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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15
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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
214 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
83 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
63 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} \frac{...
1
vote
0answers
160 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
155 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
189 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
276 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
39 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
104 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
173 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 ...
4
votes
1answer
494 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
246 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
504 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
379 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
126 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
71 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 $$ \...
12
votes
1answer
239 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 ...
12
votes
1answer
531 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
299 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 "ugly"...
3
votes
2answers
248 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
121 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 p_r^{m_r}+...
1
vote
0answers
105 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
375 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
484 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
106 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}\ \ d_n\...
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
72 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
160 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 unrelated?...
3
votes
1answer
247 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
506 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
192 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 \left\{\frac{...
1
vote
0answers
222 views

Is this a proof of the Hardy-Littlewood inequality? [closed]

V.V. Miasoyedov posted a paper to the arXiv claiming a proof of the Hardy-Littlewood conjecture $\pi(x+y) \le \pi(x)+\pi(y)$. It seems a bit off, and not only because the conjecture is widely believed ...
3
votes
0answers
117 views

exponential sum of primes

Fix $\alpha \notin \mathbb{Q}$. I would like to know a reference that shows $$\mathbb{E}_{n\leq N} \Lambda(n) e^{2\pi i \alpha n} \to 0,$$ as $N$ tends to infinity. I am familiar with Vinagradov's ...
5
votes
2answers
385 views

Does the antidiagonal in this square matrix always contain a prime?

Does the antidiagonal in the square matrix with entries $1,2,\ldots,n^2$ row by row in that order always contain a prime? For example: For n=2: $\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ ...
3
votes
1answer
152 views

Confusion regarding Riesz Function Definition

According to wikipedia: 'In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series $$ {\rm {Riesz}}(x)=...
0
votes
1answer
172 views

Pruning primitive sequences but still attaining Pillai's lower bound on sum of reciprocals

Background: The answer to a previous question I asked here specified a construction to achieve Pillai's bound on reciprocal sums of primitive sequences. A primitive sequence $1<a_1<\ldots<a_k\...
3
votes
3answers
542 views

How do estimates on $N_\chi(\alpha,T)$ lead to the Dirichlet prime number theorem for arithmetic sequences?

Let $ N_\chi(\alpha,T)$ be the number of zeros of $L(s=\sigma+it,\chi) = \sum \frac{\chi(n)}{n^s}$ where $c > 0$ and $(\sigma,t) $ are in the rectangle $ [\alpha,1] \times [-T,T]$. In various ...
14
votes
1answer
572 views

Prove $4\sum_{k=1}^{p-1}\frac{(-1)^k}{k^2}\equiv 3\sum_{k=1}^{p-1}\frac{1}{k^2}\pmod{p^2}$

Wolstenholme's theorem is stated as follows: if $p>3$ is a prime, then \begin{align*} \sum_{k=1}^{p-1}\frac{1}{k}\equiv 0 \pmod{p^2},\\ \sum_{k=1}^{p-1}\frac{1}{k^2} \equiv 0 \pmod{p}. \end{align*} ...
1
vote
0answers
142 views

The existence of solution for special equation on integer ring

I have a question which belongs to the field of number theory. Can we prove or disprove the following claim: For all prime number $p=24t+1$ and the natural number $n=6t+1$, there is at least, one ...
0
votes
2answers
455 views

Which even numbers are known to be both prime gaps and the sum of 2 primes?

Goldbach's conjecture asserts that every even integer greater than $3$ is the sum of two primes, while de Polignac's one says every even positive integer is a prime gap infinitely often. My question ...
8
votes
0answers
125 views

Equation which has nontrivial solutions modulo $N$ for every $N \ge 2$ does not have any nontrivial integer solutions

Let $\alpha = \sqrt[3]{2}$ and $K = \textbf{Q}(\alpha)$. I want to show that the equation$$\text{N}_\textbf{Q}^K\left(x + 4y + z\alpha + w\alpha^2\right) - 6(x + y)\left(x^2 + xy + 7y^2\right) = 0,$$...
5
votes
1answer
462 views

$(n+1)!_\mathbb{P}$ and the Euler-Mascheroni constant

I'm studying the following limit $$\lim_{n\to \infty} \frac{1}{n} \ln\left( \frac{(n+1)!_\mathbb{P}}{n^n}\right) $$ where $$(n+1)!_\mathbb{P} = \prod\limits_{p \in \mathbb{P}}^{} {p}^{\omega_p{(n)}},...
8
votes
0answers
211 views

Generating prime numbers

By a theorem of Mills, 1947, there is a real number $c$ such that for every $n$, $[c^{3^n}]$ is a prime number. Is there a real number $d$ such that $[d^n]$ is prime, for every $n$ ?
2
votes
0answers
112 views

Is there an odd number which has no prime to prime matchings when compared with its reverse order? [closed]

For example look at the number 9. It has prime-prime matching at 3,5, and 7. For example the sequence of 13 has matchings at 1,3,7,11,13. For example 15 has the matchings(crossings) at 3,5,11,13. ...
1
vote
1answer
187 views

Least simultaneous quadratic non-residue

Suppose $p,q$ are distinct primes with least quadratic non-residues $n_p$ and $n_q$ respectively. Can one bound the least $n$ for which $\left(\frac{n}{p}\right)=\left(\frac{n}{q}\right)=-1$ in terms ...
6
votes
2answers
274 views

Does the limit of this product over primes converge for all $\Re(s) > \frac12$?

Numerical evidence suggests that: $$\displaystyle F(s):= \lim_{N \to \infty}\, \ln^s\left(p_N\right)\, \prod_{n=1}^N \left(\dfrac{\left(p_n-1\right)^s}{p_n^s-1} -\frac{1}{p_n^s}\right)$$ with $p_n$ ...
0
votes
0answers
81 views

Probability distribution associated with total divisors of an integer

Is there a generalization to https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem which gives distribution function for $$\omega(n)=\big|\{d\in\mathsf{prime}:d|n\}\big|$$ where $\mathsf{prime}$...
0
votes
1answer
181 views

When is $a^{2^n}+1$ prime finitely often unconditionally?

Define generalized Fermat numbers following OEIS and mathworld. For natural $a,n$ and $a$ even, the generalized Fermat number (GFN) is $F_n(a)=a^{2^n}+1$. Very large GFN primes are known (in the ...
0
votes
1answer
131 views

Linear forms that avoid numbers with lot of factors

Is following true? For every given $c>0$ there is an $n_c>0$ such that for every $n>n_c$ there are integers $n<a,b<2n$ such that there are two positive integers $\frac{n}{2(\log n)^c}\...