Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a ...

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3
votes
0answers
154 views

Conjectured new primality test for Mersenne numbers

How to prove that this conjecture about a new primality test for Mersenne numbers is true ? Definition: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$ ...
10
votes
2answers
696 views

Most dense subset of numbers that avoids arbitrarily long arithmetic progressions

The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression. I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid ...
12
votes
2answers
327 views

Are there open problems for primes which are known for probable primes?

Define "probable prime" (PP) to be natural $n>1$ satisfying $2^{n-1} \equiv 1 \pmod{n}$ or $n=2$. Probable primes are the union of the primes and base two pseudoprimes. This definition is much ...
2
votes
2answers
274 views

Primality criteria for specific class of Wagstaff numbers ?

I asked this question on mathstackexchange but didn't get any answer . Definition : Let $W_p$ be a Wagstaff number of the form : $W_p=\frac{2^p+1}{3}$ , with $p\equiv 1 \pmod 4$ Next , ...
8
votes
1answer
490 views

Estimate on radical of $2^n \pm 1$

Not sure if this belongs to MO or not. Are there any lower bound on radical of $2^n \pm 1$ (Recall that radical of integer $rad(k)$ is a product of primes which divide integer $k$)? As an example If ...
5
votes
0answers
159 views

Congruences involving binary forms and primes of the form $x^2+y^2$

Let $a_s$ be \begin{align*} a_s=\sum_{k=0}^s{s+k\choose k}2^k, \end{align*} which is the coefficient of $x^s$ in \begin{align*} \frac{3-\sqrt{1-8x}}{2(x+1)\sqrt{1-8x}}. \end{align*} ( see ...
11
votes
1answer
853 views

Order of magnitude of $\sum \frac{1}{\log{p}}$

Question: What is the order of magnitude of the following sum? $$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$ Additional information: Since $$ ...
0
votes
0answers
105 views

The maximum lengthed sequence of prime numbers with certain conditions (denizens)

Definition - Denizen A sequence $\lbrace a_k \rbrace$ is a denizen if all of it's members are prime numbers, i.e $a_0, a_1, ... a_n \in \mathbb{P} $; and it satisfies the following condition; ...
7
votes
1answer
301 views

Bounded gaps between primes in arithmetic progressions

Has Zhang's work on bounded gaps between primes been extended to the following theorem? For any arithmetic progression $an+b,\gcd(a,b)=1$, there is a constant $H$ (depending only on $a$) such that ...
2
votes
1answer
336 views

A general question about strictly non-palindromic numbers

For a definition, see the wikipedia page: http://en.wikipedia.org/wiki/Strictly_non-palindromic_number So according to the wikipedia page, under properties, all strictly non-palindromic numbers with ...
22
votes
6answers
3k views

Explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros. Because there are various explicit formulae ...
1
vote
1answer
32 views

Prime constant graphicial representation [closed]

I have something to check. It is about prime constant (I don't know if it is officially so called), but it is created on following way. We start with binary point number represenation. Zero followed ...
6
votes
0answers
162 views

Intersection between the sums of the first positive integers, primes and non primes

Is the following conjecture true ? $$\left\{\sum\limits_{\begin{array}{c}k=1\\k\in\mathbb{Z}\end{array}}^nk \ |\ n\in\Bbb Z\right\} \cap \left\lbrace ...
-4
votes
1answer
183 views

Gauss-Wantzel theorem, Fermat primes and solvability of S_n [closed]

Gauss-Wantzel theorem asserts that a polygon with $n$ sides is constructible if and only if $n$ is a product of a power of $2$ and distinct prime Fermat numbers, where the Fermat number of index $k$ ...
2
votes
4answers
276 views

Prime divisors of values of a polynomial on an infinite set

This may be a well known problem: Let $f$ be a polynomial with integer coefficients. Is it possible for the set of prime divisors of values of $f$ on an infinite set of integers, to be finite? I ...
10
votes
0answers
338 views

Between Fermat's primes and the twin primes

Let me start with a curiosity. The integers $11,13,17,19$ are prime numbers, and $101,103,107,109$ are prime as well. One might wonder whether there is another occurrence where $10^m+1,10^m+3,10^m+7$ ...
4
votes
1answer
364 views

Unexpectedly prime rich cubic polynomial

We got a cubic polynomial which is unexpectedly prime rich. Let $f(x)=29160 x^3 + 30132 x^2 + 8046 x + 643$ and $\pi_f(n)$ the number of primes values of $f(x)$ for $x \in [1,n]$. Let ...
0
votes
0answers
66 views

Polynomial identities for congruent numbers and Bunyakovsky's conjecture

Bunyakovsky's conjecture states that polynomial with integer coefficients takes infinitely many prime values unless there are obvious reasons not to. It appears to imply something about polynomial ...
14
votes
1answer
542 views

A Collatz-like function that bifurcates on primes

This is likely piling one mystery on another, but ... I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows: $$ f(n) = \begin{cases} n^2 & \text{if} \;n \;\text{is ...
44
votes
3answers
7k views

Is a “non-analytic” proof of Dirichlet's theorem on primes known or possible?

It is well-known that one can prove certain special cases of Dirichlet's theorem by exhibiting an integer polynomial $p(x)$ with the properties that the prime divisors of $\{ p(n) | n \in \mathbb{Z} ...
1
vote
0answers
95 views

Can someone explain some of the steps in this paper clearly?

I'm having trouble understanding the steps this paper makes to come to the conclusion $p_{f}(d) \sim e^d\sqrt{d}$ Marek Wolf, First occurrence of a given gap between consecutive primes, preprint, ...
5
votes
1answer
440 views

Are there effective small intervals in which primes are dense?

As mentioned in Terry Tao's comment to this question, it is constructively known that there are primes between sufficiently large cubes. $\:$ According to wikipedia, "there exists a constant $\: ...
1
vote
1answer
189 views

reference on Dirichlet theorem on primes in arithmetic progression

I appreciate if you could help me to find a reference (and a proof). Combining Dirichlet theorem on primes in arithmetic progression with Chebotarev densitiy theorem, we know that given two positive ...
0
votes
1answer
226 views

Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}=n^p$ and $p$ is an odd prime? [closed]

I would like to know if there are infinitely many $k$ for which $$\sigma(k)/k=n^p$$ such that $m=k{n}^{p-1}$ with $m,n>0$ and $p$ is an odd prime? Note: $\sigma(\frac{m}{{n}^{p-1}})$ is the sum of ...
0
votes
0answers
147 views

arithmetic progressions with few primes

Is this true ? Let $\beta_0$ be a positive number. One may find $\beta>\beta_0$, $0<\lambda<1$, and infinitely many $q>1$ so that there exists an arithmetic progression of step $q$, $a_1, ...
2
votes
1answer
196 views

Counting function for prime pair with bounded gaps between them [duplicate]

I'll start by noting that I am not at all an expert on number theory. However I do use it in a proof and would like your assistance if possible. Yitang Zhang breakthrough result established that ...
13
votes
1answer
580 views

Elementary Proof of Infinitely many primes $\mathfrak{p} \in \mathbb{Z}[i]$ in the sector $\theta < \arg \mathfrak{p} <\phi $

A quick look at the primes in $\mathbb{Z}[i]$ suggests they might be evenly distributed by angle if we zoom out on a coarse enough scale. I would like ask about the much weaker statement forgetting ...
5
votes
3answers
363 views

Logarithmic integral, $π(x)$ and $x/(\ln x)$

The function $\text{Li}$ (logarithmic integral) is defined for $x>0$ by $$ \text{Li}(x)=\int_2^{x}\frac{dt}{\ln t}. $$ The prime number theorem, proven by Hadamard and de la Vallée-Poussin in 1896 ...
1
vote
0answers
158 views

Does the proof of Conjectures B and D of Hardy and Littlewood have any implication on the generalized Riemann hypothesis they used?

Does the proof of Conjectures B and D of Hardy and Littlewood have any implication on the generalized Riemannn hypothesis they used? In their paper, Some problems of 'Partitio numerorum'; III - On ...
7
votes
0answers
505 views

“Forthcoming paper” of Goldston-Graham-Pintz-Yıldırım

The above-named authors of [1] and its (significantly different) published version [2] write: In a forthcoming paper, we will show how the methods here can be extended to prove corresponding ...
9
votes
2answers
871 views

Asymptotics of the n-th prime using the gamma function

In the paper http://rgmia.org/papers/v8n2/eepnt.pdf, the author proves that proves an explicit inequality on prime numbers using the gamma function and as a corollary, he showed that. $$ p_n = n ...
6
votes
1answer
403 views

Approximating a real by a ratio of primes

Let $x$ and $y$ be positive reals in $(0,1)$ with $x < y$ and $y-x =\epsilon$. I seek smallest primes $p$ and $q$ such that $$x \le \frac{p}{q} \le (x+\epsilon) = y \;.$$ Q. What upper bound ...
3
votes
1answer
198 views

Least prime for which a square-free integer is a non-residue

Suppose $a$ is a square-free integer and $\left(\frac{a}{p}\right)=1$ for the primes $p\leq k$. I'll call $a$ a quasi-square of order $k$. What I am interested in is the maximum value of $k$ in terms ...
4
votes
0answers
110 views

Visibility in a prime orchard

This suggests a variant on Polya's orchard problem. That problem asks1 for which radius $\epsilon$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the ...
2
votes
2answers
217 views

binomial/factorial identity mod p

In trying to determine the spectrum of a well-known ergodic transformation, I came up with the following useful (for me) result. Let $p$ be a prime and $a$ a positive integer. Then for $M$ a positive ...
0
votes
1answer
155 views

A conjecture on the prime counting function

I was thinking about the Second Hardy-Littlewood conjecture for quite sometime (some of my posts are related to this). In one of my earlier post I conjectured that the inequality ($\pi(x)$ denotes the ...
13
votes
1answer
373 views

Small quotients of smooth numbers

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...
4
votes
1answer
296 views

Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip?

This question expands on this one from MSE. In the literature about Dirichlet $L$-series, I found that their Euler products: $$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$ ...
10
votes
3answers
590 views

About the prime divisors of values of polynomials

Let $P(x)$ be a polynomial having integer coefficients (and degree $\geq 3$), and let $p_1<p_2<\dots$ be the prime divisors occurring in the set of values $\{P(n):\ n\in\mathbb{Z}\}$. Is it ...
16
votes
1answer
1k views

When is the product $(1+1)(1+4)…(1+n^2)$ a perfect square?

This is a modification of an unanswered problem on the math StackExchange. When is the product $(1+1)(1+4)…(1+n^2)$ a perfect square? If $(1+1)(1+4)…(1+n^2)=k^2$ then one possibility is $n=3$, ...
7
votes
1answer
1k views

The large sieve for primes

Let $\Lambda(n)$ be the von Mangoldt function, i.e., $\Lambda(n) = \log p$ for $n$ a prime power $p^k$ and $\Lambda(n) = 0$ for all $n$ that not prime powers. Let $$S(\alpha) = \sum_{n \leq N} ...
2
votes
1answer
377 views

Number of twin primes

Consider number of twin primes less than $x$. We know that this number less than $\frac{Cx}{\log^2 x}$ for some constant $C$. Denote by $p_n$ the $n$-th prime number. Do we have the same result ...
2
votes
2answers
574 views

If $\binom{2p}{p}$ is $(-1)^{p-1} \bmod 2p+1$ is then $2p+1$ prime?

Let $p$ be a positive integer; if $2p+1$ is prime then it is easily checked that $$(2p+1)\mid\left(\binom{2p}{p}+(-1)^{p-1}\right);$$ conversely I conjecture that if the above divisibility assumption ...
4
votes
0answers
145 views

Are there infinitely many zeros of $\chi(s)+ \dfrac{2^{s}- 2^{2s-1}}{2^{s}-1}$ on the critical line?

Take $\chi(s)= 2^s\,\pi^{s-1}\,\sin\left(\frac{\pi\,s}{2}\right)\,\Gamma(1-s)$, so that $\zeta(s)=\chi(s)\,\zeta(1-s)$. The zeros of $\chi(s)=-1$ and the non-trivial zeros $\rho$ of $\zeta(s)$, seem ...
8
votes
2answers
442 views

Existence of polynomials of degree $\geq 2$ which represent infinitely many prime numbers

To my knowledge it is open so far whether the polynomial $x^2+1 \in \mathbb{Z}[x]$ takes infinitely many prime numbers as values. Is it known so far whether there is at all any polynomial $P \in ...
23
votes
8answers
11k views

What are the connections between pi and prime numbers?

I watched a video that said the probability for Gaussian integers to be relatively prime is an expression in $\pi$, and I also know about $\zeta(2) = \pi^2/6$ but I am wondering what are more ...
0
votes
2answers
85 views

Min number of primes up to n

According to the prime number theorem there are about $n/\ln(n)$ primes less than $n$. This value is a limit but it could fluctuate. My question is, is there a known bound on this fluctuation? i.e. ...
6
votes
3answers
701 views

Lower density of {primes} times themselves

We say that a set $A\subseteq \mathbb{N}$ has lower density 0 if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 0.$$ Given $A,B\subseteq \mathbb{N}$ we set $A\cdot B = \{a\cdot b: ...
34
votes
4answers
3k views

Why could Mertens not prove the prime number theorem?

We know that $$ \sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x) $$ where $c_1$ is a constant. Again Mertens' theorem says that the primes $p$ satisfy $$ \sum_{p \le x}\frac{1}{p} = \ln\ln ...
10
votes
0answers
128 views

The multiplicative group generated by shifted primes

I am asking for references about the following problem. In particular, it is still open? If not, what is the state of the art result? Problem 1. Let $\Gamma$ be the multiplicative subgroup of ...