**9**

votes

**1**answer

515 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

430 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 ...

**12**

votes

**2**answers

725 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 ...

**8**

votes

**1**answer

598 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

130 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

315 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

92 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

216 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

598 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

316 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

776 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

329 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

136 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

159 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

258 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

675 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

959 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

369 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.
...

**63**

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

578 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 ...

**8**

votes

**3**answers

525 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

225 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

232 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 ...

**4**

votes

**2**answers

289 views

### Are sums of the inverses of prime siblings finite?

PART I (Initial version)
Let $P$ be the set of all primes $2\ 3\ \ldots$. Let
$$P_d\ \ :=\ \ \{\ p\in P\ :\ \exists_{q\in P}\ \ 0 < |p-q|\le d\ \}$$
and
$$S_d\ :=\ ...

**22**

votes

**1**answer

1k views

### Permutations of $(Z/pZ)^*$

Let $p$ be a prime integer, and let $(\mathbb Z/p\mathbb Z)^*$ be the set of non-zero elements of $\mathbb Z/p \mathbb Z$.
Denote by $S((\mathbb Z/p \mathbb Z)^*)$ the group of permutations of ...

**70**

votes

**4**answers

26k views

### Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture

Yitang Zhang recently published a new attack on the Twin Primes Conjecture. Quoting Andre Granville :
“The big experts in the field had
already tried to make this approach
work,” Granville ...

**6**

votes

**6**answers

824 views

### Sequences equidistributed modulo 1

Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results.
H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidistributed modulo 1.
...

**4**

votes

**1**answer

457 views

### p such that p+1 has a large prime factor, effectively

I was reading the Boneh-Franklin IBE paper, and it seemed rather conspicuous to me that they
didn't address the question of how to find primes $p$ and $q$ satisfying what they need (on page 19).
...

**0**

votes

**1**answer

418 views

### A possible consequence of Dirichlet's theorem about primes in arithmetic progression

EDIT : I copy-paste the beginning of a previous question since Gerry Myerson suggested this question should be self-contained.
"let's consider a composite natural number $n$ greater or equal to $4$. ...

**7**

votes

**1**answer

176 views

### Composing two-term sums from the same primes

The following is an old result of Erdős and Turán (American Mathematical Monthly, 1934):
Given a set of $2^n + 1$ distinct positive integers, all of its two-term sums cannot be composed of the same ...

**0**

votes

**4**answers

320 views

### The prime number $2$ [duplicate]

Possible Duplicate:
Why is 2 so odd?
I have read few books and articles, almost all of them refer that any prime $p>2$. Just wondering why it has to be $>2$?

**2**

votes

**1**answer

470 views

### To express $e^{\sum \limits_{k=0}^\infty q^{2^k}}$ as product terms of $(1-q^k)^{c(k)}$

$|q|\lt1$
$A(q)=\sum \limits_{k=0}^\infty q^{2^k}$
Easily We can see that
$$A(q)=q+A(q^2)\tag 1$$
Let's assume we redefine $A(q)$ as below
$A(q)=-\sum \limits_{k=1}^\infty c_k \ln{(1-q^k)}$
I ...

**2**

votes

**3**answers

410 views

### Is it true that $p^2+1$ is square free if $p>7$ is a Mersenne prime

For a problem in group Theory I need some information about the Mersenne primes:
Let $p=2^a-1>7$ be a Mersenne prime and so $a$ is an odd prime. Is it true that $p^2+1$ is square free. i.e.
if ...

**5**

votes

**2**answers

281 views

### Does group of 4 equidistant successive prime exists ?

For example
Group of 2 equidistant successive primes
3,5,7 distance 2
151,157,163 distance 6
Group of 3 equidistant successive primes
...

**0**

votes

**1**answer

262 views

### Factorization of $p^2+1$ where $p$ is a Mersenne prime

Let $p=2^a-1$ be a Mersenne prime and so $a$ is an odd prime if $p>7$. We know that if $p=7$, then $(p^2+1)/2$ is equal to $5^2$.
Can we prove that if $p>7$ , then $(p^2+1)/2$ is not equal to ...

**3**

votes

**1**answer

275 views

### Least non primitive root

There is an abundant literature, and even here on MO no shortage of questions, on the question of the smallest prime primitivee root modulo $q$ (where $q$ is a prime, or more generally
an odd prime ...

**4**

votes

**3**answers

270 views

### A divergent series related to the number of divisors of of p-1

Let $d(n)$ denote the number of divisors of $n$. Is it known that the series
$$\sum_{p \text{ prime}} \frac{1}{d(p-1)}$$
diverges?
This would follow immediately from the Sophie Germain Conjecture. ...

**1**

vote

**1**answer

191 views

### What are the best known lower and upper bounds for the second Chebyshev function $\psi(x)$

I was reading through Jitsuro Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$ when $x \ge 25$.
In the paper, he uses the following bounds for the second Chebyshev function ...

**5**

votes

**2**answers

323 views

### special primes with p'=4p+1

How can I most quickly find a big prime, p, for which 4p+1 is also prime? For example, p=37 works. I wonder if these special primes have been researched and some characteristics are known. Are ...

**4**

votes

**1**answer

288 views

### A “bit” of primes

Is there anything known/proved/conjectured about the distribution of:
$$B(n) = \frac{(p_n-1)}{2} \bmod 2, \qquad p_n \mbox{ is the } n\mbox{-th prime}$$
i.e. the bit 1 of the binary representation ...

**2**

votes

**0**answers

119 views

### Two products over primes

For $k \in \mathbb{N}$ define
$$ f(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k+1)}\right)$$
$$ g(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k-1)}\right)$$
By the product for zeta ...

**6**

votes

**2**answers

599 views

### At what point would an elementary generalization of Bertrand's Postulate be interesting?

I know that in 1952 Jitsuro Nagura was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k > 24$.
At what point would an improvement on Nagura's result be interesting? ...

**6**

votes

**2**answers

732 views

### Approximate number of primes below a given integer?

The problem of the complexity of the exact counting problem for primes is interesting. The best result we have about primes is that it is hard for TC0. But counting the number of witnesses to a TC0 ...

**7**

votes

**0**answers

282 views

### Montgomery's conjecture and lower bound on certain Fourier transform.

Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heuristic) behind ...

**2**

votes

**1**answer

327 views

### At what point does Miller-Rabin become faster than trial division?

I've read in various places (and know) that Miller-Rabin is a much faster primality test than trial division for large $N$, but is much slower than trial division for small $N$.
My question is: how ...

**2**

votes

**0**answers

87 views

### Symmetric dominance regions surrounding a Gaussian prime

Let $z=a + b i$ be a complex number which is a Gaussian prime,
on neither the $x$- nor the $y$-axis.
So $a^2+b^2$ is a prime.
Construct a region $D(z)$ surrounding $z$ which is the
largest ...

**1**

vote

**2**answers

341 views

### product 1+1/p in terms of Chebyshev's theta or psi function

I would like to know if there is any formula for
$
\prod_{x<p\leq y}\left(1+\frac1p\right)
$
in terms of $\theta$ or $\psi$ functions
$
\theta(x)=\sum_{p\leq x}\log p
$
and
$
...