**16**

votes

**1**answer

259 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

**0**answers

101 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

**0**answers

51 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
$$
...

**9**

votes

**1**answer

193 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

**1**answer

411 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

**1**answer

259 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

**2**answers

225 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

**0**answers

117 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 ...

**-2**

votes

**0**answers

79 views

### Is the digital root of this number ${4^4}^n+1$ always $5$ for all $ n$? [closed]

I would like to know more about primality of this number :${4^4}^n+1$ and it's
relation to it's digital root , some computations in wolfram alpha showed
that the digital root of ${4^4}^n+1$ is ...

**1**

vote

**0**answers

86 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 ...

**4**

votes

**0**answers

261 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

**1**answer

419 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

**0**answers

174 views

### estimate sum of $(\log \log p)^2/p$ [closed]

Edited: It is well known that $$\sum_{p\leq x} \frac{\log p}{p}=\log x+c.$$
From a very nice previous answer to this question, it is also known that $$\sum_{p\leq x} \frac{\log \log p}{p} = ...

**3**

votes

**0**answers

96 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

**0**answers

47 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

**0**answers

22 views

### New largest prime number discovery - what's all the fuss [migrated]

So I've read about the latest largest prime number discovery (M74207281), but I find it hand to understand what's the dig deal because using Euclid's proof of the infinitude of primes we can generate ...

**0**

votes

**0**answers

65 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

**1**answer

145 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

**1**answer

200 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 ...

**8**

votes

**2**answers

416 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

**1**answer

171 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 ...

**1**

vote

**0**answers

211 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

**0**answers

104 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

**2**answers

336 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

**1**answer

144 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 ...

**0**

votes

**1**answer

134 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 ...

**4**

votes

**3**answers

493 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

**1**answer

539 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*}
...

**0**

votes

**0**answers

134 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

**2**answers

431 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 ...

**7**

votes

**0**answers

111 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) = ...

**5**

votes

**1**answer

396 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}}^{}
...

**8**

votes

**0**answers

198 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

**0**answers

107 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

**1**answer

169 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

**2**answers

258 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

**0**answers

77 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 ...

**0**

votes

**1**answer

176 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 ...

**1**

vote

**1**answer

124 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 ...

**4**

votes

**1**answer

197 views

### Goldbach for certain classes of $n$

Asked on MSE without response here.
$\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$.
The Wiki article on the Goldbach conjecture states that
In 1975, ...

**2**

votes

**1**answer

132 views

### Gradual monotonic morphing between two natural numbers

Let $a < b$ be two natural numbers. I will use these as an example:
\begin{align*}
a & = 2^5 \cdot 3^2 \cdot 5^2 = 7200\\\
b & = 2^3 \cdot 3^5 \cdot 7^1 = 13608
\end{align*}
I seek to ...

**0**

votes

**1**answer

47 views

### Function for number of integers having only prime factors >= x? [closed]

Is there a function for the number of integers below n having only prime factors greater than or equal to p? For example, how do i determine the number of integers below 1000 only having prime ...

**3**

votes

**2**answers

277 views

### Primes $P_{2n-1}$ that are $2$ mod $3$

Are infinitely many primes $P_{2n-1}$ expressible as $3k-1$?
The primes $P_{2n-1}$ are every other prime beginning with $2$: $2,5,11,17,23,31,\cdots$. The first few are of the form $3k-1$, but $31$ ...

**0**

votes

**0**answers

39 views

### Estimates related to sum over a primes from a fixed, possibly sparse set

Let $E$ be a fixed infinite sequence of primes such that $\sum_{p \in E} \frac{1}{p} = \infty$. Assume that $\sigma > 1$ depends on some parameter $x \rightarrow \infty$ in such a way that $\sigma ...

**1**

vote

**1**answer

184 views

### Radical of the sum $=$ radical of the product

My question is:
Has it been proved/disproved or studied the following?
For every $k\geq 4$ there are $k$ pairwise relatively prime numbers $a_1,a_2,...,a_k$ all greater than $1$ such that ...

**-5**

votes

**1**answer

202 views

### How to prove twin prime conjecture or Goldbach conjecture if we assume prime distribution is completely random? [closed]

If we assume that prime number distribution is COMPLETELY random (subject to 1/log(x) restriction), can we prove twin prime conjecture or Goldbach conjecture ?
My feeling is that, this will be ...

**5**

votes

**1**answer

260 views

### The limit of the following product? What is the closed form of the value?

Assume that $P_n$ is the $n$'th prime: Please help me solve the following $$\lim_{k\to\infty} {k}\prod_{n=1}^k \frac{P_{2n-1}}{P_{2n}}$$
I am not really sure quite where to start here as I am ...

**5**

votes

**0**answers

293 views

### A problem on prime numbers

Given integers $a,b,c,d\in[2^n,2^m]$ with $m>n>1$, how many primes $p$ are there in $[n^\alpha,n^\beta]$ for some $1<\alpha<\beta$ such that
$$0<a\bmod p<n^{\alpha/k}$$
$$0<b\bmod ...

**4**

votes

**1**answer

177 views

### What is known about the largest prime divisor of the product of $k$ consecutive integers?

Take $k$ consecutive composite integers from a prime gap. What is known about the largest prime divisor of their product?
It seems to me that except for the triplet $(8,9,10)$ and the pair $(8,9)$ , ...

**6**

votes

**0**answers

99 views

### Asymptotic density of winning positions in “Prime Nim”?

Consider a single-pile NIM variant, played under standard (not misere) objective, with the rule that you may remove any prime number from the pile. The winning positions of this game are all numbers ...