**1**

vote

**0**answers

218 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

115 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

378 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

147 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

157 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

527 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

552 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

141 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

449 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

**0**answers

123 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

457 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

207 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

108 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

182 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

262 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

79 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

178 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

**1**answer

128 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

205 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

133 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

48 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

281 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

41 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

194 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

232 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

267 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 ...

**4**

votes

**0**answers

304 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

182 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

116 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 ...

**4**

votes

**1**answer

310 views

### Upper bound for the first Hardy-Littlewood conjecture

About the Hardy-Littlewood conjecture by Terence Tao:
Conjecture 2 (Prime tuples conjecture, quantitative form) Let ${k_0 \geq 1}$ be a fixed natural number, and let ${{\mathcal H}}$ be a fixed ...

**2**

votes

**0**answers

324 views

### How big can a set of integers be if all pairs have bounded gcd

In this recent MO question, it was shown that the maximal cardinality of a subset $A(M,N)$ of $[1,N]$ where the pairwise GCD's of all set elements are upper bounded by $M,$ with $M^2\leq N$ has size ...

**4**

votes

**1**answer

395 views

### About factorization in Zhang's proof of weak Twin Prime conjecture

Why does it need to firstly factorize the number n into two factors q and r( Lemma 4 in the paper,see the following)? What's the motivation. What if it doesn't do this factorization?

**0**

votes

**2**answers

320 views

### Does theta(n)<n for all n imply the Riemann Hypothesis and/or vice versa?

I know that better and better bounds of the Chebyshev Theta and Psi functions are implied by knowing that the first (insert large number here) zeta zeroes lie on the Critical Line. These bounds, ...

**2**

votes

**0**answers

143 views

### On sets of coprime numbers

We know that from prime number theorem that the number of primes below $n$ and above $\frac n2$ (denoted by $\pi_{n,\frac n2}$ is approximately $$\pi_{n,\frac n2}\approx\frac{n}{2\ln n}.$$
Denote by ...

**2**

votes

**1**answer

158 views

### Gap between semiprimes

Is there a conjectured gap between semiprimes?
There is a conjectured gap between primes in form of Cramer's conjecture. Using this we have $p_1\leq p_0+c(\log p_0)^2$ for consecutive primes $p_0$ ...

**22**

votes

**1**answer

683 views

### How big can a set of integers be if all pairs have small gcd?

Suppose $A\subset[1,N]$ is a set of integers. If for any distinct $a,b\in A$ we have $(a,b)\leq M$ then how big can $|A|$ be?
If $M=1$ then $|A|$ is at most $\pi(N)$ since the map $a\mapsto P_+(a)$ ...

**6**

votes

**1**answer

188 views

### $N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$

Assume that $ab \neq 0$. What is $$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$$I need this result, but unfortunately I am not a number theorist. Could ...

**8**

votes

**0**answers

168 views

### Primes of the form $(2m+1)^2-2^{2s+1}$

The question is the following :
Question:
Does there exist infinitely many primes of the form $(2m+1)^2-2^{2s+1}$ with $m,s\geq 1$ ?
Why this could be true:
Bunyakowsky conjecture would ...

**2**

votes

**0**answers

202 views

### Avoiding Chinese Remainder Theorem

Given $k\in\Bbb N$ with $k<(\log_2N)^{\frac1\alpha}$ where $\alpha>2$ is fixed and $N$ being some integer such that $$N<\prod_{i=1}^k\pi_i^{a_i}$$ where $\pi_1,\pi_2,\dots,\pi_{k-1},\pi_k$ ...

**0**

votes

**0**answers

109 views

### Bounding $\dfrac{r(x)}{\pi(x+r(x))-\pi(x-r(x))}$ with $1\ll r(x)\ll \log^{4}(x)$

I would like to know whether it is possible to obtain the bounds $\sqrt{r(x)}\ll k(x)\ll r(x)$ where $k(x)={\pi(x+r(x))-\pi(x-r(x))}$ and $1\ll r(x)\ll \log^{4}(x)$ and thus ...

**2**

votes

**2**answers

344 views

### Primes $p$ for which $2p-1$ is prime

It's a well-known open problem (Sophie-Germain primes) whether there are infinitely many primes $p$, $2p+1$. What about $p$, $2p-1$?
Seemingly it's also an open problem (see here and the linked ...

**3**

votes

**1**answer

184 views

### Is $2^n -1$ finitely many times the product of consecutive primes? [duplicate]

This question was asked at MSE but recieved no attention at all.
Here it is:
Are there finitely many $(n,k) \in \mathbb{N}^2$ with $2^n-1=p_1p_2\cdots p_k$ ?
$p_1=3,p_2=5 , ...,p_k$ are ...

**4**

votes

**0**answers

100 views

### Behavior of the “mean prime factor” of numbers

This question concerns the behavior of
a function $f(\;)$ that maps each number in $\mathbb{N}$ to
its mean prime factor.
I previously posted premature questions, now deleted, which
explains the cites ...

**8**

votes

**1**answer

332 views

### Distribution of the number of prime factors

Count the number of prime factors of a number $n$
to include multiplicity,
so that
$$n=24=2^3 \cdot 3 = 2 \cdot 2 \cdot 2 \cdot 3$$
has $4$ prime factors, and
$$n =
6500 =
2^2 \cdot 5^3 \cdot 13 =
2 ...

**3**

votes

**1**answer

281 views

### Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$?

http://mathworld.wolfram.com/ChoquetTheory.html
Is the claim in the link true? Here's the reference given there:
https://www.renyi.hu/~p_erdos/1934-01.pdf
Erdős proved that there exist at least ...

**1**

vote

**0**answers

124 views

### A Question on Chinese Remainder Theorem [closed]

Let $p_1,p_2,\ldots,p_n$ be odd primes.
Let $y$ be the unique solution of the Chinese Remainder Problem ( $0 \le y < m$, $ m = p_1\cdot p_2 \cdots p_n$)
$ y = \frac{p_1-1}{2} \text{ mod } (p_1) ...

**7**

votes

**3**answers

537 views

### Quantitative and elementary proofs of the Prime Number Theorem

I would like to know two things: one, whether the best quantative bounds in the Prime Number Theorem are still basically those given by the Vinogradov-Korobov zero-free region? and two, whether there ...

**6**

votes

**0**answers

207 views

### Any ways to Simplify Daboussi's Argument for Prime Number Theorem?

One strategy to prove the Prime number theorem involves removing some factors:
$$ \limsup_{x \to \infty} \underbrace{\frac{1}{x}\sum_{n \leq x} \mu(x)}_{\color{red}{A}}\leq
\prod_{p \leq y} \left( 1 ...

**4**

votes

**1**answer

487 views

### Green-Tao theorem for 1-central numbers

This question came to my mind this afternoon while trying to figure out a possible way to tackle de Polignac's conjecture, which states that every even positive integer can be written as the ...

**17**

votes

**2**answers

3k views

### Does the equation $241+2^{2s+1}=m^2$ have a solution?

Let $p$ be a prime congruent to $1$ mod. 8.
If $p= 17$ one has : $p+ 8 = 5 ^2$.
If $p= 41$ one has : $p+ 8 = 7 ^2$.
If $p= 73$ one has : $p+ 8 = 9 ^2$.
If $p= 89$ one has : $p+ 32 = 11 ^2$.
If ...