.
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There is a prime between $n$ and $n^2$, without Bertrand

Consider the following statement: For any integer $n>1$ there is a prime number strictly between $n$ and $n^2$. This problem was given as an (extra) qualification problem for certain ...
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1answer
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Relationship between primes and practical numbers

This is my first post here. I am a musician, and not a mathematician, but I enjoy doing things to prime numbers and seeing what comes out. I have defined a sequence which takes the following values ...
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3answers
83 views

What is value of $a+b+c+d+e$?

What is value of $a+b+c+d+e$? If given : $$abcde=45$$ And $a,b, c, d, e$ all are distinct integer. My attempt : I calculated, $45 = 3^2 \times 5$. Can you explain, how do I find the distinct ...
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1answer
346 views

Bounds for $n$-th prime

In this Wikipedia page I have found that the bounds for $n$-th prime is given by, $$n(\ln n+\ln \ln n)>p_n>n(\ln n+\ln \ln n-1)$$ for all $n\ge6$. Are there even stronger bounds for the $n$-th ...
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1answer
28 views

Obscure understanding of Euclid lemma

Euclid lemma says "If $p$ is a prime that divides $ab$, then $p$ divides $a$ or $p$ divides $b$. If we suppose that $p$ does not divides $a$, then this implies there are integers $s$ and $t$ such ...
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Can Collatz's problem be used as a pseudo random prime sieve?

If you take the concept of $3x+1$, $\dfrac{x}{2}$ and starting at 2, create a tree. On the left nodes you apply the $3x+1$. On the right nodes, if the parent node is even apply the $\dfrac{x}{2}$. ...
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207 views

How to explain “why study prime numbers” to 5th Graders?

I tend to teach 5th graders math ever so often just so they can be "friendly" with math in a playful manner, instead of being afraid. However, one question that I constantly struggle with is this: ...
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78 views

Is $\sum_{n=1}^{{\infty}}\frac{1}{P_{3n}}$ convergent?

Is this sum below convergent? ($P_{n}$ is the nth prime.) $$\sum_{n=1}^{{\infty}}\frac{1}{P_{3n}}$$
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5answers
424 views

Infinite primes proof

There is a proof for infinite prime numbers that i don't understand. right in the middle of the proof: "since every such $m$ can be written in a unique way as a product of the form: ...
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1answer
37 views

More efficient method of computing the square root of $-1 \mod p$

I am currently doing collecting some preliminary data about elliptic curves over finite fields of order $p$ where $p$ is a prime congruent to 1 mod 4. Part of the data collection process requires me ...
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1answer
123 views

Wouldn't the Riemann hypothesis rule out a formula to predict primes? [closed]

Prime formula: a deterministic way to predict primes. Riemann hypothesis: implies "primes are random". If RH is true will we never have a useful prime formula?