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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0answers
150 views

Can the work of Hardy & Ramanujan about partitions shed light on Hardy-Littlewood's k-tuple conjecture? [on hold]

If I'm not mistaken, Hardy and Ramanujan produced an asymptotic formula for the number of partitions of an integer that was later shown to be an exact formula by Selberg. But Hardy also formulated ...
8
votes
1answer
186 views

Arbitrarily many primes in a Fibonacci-type sequence

It is conjectured that the standard Fibonacci sequence contains infinitely many primes. While this is perhaps too difficult, I am wondering about the following simpler version: Question. For any $K$, ...
3
votes
1answer
406 views

origin of analogy “primes as the atoms of number theory/ arithmetic”

a math student recently challenged me on the old comparison/ analogy of prime numbers to "the atoms of number theory or arithmetic" and then was wondering the origin of the phrase. where does this ...
0
votes
0answers
150 views

Asymptotic value of sum over Möbius function

Consider the sum $$ S(x)=\sum_{\substack{{d\mid P(\sqrt{x})}\\{d\leq x}}}|\mu(d)|, $$ where $P(x)$ is the product of all primes less than or equal to $x$, and $\mu(d)$ is the Möbius function. ...
3
votes
1answer
110 views

Reference Request on the existence of $k$ satisfying $P(\Phi_d(2))^k \gt \Phi_d(2)$ for all $d$

I am working my way through the literature regarding the following conjecture: There is a positive integer $k$ such that for all positive integers $d$, $$P(\Phi_d(2))^k \gt \Phi_d(2).$$ I am ...
20
votes
1answer
1k views

How many primes can there be in a short interval?

Given $n \in \mathbb{N}$, let $\pi(n)$ denote the number of prime numbers $\leq n$. What is $$ \limsup_{m \rightarrow \infty} \left( \limsup_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} ...
-1
votes
0answers
158 views

Any heuristics explaining why one seems to have $2n=p+q\Rightarrow\pi(p)+\pi(q)=Li(2n)+o(1)$?

The title explains almost everything: suppose $2n$ is a Goldbach number, i.e. an even positive integer such that there exist two primes $p$ and $q$ fulfilling $2n=p+q$. It seems that "most of the ...
1
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0answers
254 views

Controlling residue class sizes

$\forall k>10^2$ is there $m_k$ such that at infinite primes $q>m_k$ $\exists$ pairwise coprime $a,b,c$ such that $$(1)\quad q^{\frac14+\frac1k}<a,b,|a-b|<q^{\frac14+\frac2k} < ...
-1
votes
1answer
124 views

Find the values of $n$ that satisfy this inequality involving a product over prime numbers [closed]

Inequality What values of $n$ satisfy the following inequality? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the $i$th ...
4
votes
1answer
235 views

The values of $n$ which satisfy an inequality about prime numbers

For which values of $n$ does the following inequality hold for? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-1}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the ...
2
votes
1answer
87 views

Set of triple-primes satisfying a certain equation

Is there a set of triple-primes satisfying the following equation? $p_1p_2+p_2p_3+p_3p_1+p_1+p_2+p_3=2^β,p_1p_2p_3=2^α-1,α>β.$ I have checked the first 11 numbers that no one satisfy the above ...
18
votes
1answer
298 views

On random divisor sums modulo $2^k$

Let $k,n,\ell$ be positive integers with $k,n\ge 2$ and $0\le \ell \le k-1$. For each integer $2\le j \le n$, choose a divisor $d_j$ of $j$, uniformly at random from the divisors of $j$. We denote by ...
7
votes
0answers
129 views

k-Almost Primes in short intervals

According to this question every interval $[x, x + x^{0.45}]$ contains a product of two primes, and this has been improved further slightly. Are there better results available for $k$-almost primes? ...
6
votes
1answer
198 views

Can $b^4+1$ be a pseudoprime to base 2 (except for Fermat numbers)?

Cross-post: This very elementary question was first posted to Mathematics Stack Exchange but the response I got there (even after offering a bounty) was not useful. For the purpose of this question, ...
35
votes
3answers
3k views

Why such an interest in studying prime gaps?

Prime gaps studies seems to be one of the most fertile topics in analytic number theory, for long and in lots of directions : lower bounds (recent works by Maynard, Tao et al. [1]) upper bounds ...
4
votes
3answers
386 views

On a theorem of Hensel about congruence of binomial coefficient

In the paper Binomial coefficients modulo prime powers, Andrew Granville stated the following theorem: Let $n, m$ and $r=n-m$ be three given positive integer and $p^k$ is the exact power of $p$ ...
7
votes
0answers
185 views

Can primes be (almost) random sequence in von Mises sense?

Random models for primes (such as Cramer's model) have been extensively used for informal justification of various conjectures involving primes. It is crucial to understand in what sense sequence of ...
0
votes
1answer
133 views

An upper bound on $\sum_{n^{1/3}<p,q\leq n^{1/2}} \frac{n}{pq}-\lfloor \frac{n}{pq}\rfloor$

I would like to ask if there is a good upper bound on the difference $$D_2(n)=\sum_{n^{1/3}<p,q\leq n^{1/2}} \left(\frac{n}{pq}-\left\lfloor \frac{n}{pq}\right\rfloor\right)\quad (1) $$where $p$ ...
15
votes
2answers
2k views

Could this unexpected bias in the distribution of consecutive primes have any impact on the security of encryption algorithms?

In a recent paper a quite unexpected result about a new pattern in prime numbers emerged: Unexpected biases in the distribution of consecutive primesby Oliver, R. J. L.; Soundararajan, K. (Submitted ...
4
votes
0answers
201 views

Asymptotic estimate for a random model of primes

Question Let $$ \pi_{rm_c}(x) = \sum_{ \substack{ {n\leq x}\\{(n+a,P(\sqrt{n}))=1}}} 1-1, $$ where $P(x)$ is the product of all primes less or equal to $x$ and $a$ is a random integer constrained to ...
4
votes
0answers
81 views

Large Gaps Between Almost Primes

What is the best lower bound for the longest interval contained in $[1,x]$ free of primes and products of two primes? In other words I am asking for the best lower bounds in a variant of the ...
1
vote
0answers
60 views

Behavior of partial Euler product in the critical strip (with Dirichlet Character)

Consider a primitive Dirichlet Character $\chi$ (non principal) and the partial Euler product attached to the L-function $L(\chi,s)$ ($p_i$ are the prime numbers) : $$P(\chi,N)=\prod_{i=1}^{N} ...
1
vote
0answers
152 views

Is the following claim true about systems of quadratic congruences modulo consecutive prime numbers

Is the following true? Choose any value for $y : y \in \mathbb{N}$ If $N(y)$ is the smallest natural number that satisfies the following system of quadratic congruences: $N(y)^2 \not\equiv 1$ ...
4
votes
2answers
151 views

Evolution of partial sum of a sequence of induced Dirichlet characters

Let's consider the Dirichlet Character $\chi_3(n)$ modulo 3 given by $\chi_3(1)=1$, $\chi_3(2)=-1$ and $\chi_3(3)=\chi_3(0)=0$. Lets consider the sequence of induced characters $\chi^{P_N} $ obtained ...
3
votes
1answer
173 views

On partial sum of non-primitive Dirichlet characters

Consider a Dirichlet character, $\chi(n)$, and the partial sum : $$S(\chi,x)=\bigg |\sum_{n=1}^{x} \chi(n)\bigg|$$ There are many works to bound this sum when $\chi$ is a primitive character, but ...
9
votes
0answers
254 views

Independence between the number of prime factors of $n$ and $n+2$

I am interested in having an upper bound for the cardinality of $\#\left\{n\leq x\,:\quad\omega(n)=k, \omega(n+2)=\ell\right\}$ for $k,\ell\geq 1$, where $\omega(n)=\sum_{p\vert n}1$ counts the ...
1
vote
0answers
36 views

Distribution of primes and near-primes among $\prod p_k \pm 1$

For $n\in \Bbb{Z}^+$ define the statement "$n$ is $k$-social" to mean that $$ \prod_{i=1}^n p_i +1 \mbox{ has exactly } k \mbox{ prime factors} $$ where $p_i$ is the $i$-th prime. So for example $5$ ...
0
votes
1answer
100 views

What is the relative size of the radical of an ABC-triple relative to the number of primes up to its largest element?

Write $\bf N$ for the set of natural numbers, and $P$ for the set of primes. For $x$ in $\bf N$ let $p(x)$ be the product of the primes dividing $x$ (that is, the "radical" of $x$). Also write $\#(x)$ ...
0
votes
1answer
170 views

Is it consistent with Cramer's conjecture to conjecture that $x/\pi_{2}(x)>2B_{2}/6\times\log^{2}x$?

Brun's constant $B_{2}$ is defined as $B_{2}=1/3+1/5+1/5+1/7+1/11+1/13+...$ where the sum is taken on $p$ such that $p$ is an element of a couple of twin primes. The number of twin primes below is ...
3
votes
1answer
466 views

Use of infinitude of primes in the Green-Tao theorem [closed]

In a video I watched last night on nuking mathematical mosquitos, Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The Green-Tao theorem says ...
4
votes
1answer
245 views

Can we always attain another prime via inserting digits between the digits of a fixed prime?

The sequence OEIS A080437 is For n > 10, let m = n-th prime. If m is a k-digit prime then a(n) = smallest prime obtained by inserting digits between every pair of digits of m. I don't see why ...
3
votes
2answers
500 views

Primary structures in $\mathbb Q$

I'll formulate a topic restricted here to the positive rational numbers $\ \mathbb Q_{_{>0}},\ $, then will pose a question (Q2) plus some related, to which I don't know the answers nor reference. ...
18
votes
1answer
376 views

How big are the prime factors of $2^kp - 1$?

I have already asked this question here. No answers despite the bounty (which has now ended) Let $p$ be a prime number, $p > 3$. Does there always exist $k \in \mathbb N_{\ge 1}$ such that the ...
4
votes
0answers
125 views

Are prime gaps of even index essentially larger than those of odd index?

Let $g_{n}:=p_{n+1}-p_{n}$ be the $n$- th prime gap, and let's introduce the following summatory functions: $$G_{1}(x):=\sum_{1\leq n\leq x}g_{2n-1}$$ $$G_{2}(x):=\sum_{1\leq n\leq x}g_{2n}$$. Let's ...
3
votes
0answers
67 views

Square integral of finite Euler product

Consider the finite Euler product $$ P(t) = \prod_{r=1}^R \left(1 + p_r^{i t} \right). $$ (Here $p_1, p_2, \dots$ are of course the primes.) Question: What is a good asymptotic upper bound for $$ ...
11
votes
1answer
233 views

Elementary prime-generating sequences

A student of mine keeps coming again and again and telling "I've found a formula $n\mapsto f(n)$ giving all primes" or sometimes "infinitely many primes", where $f$ is a classical function (I mean ...
13
votes
1answer
519 views

Does the sum $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converge?

Prove, if possible in an elementary way, that $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converges/diverges, where $p_n$ denotes the $n^{\textrm{th}}$ prime.
1
vote
1answer
287 views

Asymptotics of “ugly” function elucidate Goldbach's conjecture?

Question We now define the following "ugly" function: $$ A_c(s,r,n,m) = \begin{cases} 1 & \text{ if only $sr+nm=2c$ } \\ 0 & \text{otherwise} \end{cases} $$ How does the ...
3
votes
2answers
236 views

Unfamiliar prime-generating polynomials related to Heegner numbers

I just stumbled on a set of prime-generating polynomials of the form $$9 n^2-3 H n+H (H+1)/4$$ (where $H$ is a Heegner number $>11$), which generate the same number of distinct primes as their more ...
1
vote
0answers
120 views

Primes of the form $2^{m_0}p_1^{m_1}\ldots p_r^{m_r}+1$

Is it known any example of a set of primes $\{p_1,\ldots,p_r\}$ with the following property: there are infinitely many $(m_0,\ldots,m_r)\in\mathbb N^{r+1}$ such that $2^{m_0}p_1^{m_1}\ldots ...
1
vote
0answers
102 views

Density of ratios of an arbitrary increasing sequence of prime numbers

It is well known that the set $\left\{ \frac{p}{q} : p,q \textrm{ prime numbers }\right\}$ is dense in the positive real numbers $\mathbb{R}_{>0}$. Not having a background in number theory, I ask ...
5
votes
1answer
368 views

natural radical and an algebraic expression in $\pi$ and/or $e$

Let $\ \mathbb N:= \{1\ 2\ \ldots\}\ $ be the set of natural numbers. Let $\ \mathbb P:=\{2\,\ 3\,\ 5\,\ 7\,\ 11\,\ \ldots\}\ $ be the set of primes. Then natural radical $\ rad(n)\ $ is $$ rad(n)\ ...
10
votes
1answer
470 views

Why do the Maynard-Tao weights work so well?

I am looking for an intuitive reason for why the Maynard-Tao weights work well to capture many primes of the form $n+h_1, \ldots , n+h_k$, where $(h_1, \ldots , h_k)$ is any admissible $k$-tuple. For ...
3
votes
0answers
102 views

Distribution of the inbetween prime

Let $\ \mathbb J_n\,:=\,\{1\ \ldots\ n\}\ $ be the initial interval of natural numbers, and $$2=p_0<p_1<\ldots$$ be the increasing sequence of all primes. Let $$ \forall_{n=1\ 2\ \ldots}\ \ ...
1
vote
0answers
48 views

On the sum of digits of primes in binary form [duplicate]

Let $s_2(m)$ be the sum of digits of $m$ in binary form. I would like to ask the following question: Is it true that for every $n\in \mathbb{N}$ there is at least one prime $p$ which has ...
0
votes
0answers
70 views

Explicit formula for $\vartheta(x)$

Is there an explicit formula for the Chebyshev Theta Function like there is for the Psi Function in terms of the zeta zeroes? I know one for theta can be derived from the one with psi using mobius ...
2
votes
1answer
156 views

Bounds on $\pi(x)$ vs. bounds on $\vartheta(x)$

If $\pi(x) > \operatorname{Li(x)},$ is $\vartheta(x) > x$? Are the two inequalities (solutions to both of which are known to exist but not known exactly) equivalent, similar, or mostly ...
3
votes
1answer
233 views

Integral involving the gamma function

If we define $$f(x) = 1 + \frac{\cos\big(\pi\frac{\Gamma(x) + 1}{x}\big)}{2 - \cos(2\pi{x})}$$ how would one go about evaluating $$ \int_1^R \frac{1}{x} \log{f(xe^{i\alpha})} dx$$ for some parameter ...
7
votes
2answers
497 views

How do i show that:$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$ without using properties of Riemann zeta function? [duplicate]

In order to know more about product over primes ,I would like to know how do I show that :$$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$$ without using properties of Riemann zeta function ? Note01 : it ...
1
vote
1answer
187 views

Convergence of a double sum involving prime numbers

This has been moved from math.stackexchange; I am attempting to prove/disprove convergence of the following sum $$ \lim_{n \to \infty} \frac{1}{n} \sum_{p \leq n} \sum_{k=0}^\infty \ln p ...