0
votes
1answer
49 views

Is there a 10-digit emirp?

Does a 10-digit emirp exist? Unfortunately, the lists of emirps I could find on the Web are quite small and my programming skills aren't good enough to write a program to check all the primes up to ...
1
vote
2answers
102 views

Property of set of prime numbers

let $\{p_1,p_2,p_2,\cdots ,p_r\}$ be the set of $r$($\ge2$) pair wise distinct prime numbers i.e.., $(i\ne j \implies p_i \ne p_j)$ for all $1\le i,j\le r$ ${Statement}$ : For any such ...
1
vote
1answer
33 views

Generalizing primality to other operations

(By "number" below I always mean an element of $\mathbb{Z}^+\setminus $$\left\{1\right \}$.) We all know that a number $p$ is prime iff it cannot be represented as $ab$ for any two numbers $a$ and ...
6
votes
1answer
144 views

Does $\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ converge to the golden ratio?

The sum $\displaystyle\sum\limits_{n=2}^{\infty}\frac{1}{n\ln(n)}$ does not converge. But the sum $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ where $P_n$ denotes the $n$th prime ...
3
votes
0answers
62 views

Resources about infinite primes of form $n^2 + 1$

Where can one find existing work on the following problem? Prove there are infinitely many primes of the form $n^2 + 1$. Resources about related work would also be appreciated.
0
votes
1answer
35 views

References for Legendre's prime-counting function

This question is about Legendre's prime-counting function, the one that can be used to calculate the exact amount of prime numbers that are less than or equal to a given number (as long as the number ...
2
votes
1answer
32 views

Is a Lucas Number with either a power of 2 or a prime index always coprime with all previous Lucas Numbers?

I was looking at this webpage which lists the first 200 Lucas Numbers color-coded with their prime factors and I noticed that all the Lucas numbers with power of two or prime indexes were relatively ...
-3
votes
1answer
65 views

What's the Shannon entropy of the prime numbers?

Here's a note that calculates it as 1. Do you know of any other calculations? http://www.math-math.com/2014/05/shannon-entropy-shannon-entropy-of.html
1
vote
1answer
57 views

Is there any prize for proving conjecture on Fermat's prime ?-+

I know this site is for mathematical questions and answer places, but I need a little help from you in some other aspect. I have searched in google but didn't get any satisfactory answer for it. This ...
0
votes
1answer
44 views

A question about an asymptotic formula

I've been told that the asymptotic formula $\pi(x+y)-\pi(x)\sim y/\ln x$ holds for $y\ge x^{1/2+\varepsilon}$ if Riemann's hypothesis is true, but I was unable to find a journal reference for this. ...
2
votes
0answers
41 views

Renyi entropy of prime gaps

Denote with $p_n$ the $n$-th prime number and let $$ h_N(d) = |\{ n : p_{n+1} < N, p_{n+1} - p_n = d \}| $$ be the number of times that prime gap $d$ happens for primes less than $N$. Let $H = ...
0
votes
0answers
29 views

Almost primes - k as function of x

Below is the plot $N_k(x)$, where $N_k(x)$ is the counting function of numbers with $k$ prime factors (counted with multiplicity), and $k=\lfloor\log_2(x)\rfloor-2$, which generates a repeating ...
1
vote
1answer
38 views

Reference request for proof of Landau's generalised PNT

Could someone please point me in the direction of a proof for Landau's asymptotic formula for k-almost primes: $$\pi_k(n) \sim \left( \frac{n}{\log n} \right) \frac{(\log\log n)^{k-1}}{(k - 1)!}$$ I ...
1
vote
1answer
39 views

Are the conjectural values of $H_{k}:=\lim\inf_{n\to\infty}p_{n+k}-p_{n}$ available somewhere?

The question is in the title. It can be found on the current Polymath 8b project page that one expects to have $H_{1}=2$, $H_{2}=6$, $H_{3}=8$, $H_{4}=12$ and $H_{5}=16$ but I'm interested in larger ...
0
votes
0answers
33 views

Landau's PNT extension

In Tenenbaum's book An Introduction to Analytic and Probabilistic Number Theory, he states in chapter 2.6: From the prime number theorem, it is easy to show by induction on the integer $k\geq1$ ...
3
votes
1answer
85 views

Fermat-quotient of “order” 3: I found $68^{112} \equiv 1 \pmod {113^3}$ - are there bigger examples known?

I'm rereading an older text on fermat-quotients (see wikipedia) from which I have now the Question for $$ b^{p-1} \equiv 1 \pmod{ p^m} \qquad \text{ with $p \in \mathbb P $, $1 \lt b \lt p$ and ...
12
votes
2answers
400 views

On prime factors of $n^2+1$

It is a well-known conjecture that there are infinitely many primes of the form $n^2+1$. However, there are weaker results that one can prove. For example, There are infinitely many positive ...
12
votes
2answers
319 views

Are there Groups of Strictly Primes

Motivation Since Euclid's proof of the infinitude of the primes, the structure and properties of primes has always fascinated mathematicians. This lead to great work in their properties and ...
4
votes
1answer
104 views

Are most numbers of the form $a\cdot b^n+c$ composite?

It seems evident that for $a,b,c$ with $a>0$ and $b>1$ that there are only $o(x)$ primes of the form $a\cdot b^n+c$ with $n\le x.$ Has this been proven? Hooley (Applications of Sieves to the ...
6
votes
1answer
218 views

Are differences between powers of 2 equal to differences between powers of 3 infinitely often?

Consider the equation $2^a-2^b=3^c-3^d$ where $a>b>0$, $c>d>0$, and $a,b,c,d$ are all integers. A computer search for solutions with $a,c\le20$ only finds 8-2=9-3, 32-8=27-3, and ...
3
votes
1answer
97 views

Every two positive integers are related by a composition of these two functions?

How would one prove/disprove this? ... Conjecture: Suppose $p$, $q$ are distinct primes, and define $\ f(n) = n p, \ g(n) = \left \lfloor \frac{n}{q} \right \rfloor$ for all $n \in ...
1
vote
1answer
66 views

What is the well-known result used to prove primality of $n=2pq+1$ under certain conditions?

On Henri Lifchitz's website, we find: If $n=2pq+1$, $p$ and $q$ primes and $q>2p$, if there is an integer $a$ such $a^{n-1} \equiv 1 \pmod n$ and $\gcd(a^{2p}-1,n)=1$ then $n$ is prime. It is ...
0
votes
1answer
46 views

Have either of these sequences been cataloged?

Let $x(n)$ be the remainder when $p(n + 2)$ is divided by 3, where $p(n)$ is the $n$-th prime. Let $y(n)$ = $x(n)$ - 1. Then $\{y(n)\}$ is a binary sequence, that is, is a sequence of $0$'s and ...
5
votes
0answers
268 views

Logical consequence of Euclid's theorem

Are there any far reaching non-trivial consequences of Euclid's infinitude of primes where theorems make use of it? Wikipedia does not have the list of applications of this theorem, rather modern ...
8
votes
2answers
168 views

Extending the primes

I had an idea and I'd like to find out whether it has a name or has been studied before. Imagine the natural numbers and the operations of addition and multiplication, but with the following ...
16
votes
1answer
481 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
vote
1answer
48 views

Information on “stronger form” of Dirichlet's Theorem on Arithmetic Progressions

From Wikipedia: "Stronger forms of Dirichlet's theorem state that, for any arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges." Can anyone direct me ...
5
votes
1answer
109 views

Does this have a name: If an odd prime $p$ does not divide $a$, then $p$ divides $a^n + 1$ or $a^n - 1$

After seeing and doing a bunch of proofs like "For all $a$ in the natural numbers, then if $7$ does not divide $a$, then $7$ divides $a^3+1$ or $a^3-1$," I conjectured the following, but got stuck in ...
1
vote
1answer
85 views

Research on $\log(p)$? [closed]

We are searching for articles, research work or interesting interpretations/applications of $\log(p)$ where $p$ a prime. It should not be only limited to math, contributions from physics are also ...
1
vote
1answer
84 views

Dirichlets theorem on primes

I want to use Dirichlets theorem on primes for my diploma thesis. I want to use following form Let $a,b\in\mathbb{N}$, such that $\gcd(a,b)=1$. Then the set $\{a\cdot n+b| n\in\mathbb{N}\}$ contains ...
1
vote
1answer
330 views

Is there a list of all known Sophie Germain prime numbers?

Is there a list of all known Sophie Germain prime numbers available anywhere for download? I found a small list from OEIS and the top 20 biggest of such primes, but I can't find a list that would ...
0
votes
2answers
80 views

What is the biggest known safe prime number?

I am looking for the biggest known safe prime number. Can someone provide some reference to what that number is and a proof that it is indeed a safe prime number?
3
votes
2answers
1k 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
635 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?
6
votes
3answers
210 views

Good introductory readings to topics related to prime numbers for non-mathematicians

I'm a maths hobbyist who is fascinated by prime numbers. My quest to delve into the interesting parts of the topic is always hindered by my inability to understand the notation and concepts I ...
2
votes
1answer
92 views

Reference for Brun-Titchmarsh inequality

Does anyone know a proof of the Brun-Titchmarsh inequality in the following form starting from the large sieve inequality? Brun-Titchmarsh inequality: Let $\pi(x;q,a) = |\{p \text{ prime}: p\equiv ...
9
votes
2answers
213 views

“Dirichlet's theorem” on pairs of consecutive primes

The number of primes in each of the $\phi(n)$ residue classes relatively prime to $n$ are known to occur with asymptotically equal frequency (following from the proof of the Prime Number Theorem). ...
4
votes
2answers
239 views

Is there a function that only generates primes?

The title sums it up: does there exist a "nice" injective function $f(n)$ such that $f(n)\in\mathbb P$ for all $n\in\mathbb N$? I'm having difficulty specifying exactly what I want "nice" to mean, ...
1
vote
3answers
76 views

Questions about assigning a probability to a randomly chosen large integer $n$ being prime

I heard this question a few days ago, so reciting from memory: If I were to randomly choose an arbitrarily large positive integer $n$, could I write a function that determines the likelihood of it ...
2
votes
0answers
66 views

$\sum_p z^p$ where $p$ is prime

I've started reading Shakarchi's Complex Analysis, and I thought about something interesting. If I haven't mistaken, for any subsequence $A\subset \mathbb{Z}^+$, $\sum_{n\in A} z^n$ has radius of ...
1
vote
0answers
136 views

Matiyasevich polynomial proof

Can someone provide a proof, or a link to a proof, of why does the Matiyasevich polynomial always generate primes for the nonnegative results? Any help will be appreciated.
3
votes
1answer
345 views

Longest Odd/Even Sequence in Composite Patterns

NOTE I have completely reworded this because I made a complete hash of it the first time, it got worse as I added to it. I apologize to anyone who might have been confused, and hope that this will be ...
8
votes
1answer
2k views

Pollard-Strassen Algorithm

I'm aware that the Pollard-Strassen algorithm can be used to find all prime factors of $n$ not exceeding $B$ in $O\big(n^{\epsilon} B^{1/2}\big)$ time. This is really useful because I need to find all ...
7
votes
1answer
75 views

Parameters giving maximal-length Collatz-like sequence

In a recent question the following recursive sequence was considered: $$ a_{n+1} = \cases{\frac{a_n}{2} & $a_n$ is even \\ a_n +d & $a_n$ is odd}, \quad a_1 = d + 1 $$ where $d$ is an odd ...
4
votes
1answer
176 views

infinitely many prime numbers with prescribed digits

My main question is the generalization, though one can answer the first one and it will get accepted. Are there infinitely many primes involving $3,7$ only? Generalization: For what sets of given ...
4
votes
1answer
163 views

Primes of the form $\lfloor x^k\rfloor$

I'm looking for a result (embarrassingly enough, a somewhat famous result) which shows the infinitude in some sense I don't recall of primes of the form $$ \lfloor x^k\rfloor $$ for $k$ fixed and ...
2
votes
4answers
195 views

A good introduction to Prime Numbers

I'm looking for a good introduction to Primes Numbers, their properties, and some of the better known theorems concerning them. I would prefer references assume knowledge of undergraduate level real ...
1
vote
2answers
392 views

Discussion on twin prime conjecture

I understand where I am wrong in my previous post. Also, I am very thankful to all members, who answered and showed my errors in post. Now, I would like to know the proof for the following. "The ...
5
votes
1answer
153 views

A trivial but maybe nonetheless non-trivial method of inferring primality

The topologist J. H. C. Whitehead (not to be confused with his famous uncle) said it is naive to think a theorem is trivial merely because its proof is trivial. Hence I'm wondering if a certain ...
34
votes
2answers
1k views

Small primes attract large primes

$$ \begin{align} 1100 & = 2\times2\times5\times5\times11 \\ 1101 & =3\times 367 \\ 1102 & =2\times19\times29 \\ 1103 & =1103 \\ 1104 & = 2\times2\times2\times2\times ...