**0**

votes

**0**answers

120 views

### Does Riemann's explicit formula imply invariance of the prime gaps distribution under a Fourier-like transform?

Loosely speaking, Riemann's explicit formula states that there exists a Fourier-type duality between the primes and the non trivial zeroes of the Riemann zeta function. Does this mean that the ...

**5**

votes

**1**answer

486 views

### Has this strengthening of the PNT already been conjectured?

Suppose $f:\mathbb{Z}_{\geq 0}\to\mathbb{Z}_{\geq 0}$ is an arithmetic function that grows slower than the identity map. Has it already been conjectured that, under this general hypotheses, ...

**34**

votes

**1**answer

3k views

### Are the primes normally distributed? Or is this the Riemann hypothesis?

Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes.
Let $\mathrm{Li}(x)$ be the offset logarithmic ...

**2**

votes

**1**answer

430 views

### On quantities with no very small odd prime factors; a response to Wlodzimierz Holsztynski

In response to a comment posted under
Powers of $2$ and the products of initial odd primes , I shall raise some questions about quantities near $O_n= P_{n+1}/2$, the product of the first $n$ odd ...

**1**

vote

**3**answers

226 views

### Powers of $2$ and the products of initial odd primes

NOTATION: $O_x$ -- the product of all odd primes $\le x$.
E.g. $O_7=3\cdot 5\cdot 7 = 105$.
QUESTION: Are the three ordered pairs $\ (d\ p)=(1\ 3)\ \ (2\ 3)\ \ (4\ 5)\ $ the only solutions of the ...

**11**

votes

**1**answer

399 views

### Roots of unity near 1 in Z / p Z

Let $r \ge 3$ be a fixed integer. I'm interested in primes p such that no integer in the interval $(-\sqrt{p}, \sqrt{p})$, except $1$ (and $-1$ if $r$ is even), is an r-th root of unity modulo p.
The ...

**0**

votes

**2**answers

495 views

### Yitang Zhang's paper [closed]

I just want make thing clear for myself. Others may have asked before in different ways. Does Yitang Zhang's paper prove that for any given length gap $g_n > N$ there is a prime $p_n$ for which ...

**2**

votes

**1**answer

332 views

### Looking for paper: Weil's original 1952 “Sur les formules explicites de la théorie des nombres premiers”

I am looking for a source (preferably online) for Weil's original 1952 paper on the explicit formula. I am aware of an english translation available here, but would like to have access to the original ...

**3**

votes

**1**answer

163 views

### Congruences among primes modulo which a given polynomial has roots

Suppose $f(x)\in\mathbf Z[x]$ is nonconstant. I would like to know if either of the following statements is true.
If $a$ and $b$ are coprime integers (probably with some additional restriction), ...

**6**

votes

**0**answers

233 views

### Does the Maynard-Tao Theorem apply to general tuples of linear forms?

In the paper http://arxiv.org/pdf/1311.5319v1.pdf the author states the following theorem, which he attributes to Maynard and Tao.
For any integer $m > 2$, there exists an integer
$k = k(m)$ such ...

**4**

votes

**0**answers

157 views

### Can the following quantitative version of Chen's theorem be obtained?

The theorem of the Chinese mathematician Chen Jingrun is currently one of the best results with respect to the binary Goldbach problem. It asserts that every even integer $n$ is in the sumset $P + ...

**5**

votes

**0**answers

238 views

### Primes for which 2 is a primitive root

I am writing a paper in which I keep referring to primes p for which 2 is a primitive root mod p and so I want to give a name for these primes. Is there a name for these primes in the literature ...

**1**

vote

**1**answer

157 views

### First Parameterized Subset of Primes that was Related to a Mathematical Result

To my knowledge, Fermat primes, i.e. primes of the form $2^{2^n}+1$ were the first to play a role in a mathematical result, namely in the characterization of constructible regular n-gons. Gauss ...

**6**

votes

**1**answer

561 views

### What keeps asymptotic Goldbach's conjecture out of reach of current technology?

Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...

**6**

votes

**0**answers

291 views

### Twin Primes that are Sophie Germain Primes

Suppose $p$ is a prime such that $p + 2$ is also prime, and nothing else is known about $p$.
Is there any reason to think that this affects the probability that $p$ is also a Sophie Germain prime? ...

**3**

votes

**1**answer

133 views

### Decidability of prime gap sequences

Is the following problem undecidable?
Given a sequence of $n$ gaps $d_1,d_2,...,d_n$, does there exist a sequence of $n+1$ primes $p_1,p_2,...,p_{n+1}$ such that $p_{i+1} - p_i = d_i$ ?
If not, is ...

**0**

votes

**0**answers

147 views

### Average order and upper bound of $r_{0}(n)$

Assume Goldbach's conjecture. Then for every integer $n>1$ there exists a non-negative integer $r$ such that $n-r$ and $n+r$ are both primes. For a given $n>1$, the smallest such $r$ will be ...

**26**

votes

**4**answers

3k views

### What is exceptional about the prime numbers 2 and 3?

Admittedly this question is vague. But I hope to convey my point. Feel free to downvote this.
Permit me to define prime number the following way:
A number $n>1$ is a prime if all integers $d$ ...

**6**

votes

**1**answer

213 views

### Question about a certain class of primes

I've come across a set of primes in a problem I'm working on, and I'm wondering if there's more information available about them. I'm guessing not much, particularly since the question of infinitude ...

**1**

vote

**0**answers

195 views

### An inequality about Goldbach conjecture

Let $N$ a large natural number, let $\forall n\leq N,\, R_{2}\left(n\right)=\underset{p_{1}+p_{2}=n}{\sum}\log\left(p_{1}\right)\log\left(p_{2}\right)$ and let $S\left(\alpha\right)=\underset{p\leq ...

**1**

vote

**2**answers

243 views

### Infinite play with tape, or covering the integers with prime arithmetic progressions

It is possible that a more technical version of this question has been
asked and answered in the literature. If so, then a reference is much
appreciated. I will phrase it in terms of colored tapes ...

**2**

votes

**1**answer

79 views

### Principally split primes with factors in arbitrarily small angular sectors

I wonder if the following is known:
let $n$ be a (square-free) positive integer. Is there ever/always a sequence of prime numbers $p$ that can be written in the form $$p = x^2 + ny^2,$$ where
$x, y$ ...

**3**

votes

**0**answers

129 views

### An estimate for dividing n^2 by each of the primes up to and including n, and then summing the results [closed]

I know that the asymptotic for the sum of all the primes up to n is $n^2/2\log n$. But I'm trying to find the formula (an estimate) for when $n^2$ is divided by each of the primes up to $n$, in turn ...

**3**

votes

**1**answer

433 views

### When does Merten's product theorem accurately estimate the number of coprimes in an interval?

Assume an arbitrary $x$ and let $z$ be smaller than $y$, where $y$ is the length of the interval $[x,x+y]$. What I would like to know is:
Let $W(z)=\prod_{p\leq z}\left(1-\frac{1}{p}\right)$. For ...

**5**

votes

**2**answers

271 views

### Sum of digits of repeating end of reciprocal of prime over period is $\frac{9}{2}$

Take a prime other than 2,3 or 5 and look at the part of it that repeats in base 10. Is it true that the sum of the digits in the end divided by the period(number of repeated digits id always ...

**2**

votes

**3**answers

358 views

### Does this 'alternating' Euler product converge for all $\Re(s) > 0$?

Does the following 'alternating' Euler product, with $p_n$ the $n$-th prime number, converge for $\Re(s)>0$ ?
$$\displaystyle \prod_{n=1}^\infty \left( \dfrac{1}{1-\frac{1}{p_{n}^{s}}} ...

**0**

votes

**3**answers

569 views

### Definition of Prime Numbers [duplicate]

The first time I heard of prime numbers, they were defined as natural numbers $n$ that can only be divided by 1 and themselves without remainder; later, when prime factorization was introduced, I ...

**2**

votes

**2**answers

233 views

### What is the minimal range $[f(n),g(n)]$ that contains a prime number for every integer $n>0$?

I know the following:
Proven: There is a prime number between $n$ and $2n$ for every integer $n>0$.
Conjectured: There is a prime number between $n^2$ and $(n+1)^2$ for every integer $n>0$.
...

**1**

vote

**2**answers

246 views

### prime zeta function when $0<s<1$ [closed]

I will not be surprised if this question seems trivial in MO but i asked it first in MathSE and i did not get an answer.
So, here it is:
I would like to know if there is a good estimate for the sum ...

**5**

votes

**1**answer

451 views

### Lines in image; are they significant to prime numbers if so how?

Amateur math question. I was playing around generating some 2D images, and wondered what it would look like if placed $P_{i}$ dots on a circle with diameter of $i$ for increasing values of $i$, where ...

**3**

votes

**1**answer

322 views

### Giuga's Conjecture: Central or Peripheral?

An earlier MO question
highlighted
Giuga's Conjecture:
A positive integer $n>1$ is prime if and only if
$$\sum_{k=1}^{n-1} k^{n-1} \equiv -1 \pmod{n}$$
For example, for the prime $n=5$, ...

**3**

votes

**0**answers

183 views

### Logarithms of ratios of squarefree numbers

Let $M \geq 1$, and $N=2^M$. Let $a_1, \dots, a_N$ be the set of all the numbers that you get when forming all square-free products of the first $M$ primes.
For example, for $M=2$ and $N=4$ you get ...

**10**

votes

**1**answer

500 views

### Chebotarev density theorem for $k$-almost primes

Consider a finite Galois extension $L$ of $\mathbb Q$, of Galois group $G$. Let $k \geq 1$ be a fixed integer. Let $D$ be a subset of $G^k$ invariant by conjugation and by the natural action of the ...

**4**

votes

**1**answer

360 views

### Does this prime-gaps pattern occur infinitely often?

Let $p_n$ be the $n$-th prime.
For each integer $k \ge 0$, do there exist
an infinite number of $k+3$ consecutive primes
$(p_n, p_{n+1}, \ldots, p_{n+2+k})$
so that
(1) The gap between the 1st and ...

**-2**

votes

**1**answer

196 views

### Giuga and Carmichael numbers

If $p$ is both Giuga and Carmichael number
then its known that
$1^{p-1}+2^{p-1}+3^{p-1}+\cdots+(p-1)^{p-1} \equiv -1\pmod{p}$
is it true that
if $p$ is both Giuga and Carmichael number then
...

**0**

votes

**0**answers

161 views

### Interval containing prime numbers

Let $\varepsilon$ be an arbitrary small positive number. Can we prove that there exist an $n\in Z$ such that the interval $[2^n,(1+\varepsilon)2^n]$ contain a prime number?

**5**

votes

**0**answers

243 views

### Should I expect to see numbers this smooth?

I have a sequence $N_k$ of numbers whose growth I wish to determine, or at least
approximate nicely. When I look at the ratios of consecutive members,
I find some interesting simplifications ...

**-1**

votes

**1**answer

112 views

### Fermat pseudo prime base-3 [closed]

Good morning!
I have checked the following statement by random numbers of my choice. I am seriously looking for proof of the statement.
Statement: $m$ is said to be Fermat pseudo prime in base-3, ...

**2**

votes

**0**answers

121 views

### Analytic varieties for the primes and the twin primes

I am wondering what real and complex analysis say
about the primes and twin primes.
According to Wikipedia
analytic variety is defined locally as the set of common zeros of finitely many analytic ...

**5**

votes

**3**answers

322 views

### Weak versions of Bertrand's postulate

We are interested in the following statement:
For each $n>1$ and $x>2$ there is at least one prime $p$ satisfying $x<p<n x$.
For $n=2$ we get precisely the Bertrand's postulate which is ...

**0**

votes

**0**answers

65 views

### existence of elements with specific norms in pure cubic fields

Is there any specific way to find an element with a given norm in pure cubic field? say for an example an element of norm 5 in pure (monogenic) cubic field of 11.
it is easy to check that 5 as an ...

**4**

votes

**1**answer

163 views

### Consecutive Primes mod 3

Is anything known asymptotically about the binary "primes mod 3" sequence besides Dirichlet's result that 1 and 2 occur half of the time? For example, can you prove that it does not eventually cycle ...

**4**

votes

**2**answers

313 views

### Can a polynomial be almost always divisible by a member of a finite set of primes?

Special case of Bunyakovsky conjecture
Let $f(x)$ be non-constant irreducible polynomial with integer
coefficients, no fixed prime factor and positive
leading coefficient. Let $S$
be a finite set of ...

**0**

votes

**1**answer

82 views

### Joint Modular Distribution of Primes

Dirichlet's theorem shows that, for any fixed prime integer a,
"big prime numbers mod a" are uniformly distributed between
1 and a-1. If we similarly pick different prime integers
b,c,..., are these ...

**1**

vote

**0**answers

125 views

### Updated tables of maximal prime gaps? [closed]

The website http://www.trnicely.net/gaps/gaplist.html contains a long list of maximal and nonmaximal prime gaps. In this list, the largest maximal prime gap is one of length 1476:
Size: 1476
Gap ...

**7**

votes

**1**answer

315 views

### Are primes of density 0 in $a\cdot b^n+c$?

Hooley proves in Applications of Sieves to the Theory of Numbers that there are only $o(x)$ numbers $n\le x$ such that $n\cdot2^n+1$ is a (Cullen) prime. The proof generalizes to forms ...

**1**

vote

**0**answers

86 views

### For which types of problems can one expect to use Bombieri-Vinogradov in place of GRH?

I must profess a general ignorance of problems that were once known to be true under GRH but has since became unconditional due to the Bombieri-Vinogradov theorem, but I am aware of the heuristic that ...

**4**

votes

**1**answer

155 views

### Higher dimensional generalization of the Hardy-Littlewood conjecture?

The famous Hardy-Littlewood conjecture on prime-tuples states that if $\{h_1, \cdots, h_k\} = \mathcal{H}$ is an admissible set, that is, for every prime $p$ the set $\mathcal{H}$ does not contain a ...

**15**

votes

**1**answer

760 views

### Sums of primes that are themselves prime

I'm not a math expert so this may be a trivial question; if $p_i$ is the $i$-th prime, let:
$$S(n) = \sum_{i=1}^n p_i$$
be the sum of the first $n$ primes and
$$P(n) = | \{1 \leq i \leq n \mid ...

**0**

votes

**1**answer

141 views

### Tail of singular series of Goldbach problem

Let $N$ a large number and $P=P(N)$. We know that the "tail" of singular series of Goldbach problem is $$ ...