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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11
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1answer
217 views

Is $\sum_{n \ge 1}{\frac{p_n}{n!}}$ irrational?

Is $\sum_{n \ge 1}{\frac{p_n}{n!}}$ irrational, where $p_n$ is the $n^{\text{th}}$ prime number? This question is spurred by the comment thread on this question where I presented a rough idea of a ...
0
votes
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 ...
1
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0answers
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 $a<p_n<b$?
1
vote
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 $q$.
1
vote
1answer
31 views

How many times do I loop Solovay--Strassen primality test

First, I am aware of this former thread: math.stackexchange Yet it doesn't answer my question. If I want to check if an integer $n$ is prime using the Solovay--Strassen test, how many times do I ...
2
votes
1answer
73 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 ...
3
votes
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 ...
1
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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 ...
2
votes
0answers
53 views

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 ...
1
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0answers
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 ...
0
votes
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?
3
votes
0answers
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 ...
7
votes
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 ...
1
vote
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 ...
3
votes
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 ...
1
vote
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 ...
21
votes
1answer
552 views

Is this a way to prove there are infinitely many primes?

Someone gave me the following fun proof of the fact there are infinitely many primes. I wonder if this is valid, if it should be formalized more or if there is a falsehood in this proof that has to do ...
0
votes
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 ...
2
votes
0answers
31 views

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 ...
1
vote
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 ...
1
vote
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 ...
13
votes
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 ...
4
votes
1answer
67 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?
1
vote
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 ...
4
votes
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.
3
votes
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 ...
4
votes
0answers
67 views

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 ...
3
votes
3answers
79 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 ...
2
votes
3answers
88 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 $n_k$ ...
1
vote
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 ...
0
votes
0answers
45 views

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 $\dfrac{x}{2}$. ...
0
votes
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 ...
9
votes
1answer
92 views

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 ...
5
votes
5answers
423 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: ...
6
votes
2answers
78 views

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

Is this sum below convergent? ($P_{n}$ is the nth prime.) $$\sum_{n=1}^{{\infty}}\frac{1}{P_{3n}}$$
10
votes
2answers
148 views

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 ...
4
votes
0answers
38 views

Evaluating a double sum involving prime numbers

Evaluate $$ \lim_{n \to \infty} \frac{1}{n} \sum_{p \leq n} \sum_{k=0}^\infty \ln p \left\{\frac{n}{(p-1)p^k} \right\}$$ where $\{ x\}$ denotes the fractional part of $x$, and $p \leq n$ denotes all ...
1
vote
1answer
37 views

Is it rational or not?

I have two interesting question : Is this number rational or not: $$0.F_{1}F_{2}F_{3}...$$, where $F_{i}$ - Fibonacci number. And is this number rational or not: $$0.p_{1}p_{2}...$$
-2
votes
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?
4
votes
1answer
49 views

Help with a proof of a special case of Dirichlet's Theorem

So I am reading through a proof of a special case of Dirichlet's theorem on arithmetic progressions, specifically a proof that if $p$ is prime, then there are infinitely many primes congruent to $1$ ...
6
votes
4answers
204 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: ...
1
vote
1answer
45 views

On an exercise from a journal using Hölder and Stoltz theorems, now with twin primes

I use [1] (in spanish) for the sequence of positive terms defined by $$ a_k = \begin{cases} \frac{1}{k}(\frac{1}{p_k}+\frac{1}{p_k+2}), & \text{for the kth twin prime pair} \\ 0, & \text{if ...
2
votes
1answer
68 views

A conjecture about the prime counting function

Using this lemma it can be proved that $\Delta(m,n)=\pi(m\cdot n)-\pi(m)\cdot\pi(n)+1$ (where $\pi$ is the prime counting function) is a function $\Delta:\mathbb N\times\mathbb N\to\mathbb N$. ...
2
votes
0answers
26 views

The nth roots of $z_n=p_n\cdot(i)^{p_n}$, where $i=\sqrt{-1}$ and $p_n$ is the nth prime number

I want refresh some basics too in Complex Analysis. Let $p_n$ the sequence of prime numbers $2, 3, 5, 7\ldots$, thus $p_n$ is the general term of this sequence, and $i=\sqrt{-1}$ is the complex ...
-5
votes
1answer
105 views

Twin prime conjecture hypothesis

Let $c$ be a positive integer and fix $a=c-1$, and $b=c+1$. Challenge: Find the largest value of $c$ such that $ac\pm1$ and $bc\pm1$ are pairs of twin primes. For example, with $c=6$ we have ...
4
votes
1answer
60 views

Proving for $n \ge 25$, $p_n > 3.75n$ where $p_n$ is the $n$th prime.

The elements of the reduced residue system modulo $30$ are $\{1, 7, 11, 13, 17, 19, 23, 29\}$ If we order them as $e_1, e_2, e_3, \dots$ so that $e_1 = 1, e_2 = 7, \dots$, it follows that $3.75(i-1) ...
4
votes
2answers
130 views

Infinite product involving primes

I just had my first analysis course as an undergraduate, and I'm trying to learn more about analytic number theory. Right now I'm looking at prime numbers in particular--I'm studying (mostly just ...
2
votes
1answer
34 views

Periodicity of fractional part of a sequence

Let $u_n = \mathrm{frac}(a n^2)$, where $a$ is some real number and $\mathrm{frac}$ denotes the fractional part. Question 1) Can $u_n$ be eventually periodic even if $a$ is irrational? Question ...
2
votes
2answers
38 views

Prove that $\sigma(n)\le \lceil\log_2(n)\rceil$

Let $f:\mathbb{N}\to\mathbb{N}$ be defined as $f(1)=1$ and if $n=\prod_{r=1}^{k}p_r^{\alpha_r}$ is the prime decomposition of $n$ then: $$ f\left(n\right)=\prod_{r=1}^{k}(p_r-1)^{\alpha_r} $$ Let ...
0
votes
3answers
48 views

Generate prime of x decimal digits using bit-oriented prime generator

I've got a question on stackoverflow where somebody asks to generate a random 18-digit prime. Unfortunately, the only prime generator is the one from OpenSSL. This prime generator is however geared ...