**7**

votes

**1**answer

492 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

**1**answer

213 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

**1**answer

249 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

**0**answers

150 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

**1**answer

208 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

**1**answer

623 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 ...

**1**

vote

**0**answers

177 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

**0**answers

557 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 ...

**7**

votes

**1**answer

419 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

**1**answer

211 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 ...

**5**

votes

**0**answers

117 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

**2**answers

236 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

**1**answer

179 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 ...

**5**

votes

**3**answers

391 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 ...

**4**

votes

**1**answer

320 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:
$$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$
...

**2**

votes

**1**answer

392 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

**2**answers

591 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 ...

**5**

votes

**0**answers

158 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 ...

**0**

votes

**2**answers

93 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. ...

**8**

votes

**2**answers

569 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 ...

**11**

votes

**3**answers

741 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 ...

**6**

votes

**3**answers

722 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: ...

**10**

votes

**0**answers

133 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 ...

**44**

votes

**4**answers

2k views

### How did Cole factor $2^{67}-1$ in 1903

I just heard a This American Life episode which recounted the famous anecdote about Frank Nelson Cole factoring $N:=2^{67}-1$ as $193707721 \times 761838257287$. There doesn't seem to be a historical ...

**6**

votes

**0**answers

73 views

### Prime divisors of the norm of the first coefficient of an elliptic newform at width-1 cusps.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ and let $f$ be its newform.
Suppose $p \geq 5$ is a prime such that $p^2 \mid N$. We assume $f$ is $p$-minimal, which is equivalent to that the ...

**4**

votes

**2**answers

390 views

### Non-standard Gauss sums

I have the following problem. Let $p$ be some prime. What is the value of
\begin{equation}
\sum_{k=1}^{p-1} \left(\frac{k+1}{p}\right) \omega_p^{kl},
\end{equation}
where $\left(\frac{k+1}{p}\right)$ ...

**0**

votes

**2**answers

172 views

### Smallest constant so that there are at least $n/\log_2{n}$ primes between $n$ and a constant multiple of $n$

What is the smallest known $c$ so that for any $n\geq 2$ there are at least $n/\log_2{n}$ primes between $n$ and $cn$ (inclusive)?
The prime number theorem seems to give an asymptotic result so I am ...

**0**

votes

**0**answers

87 views

### Expliciting the distance between consecutive Goldbach numbers assuming it's finite

In this paper, the author shows unconditionally that at least one of the following statements holds:
i) the distance between two consecutive Goldbach numbers is finite, i.e. there exists an ...

**1**

vote

**0**answers

50 views

### Asymptotics on number of bounded prime gaps [duplicate]

It's been over 2 years since the groundbreaking paper by Yitang Zhang in which he has shown that infinitely many prime pairs are by some constant $H$, with $H\leq 70000000$. Over the course of the ...

**5**

votes

**0**answers

294 views

### Are there always at least *five* divisions?

@JosephO'Rourke asked a question about a Collatz like function related to primes:
$f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is prime} \\
\lfloor n/2 \rfloor & \text{if} \;n ...

**3**

votes

**3**answers

403 views

### Are There Infinitly Many $n$ Which $a\times n!+1$ Be Composite?

$ax+1$ is a linear polynomial with integral coefficients.
Are there infinitly many $n$ which $a\times n!+1$ be composite?
As I know this problem is true for polynomials with degree greater that 1, ...

**15**

votes

**1**answer

593 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 ...

**1**

vote

**4**answers

667 views

### Distribution of composite numbers

I have moved this question to math.stackexchange.com. People who are interested in this question can discuss at :http://math.stackexchange.com/questions/1272431/distribution-of-composite-numbers
...

**4**

votes

**0**answers

159 views

### Farey Fractions Estimate Equivalent to the Prime Number Theorem?

Wikipedia's article on Farey Fractions points to an article of Jerome Franel that some averages are equivalent to the Riemann hypothesis.
Let $F_n$ be the $n$-th Farey sequence, then the number of ...

**0**

votes

**1**answer

179 views

### Is the following conjecture equivalent to the Second Hardy-Littlewood Conjecture? [closed]

Let $y$ be an arbitrary positive real number such that $y\ge 2$. Then if we can prove that, $$\lim_{x\to\infty}\dfrac{\pi(x)+\pi(y)}{\pi(x+y)}=1$$will it imply that for all sufficiently large $x$ ...

**1**

vote

**3**answers

334 views

### Has this formula for $G_{k}:=\lim\inf_{n\to\infty}p_{n+k}-p_{n}$ been conjectured?

I give here a heuristics that suggests that the quantity $\displaystyle{G_{k}:=\liminf_{n\to\infty}p_{n+k}-p_{n}}$ should be approximately equal to $k(1+H_{k})$, where $H_{k}$ is the $k$-th harmonic ...

**12**

votes

**0**answers

556 views

### Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system,
$$x_1^2+x_2^2+\dots+x_n^2 = y^2$$
$$x_1^3+x_2^3+\dots+x_n^3 = z^3$$
and define the function,
$$F(s_m) = x_1+x_2+\dots+x_n$$
For $n\geq3$, using an ...

**6**

votes

**1**answer

458 views

### Are reduced residue systems relative primorials an active area of research? If not, why not?

As a math amateur, I am finding the study reduced residue systems relative a primorial a very interesting way to understand the distribution of primes. For example, it is fascinating to me that it is ...

**3**

votes

**4**answers

283 views

### Uniform upper bound for the sum over primes $\sum_{p \leq x} p^{-1+\varepsilon}$

I am reading the article D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprime, Math. Comp. (1991) (click).
In equation (27) the authors, apparently, used the following ...

**8**

votes

**1**answer

283 views

### Asymptotic limit of truncated Legendre sieve

Consider the truncated sum
$$
S(x):=\sum_{\substack{{d\mid P(\sqrt{x})}\\{d\leq x}}}\mu(d)/d,
$$
where $P(z)$ is the product of all primes less than or equal to $z$, and $\mu(d)$ is the Möbius ...

**20**

votes

**3**answers

1k views

### A Polynomial With Positive Prime Density

Let $P(x)$ be a non-constant polynomial with real coefficients.
Can natural density of
$$\{n\ |\ \lfloor P(n)\rfloor \ \text{is prime.}\}$$
be positive?

**2**

votes

**0**answers

155 views

### Squarefree part of a Mersenne number

Consider the Mersenne number; $M_p=2^p−1$.
Let $M_p=a_pb^2_p$ where $a_p$ is positive, squarefree, and $p$ is prime.
A chinese paper written by Le Maohua "“On Mersenne Numbers”" states that the ...

**6**

votes

**1**answer

314 views

### Primes isolated by large gaps to either side

Say that the $n$-th prime $p_n$ is isolated to degree $k$
(my notation) if
the prime gap to either side is larger than $\log p_n$ to the $k$-th power:
\begin{eqnarray*}
p_n - p_{n-1} & > & ...

**2**

votes

**2**answers

367 views

### Primes as uncorrelated random variables [closed]

The heuristic justification section of the Wikipedia article about Goldbach's conjecture says that the argument that suggests that
the number of twin primes below $x$ should be roughly ...

**14**

votes

**3**answers

2k views

### A variant of Goldbach Conjecture

I'm asking if this variant of weak Goldbach's Conjecture is already known.
Let $N$ be an odd number. Does there exists prime numbers $p_1$, $p_2$ and $p_3$ such that $p_1+p_2-p_3=N$? Ideally, can we ...

**22**

votes

**4**answers

1k 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 ...

**8**

votes

**2**answers

953 views

### divisible by all standard prime numbers

This question is about prime numbers in nonstandard models of Peano Arithmetic. Every such model looks like N+AxZ, where A is a dense linear order without end points.
There are many nonstandard ...

**6**

votes

**0**answers

257 views

### Can integers be distorted to make primes more regular?

Given a set $P$ of real numbers $\ge 1$, define the gap among different products in $P$ as
$$g(P) = \inf \big\{\prod_{i=1}^n p_i^{a_i} - \prod_{i=1}^n p_i^{b_i} \mid p_i\in P;\,\, p_i\ne p_j \,\text{ ...

**1**

vote

**0**answers

141 views

### Generalization of Little Fermat Theorem for a particular $a$ and perfect shuffles

I'm looking for the smallest $n\in \mathbb{N}$ that solves the following equation:
$$2^n=1 \mod m$$
For an odd $m$. I know that Little Fermat Theorem and Euler Totient give me a solution but they ...

**3**

votes

**1**answer

119 views

### What is the Complexity Class of the “Function Variant” of the Integer Factorization Problem?

I've been reading up a lot Prime Factorization and it's complexity, including a fair number of questions on this very site. However, I still feel there is a question still left unanswered.
So, ...