**9**

votes

**0**answers

444 views

### Are the twin primes the only positive double zeros of this real function?

Agno's answer
was extremely helpful.
For $x \in \mathbb{R}, x \ge 1$ define
$$ f(x) = \sin\frac{\pi(\Gamma(x)+1)}{\lfloor x \rfloor}$$
By Wilson's theorem the positive integer zeros of $f(x)$ are ...

**5**

votes

**2**answers

307 views

### Function with zeros plus/minus the primes

While playing with Cohen's pari script prodeulerrat found a function.
For $s \in \mathbb{C}$ define
$$ f(s) = \prod_{p \text{ prime}} (1-\frac{s^2}{p^2})$$
The product converges everywhere, no poles ...

**4**

votes

**3**answers

307 views

### Set of primes dividing polynomials and composition

For a non-constant polynomial $A \in \mathbb{Z}[x]$, let $\mathcal{P}(A)$ denote the set of prime numbers $p$ which divide $A(n)$ for some integer $n$. If $\mathcal{P}(A) \subseteq \mathcal{P}(B)$ for ...

**5**

votes

**1**answer

564 views

### An interaction between prime numbers

Let $p_1\ p_2\ \ldots$ be the sequence of all natural prime numbers. There is a slight (just slight) but clear tendency for imitating the number of primes in an interval $(p_k;\ p_n)$ by ...

**1**

vote

**0**answers

291 views

### Green-Tao style theorem for quadratic regressions (Ulam Spiral)

This is a naive question about number theory.
Looking at an Ulam spiral which illustrates primes of the form e.g. $4x^2-2x+c$ and other quadratic equations $ax^2+bx+c$, with $c>0$, there appears a ...

**5**

votes

**1**answer

441 views

### Other implications of Zhang's method

I have been reading a bit about Zhang's proof and the associated Polymath8 project.
Though Tao's high level summary
...

**0**

votes

**0**answers

86 views

### Sequences sharing some primitive prime divisors

Let $q=p^\alpha$ and $q'=p'^\alpha$. Moreover, define $r_i$ and $u_i$ as primitive prime divisor of $q^i-1$ and $q'^i-1$, respectively. Let $\{r_1\}=\{u_1\}$, $\{r_2\}=\{u_2\}$, $\{r_3\}=\{u_3\}$, ...

**6**

votes

**3**answers

589 views

### a question for the prime counting function

A famous inequality that has been proved by J.B. Rosser and L. Schoenfeld says that
$\frac{n}{\ln n-1/2}$ < $\pi(n)$<$\frac{n}{\ln n-3/2} , n\ge 67$.
Using this inequality we can prove ...

**1**

vote

**0**answers

169 views

### apparent contradiction from result on prime gaps [closed]

I'm looking at Theorem 3 of this paper, which is
If $\:x\geq \exp(\exp(45))\:$ and $\;\;h\:\geq\:3\cdot x^{\frac23}\;\;$ then $$\pi(x+h)-\pi(x) \; \geq \; h\cdot \left(1-\left(3192.34\cdot ...

**0**

votes

**0**answers

102 views

### Composite Euclid Numbers

Is there any way (general procedure, i mean) to determine if a Euclid Number (En = pn# + 1) is prime or composite? Any research papers exploring this theme are also welcome. Thanks!

**8**

votes

**0**answers

112 views

### Does the given operation on pairs of primes always repeat?

Let $p$ and $q$ be two distinct primes. The set
$$A(p,q) =\{m+n : mp+nq=1 \textrm{ and } m,n \in \mathbb{Z}\}$$
is an arithmetic progression. Its step size $p-q$ is coprime to a fixed $m+n$ because
...

**8**

votes

**1**answer

302 views

### How constructive is Dirichlet on primes in progressions?

Is there a known elementary function bound in terms of $a,b,n$ for the $n$-th prime equal to $b$ modulo $a$ (coprime to $b$)?
Bounds on Linnik's constant answer this for the first prime in each ...

**7**

votes

**1**answer

249 views

### Class number of real maximal subfield of cyclotomic fields

Let $p$ be a prime number and $h_p^+$ the class number of $\mathbb{Q}(\zeta_p + \zeta_p^{-1})$. What is known about the values of $p$ for which $h_p^+ = 1$?
Are there infinitely many? Finitely many? ...

**8**

votes

**0**answers

316 views

### Sieve bound for prime $k$-tuples

Let $d_1<d_2<\dots<d_k$ be integers. Then the number of integers $n\leq x$, such that $n+d_1, n+d_2, \ldots, n+d_k$ are simultaneously prime, is bounded above by
$$
\mathfrak{S}(d_1, \ldots, ...

**7**

votes

**1**answer

410 views

### The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$

The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
...

**2**

votes

**0**answers

111 views

### Steps required to recognise a $z$-smooth number

I am currently reading section 5 of Pomerance's paper The Number Field Sieve and I have a few questions about smooth numbers.
A number $x\in\mathbb Z_{\ge1}$ is called $z$-smooth if every prime ...

**1**

vote

**0**answers

148 views

### Proof of the Infinitude of Odd Primitive Pseudoperfect Numbers

I'm interested in the infinitude of odd primitive pseudoperfect numbers. Richard K. Guy's book "Unsolved Problems in Number Theory 3rd edition" says that P. Erdős proved the infinitude of odd ...

**1**

vote

**1**answer

73 views

### computing $c_5$ in “Primality testing with Gaussian periods”

As far as I know, the April 2011 version of #143 on this page has not been improved upon.
On page 10 of that paper, the authors give an algorithm that uses a constant $\:c_{\hspace{.01 in}5}\:$.
...

**8**

votes

**1**answer

331 views

### Least prime $p$ such that an irreducible polynomial of degree $n$ has no root modulo $p$?

This question is inspired by an old question of Greg Kuperberg, about how small is the first prime $p$ which makes a given monic polynomial $P$ with integral coefficient have a (simple) root modulo ...

**2**

votes

**0**answers

140 views

### Best known Upper bound on Twin Primes [duplicate]

I know that there is a result from J Wu that the number of twin primes less than a given magnitude $N$ does not exceed
$$\frac{2aCN}{\log^2{N}}$$
Where $C=\prod \frac{p(p-2)}{(p-1)^2}$ and $a$ is ...

**30**

votes

**2**answers

1k views

### A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself

Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...

**16**

votes

**1**answer

710 views

### Infinitely many primes, and Mobius randomness in sparse sets

Problem 1: Find a (not extremely artificial) set A of integers so that for every $n$, $|A\cap [n]| \le n^{0.499}$, ($[n]=\{1,2,...,n\}$,) where you can prove that $A$ contains infinitely many primes.
...

**2**

votes

**1**answer

451 views

### Prime numbers of a special kind

Is it proved that there is only a finite number of prime numbers of the form $2^{2n} + 2^n + 1$ (where $n \in \mathbb{N}$)?

**9**

votes

**1**answer

492 views

### A conjecture by Euler about $8n+3$

Euler's conjecture: For any positive integer $n$, $8n+3$ can be represented as a sum
$$8n+3=(2k-1)^2+2p,$$
where $k$ is a positive integer, and $p$ is a prime.
I want to know whether there has been ...

**5**

votes

**2**answers

411 views

### General Euclid-Fermat sequences

Question
Fermat sequence
$$ \forall_{n=1\ 2\ \ldots}\quad F_n:=2^{2^{n-1}}+1$$
has every two different terms relatively prime, and $1$ does not appear as any of them. Thus it would be ...

**13**

votes

**2**answers

702 views

### An Euler-proof that cannot be repaired?

Ed Sandifer writes on Euler's wonderful work on prime numbers: The sum of the series of reciprocals of prime numbers $\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$ is infinitely large, and is infinitely ...

**7**

votes

**1**answer

499 views

### Density of prime pairs whose gap is less than the average gap

By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap ...

**1**

vote

**0**answers

129 views

### Division between deformed Gausian integers

Let $P_3=${$z\in Z[\sqrt{-3}],|z|^2 \text { is a prime number}, >3$}
Let $\alpha_1,...\alpha_n$ be distinct elements in $P_3$, and $l_1,...l_n\in Z^+$. Set ...

**2**

votes

**0**answers

296 views

### On the primitive prime divisors of $q^n-1$

Let $q=p^\alpha$ be a prime power.
We call $r$ a primitive prime divisor of $q^n-1$ where $r\mid (q^n-1)$ but $r\nmid (q^i-1)$ for each $1\leq i\leq n-1$. The set of all primitive prime divisors of ...

**1**

vote

**1**answer

89 views

### On the set of divisors of $q-1$ and $q'-1$

For a natural number $n$ we denote by $\pi(n)$, for the set of prime divisors of $n$.
Let $q=p^\alpha$ and $q'=r^\beta$ where $r$ and $p$ are odd prime numbers and $\alpha,\beta$ are natural numbers.
...

**0**

votes

**1**answer

205 views

### For any n and some prime p there is an elemnet in Zp* of order n [closed]

How can I prove, that for any positive integer $n>0$ there is a prime $p$, such that the multiplicative group of the residue ring $Z_p^*$ contains an element $a$ of order $n$? No ideas at all...

**6**

votes

**0**answers

565 views

### Would the following conjectures imply Cramer's conjecture?

Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, ...

**-1**

votes

**1**answer

279 views

### Mersenne Prime Sequences

Hi.
Given the following sequence (of Mersenne primes):
$ A_{1} = 2 $
$ A_{n} = 2^{A_{n-1}} - 1 $
The first five elements are all prime numbers:
$ 2 $
$ 2^{2}-1=3 $
$ 2^{3}-1=7 $
$ 2^{7}-1=127 ...

**2**

votes

**2**answers

642 views

### Finding primes using Euler's sum of divisors recurrence relation

Euler came up with following recurrence relation for the sum of divisors (refer to http://arxiv.org/abs/math/0411587)
$$\sigma(n) = \sigma(n−1) + \sigma(n−2) − \sigma(n−5) − \sigma(n−7) \dots$$
Since ...

**3**

votes

**1**answer

306 views

### The Nth number with M prime factors

Hi.
Suppose we arrange all natural numbers in a matrix P defined as follows:
P[I][J] = The Jth number with I prime factors. So P looks something like:
1
2 , 3 , 5 , 7 , 11 , 13 , 17 ...

**0**

votes

**0**answers

130 views

### Prime-Counting Function

Would a summation of floor[cos^2π(((j-1)!+1)/j)] from j=1 to x be the same as π(x)?

**1**

vote

**1**answer

147 views

### Behavior of a quantity related to Fermat's 4n + 1 Theorem

One of Fermat's theorems states that if $p = 4n + 1$ for some integer $n$, then $p$ can be expressed uniquely as a sum of two squares, $p = a^2 + b^2$. I am working on a problem and I would like to ...

**3**

votes

**1**answer

237 views

### Primes in short intervals with a preassigned frobenius

Edited after mistake in the first version.
It is known since Selberg that under the Riemann Hypothesis, given an $\epsilon>0$, there is a prime between $x$ and $x+O(x^\epsilon)$ for all $x$ in a ...

**8**

votes

**0**answers

530 views

### Would the following conjectures imply $\lim\inf_{n\to\infty}p_{n+k}-p_{n}=O(k\log k)$?

Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, ...

**1**

vote

**1**answer

802 views

### The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus?

After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial.
In particular, it's the case ...

**3**

votes

**1**answer

346 views

### Heuristic for Montgomery's conjecture

This is my third question on this site regarding Montgomery's conjecture -- and I apologize
if this is too much -- but I am still not understanding well why this conjecture is believed to be true.
...

**61**

votes

**6**answers

6k views

### Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?

Let $p_n$ be the $n$-th prime number, as usual:
$p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$, etc.
For $k=1,2,3,\ldots$, define
$$
g_k = \liminf_{n \rightarrow \infty} (p_{n+k} - p_n).
$$
Thus the twin ...

**16**

votes

**1**answer

4k views

### Tightening Zhang's bound

Inspired by a blogpost by Scott Morrison and ongoing discussion there I decided to create this community wiki to track progress on the original bound of Yitan Zhang.
The original bound was ...

**4**

votes

**1**answer

546 views

### Goldbach's conjecture and Euler's idoneal numbers

Recently, I stumbled upon an interesting statement regarding Quadratic Forms. It is quite well-known and, as I will describe briefly, equivalent to Goldbach's conjecture.
Let $p,q$ be odd primes and ...

**19**

votes

**4**answers

4k views

### How does Yitang Zhang use Cauchy's inequality and Theorem 2 to obtain the error term coming from the $S_2$ sum

I have been reading Yitang Zhang's paper now for one and a half weeks and also volunteered to give a popular talk on the paper next week at Stockholm University.
Today I found a detail in the proof ...

**12**

votes

**2**answers

1k views

### Distinctive property of the primes 17 and 19?

Consider the question whether it is true that a prime number $p$ divides
$1^1+2^2+3^3+....+(p-1)^{p-1}$ if and only if $p \in \{17,19\}$.
For the obvious heuristic reasons, for large $n$ one would ...

**0**

votes

**0**answers

177 views

### Period of decimal for $1/n$, odd part of $n+1$, and primes.

Let $n$ be a nature number is relatively prime to 10,such the period of the decimal expansion of $1/n$ is $n-1$ or a divisor of $n-1$, and let $c$ be the "cycle length of $n$" (defined below). If ...

**7**

votes

**3**answers

455 views

### Asymptotic bounds on $\pi^{-1}(x)$ (inverse prime counting function)

What are the current best asymptotic bounds on $\pi^{-1}(x)$, where $\pi(x)$ denotes the prime counting function (number of primes at most $x$)?
In other words, I am curious about the state of the ...

**2**

votes

**1**answer

197 views

### Given an even integer N, what is the minimum set of primes such that any even number x <= N can be expressed as the sum of two primes from the set?

Given an even integer N, what is the minimum set of primes such that any even number $x \leq N$ can be expressed as the sum of two primes in the set?
Goldbach's conjecture said Every even integer ...

**4**

votes

**1**answer

199 views

### Estimate on the prime-counting function $\psi(x)$.

There is an elementary statement that I believe I have read somewhere, but I can't remember where. I'd like to know if the statement is correct (in which case it is surely standard) and if so, where I ...