**4**

votes

**1**answer

88 views

### References to proofs of upper and lower bounds on the number of coprimes in an interval?

On the first page of the article "When the sieve works", the authors present upper and lower bounds for $S(T,T+x;\mathcal{E})$; the number of integers in the interval $(T,T+x]$ that are coprime to all ...

**2**

votes

**0**answers

93 views

### Siegel Walfisz Theorem for algebraic number fields

Is there a generalization of the Siegel Walfisz to algebraic number fields? This has been done for the prime number theorem in the prime ideal theorem.

**-1**

votes

**0**answers

78 views

### Fermat's little theorem question [closed]

I'm studying Number theory (in my spare time) and I need to prove a lemma in order to prove the exercise. The topic is Fermat's little theorem.
Well the lemma goes like this:
Let's say we have ...

**0**

votes

**0**answers

93 views

### E- and A-algorithms for finite arithmetic prime progressions and other sets

There is certain Eratosthenes spirit to my problem (See below). First of all I'd like to stress the mathematical aspect of my question. Also, my question does not amount to the divide and conquer ...

**3**

votes

**1**answer

239 views

### Sum of two squares and implication of Bunyakovsky conjecture

Bunyakovsky conjecture states that a polynomial with integer
coefficients takes infinitely many prime values at integers,
unless this is impossible for trivial reasons.
Let $a_1(x), a_2(x), a_3(x), ...

**0**

votes

**0**answers

107 views

### every arithmetic progression contains a sequence of $k$ “consecutive” primes for possibly all natural numbers $k$?

I ask the same question here:http://math.stackexchange.com/q/1019404/192097
writing a little better the previous question: it´s true that if we let $a$ and $b$ be coprime integers, then the ...

**5**

votes

**1**answer

378 views

### Are there infinitely many primes p such that both p-1 and p+1 have at most 3 prime factors, counted with multiplicity?

This is a subsequence of primes $p$ for which $p^2-1$ has at most 6 prime divisors counted with multiplicity.
This sequence described in the question is the sequence A079153 in OEIS.
I could not ...

**5**

votes

**0**answers

145 views

### Is ${\rm lcm}\{{\rm ord}_p(q)\colon q\mid p-1,\ q>2\}>\sqrt p\ \,$?

The following question is "ideologically related" to the one I recently asked here.
For a prime $p$, let $M_p$ denotes the least common multiple of the orders modulo $p$ of all odd prime divisors of ...

**-2**

votes

**1**answer

86 views

### Looking for the name of an infinite sequence [closed]

I am looking for information about a sequence that seems like it
should converge. The sequence is textually described as:
...

**5**

votes

**0**answers

171 views

### A conjecture of Erdos on consecutive differences of primes

Let $d_k = p_{k + 1} - p_k$ be the difference between consecutive primes and define
\begin{equation}
e_k = \left\{\begin{array}{c l} 1 &, d_{k + 1} > d_k \\ 0 &, \text{otherwise} ...

**0**

votes

**0**answers

117 views

### Estimating the number of twin primes of given natural configuration order

This question is a follow-up from About Goldbach's conjecture. Let
$$\mathrm{Co}_{k}(x):=\{n\le x:\mathrm{ord}_{c}(n):=\pi(\sqrt{2n-3})=k\},~~~\mathrm{co}_{k}(x):=\vert\mathrm{Co}_{k}(x)\vert$$
...

**0**

votes

**0**answers

93 views

### Lower bound on number of smooth values of polynomial at primes

Given a polynomial $f$, it is known believed that the number of smooth values of $f$ has a positive proportion (for fixed $u$, $\lim_{X\rightarrow\infty} \frac{|\{ n < X\ :\ f(n)\ is\ X^u\ smooth ...

**10**

votes

**1**answer

386 views

### Parity of the Prime Counting Function

I am interested in the distribution of the parity of $\pi(x)$, the prime counting function, over the natural numbers.
Let:
$\ \ E_n := \left\{ k \in \left\{1,\dots,n\right\} : \pi(k) \equiv 0 \mod 2 ...

**11**

votes

**0**answers

200 views

### Why $\gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}$ likes to be large?

For a prime $p$, let $F_p$ denote the greatest common divisor of the orders modulo $p$ of all prime divisors of $p-1$:
$$ F_p = \gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}; $$
thus, for instance, ...

**0**

votes

**1**answer

231 views

### Is there a formula that can predict the primes in the sequence of ratios of consecutive superior highly composite numbers? : $2, 3, 2, 5, 2, 3, 7,…$

This is the sequence of prime numbers which are the elementary building blocks for the superior highly composite numbers:
$2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, ...$
The $n^{th}$ ...

**2**

votes

**0**answers

126 views

### Effective version of the Bombieri-Vinogradov theorem

Is there an effective version of the Bombieri-Vinogradov Theorem, in that have bounds on the implied constant been found?

**2**

votes

**0**answers

183 views

### What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?

The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log ...

**3**

votes

**0**answers

524 views

### Second Hardy-Littlewood Conjecture theme

If Second Hardy-Littlewood Conjecture is true then we can claim that $\pi(x)-\pi(y)\leq \pi(x-y)$. Thus the conjecture gives an upper bound for the number of primes between $x$ and $y$. I have found ...

**5**

votes

**1**answer

265 views

### Parity of primes [duplicate]

While working on a completely different (combinatorial) problem, I ran a simple program to calculate the parity of the first ~50000 primes (number of 1s in their binary representation modulo 2). The ...

**5**

votes

**1**answer

264 views

### What does the sum of the reciprocals of all the highly composite numbers converge to?

I've calculated the sum of the reciprocals of all the $156$ first highly composite numbers up to $10^{18}$:
$\sum \dfrac{1}{HCC(n)} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + \dfrac{1}{12} + ...

**2**

votes

**0**answers

94 views

### The behavior of series involving special subsets of the prime numbers

It is well known that the series $\sum_{p\in \mathbb{P}} \frac{1}{p}$ diverges where $\mathbb{P}$ denotes the set of primes. Brun proved that $\sum_{p\in \mathbb{P_2}} \frac{1}{p}$ converges where $ ...

**7**

votes

**0**answers

213 views

### On the ratio of Gilbreath sequences

Definitions: let $n \in \mathbb{N}_{>0} \cup \{ \infty \}$ and let $E_n$ be the set of sequences $(d_i)_{i=1}^n$ such that $d_1=1$, $d_i$ is an even integer (for $i > 1$) and $0<d_i \le i$. ...

**1**

vote

**0**answers

72 views

### Definition of primitive divisor of a Lucas sequence

If $\{a_n\}_{n=1}^\infty$ is a sequence of integers, then the definition of primitive divisor of one of its terms is quite natural:
Def. 1. A prime number $p$ is a primitive divisor of $a_n$ if $p ...

**3**

votes

**1**answer

433 views

### what would be the consequences on the distribution of primes of $\Lambda=\infty$?

It is widely believed that the quantity $\Lambda:=\lim\sup\dfrac{t_{n+1}-t_{n}}{2\pi/\log t_{n}}$, where $t_{n}$ is the imaginary part of the $n$-th non-trivial zero on the critical line of the ...

**2**

votes

**3**answers

276 views

### Prime factors of the members of a certain recurrence

It is possible to prove elementarily that there are infinitely many primes that divide some element of the sequence $a_0 = k\ge 0$, $a_n = a_{n-1}^2+ 1$ for all $n\ge 1$ by showing that for all $m$, ...

**0**

votes

**1**answer

304 views

### The periodic architecture underlying the natural numbers [closed]

EDIT In the original version of this post I did not include a well specified mathematical question, and learning by failing, I realize there should have been. Closing or deleting the question is the ...

**14**

votes

**0**answers

375 views

### Does $\binom{2n}{n} \equiv 2 \pmod p$ ever hold?

Well, the title does not tell the whole story; the complete question is:
Are there any primes of the form $p=2n(n-1)+1$, with integer $n\ge 1$, such that
$$ \binom{2n}{n} \equiv 2\pmod p ? $$
...

**1**

vote

**1**answer

232 views

### Conjecture on prime numbers

Given a prime $p$, let $a_n=pn+n-1$.
I have noticed that $\forall{p}\exists{n}\in[2,p]:a_n\in\mathbb{P}$.
For example: $p=7,a_3=23,a_4=31,a_6=47$.
What is this conjecture called, and has it been ...

**-3**

votes

**1**answer

171 views

### Andrica's and Legendre's Conjectures [closed]

My question is, which of these two conjectures is stronger, Andrica's or Legendre's? Could proving one prove the other? If the upper bound for the prime gap above any given natural number $n$ were to ...

**3**

votes

**1**answer

175 views

### Prime divisors of the respectively minimal binomial coefficients

In view of Chebyshev's approach to prime numbers, I would like to ask about the regularities and peculiarities of the two sequences $\ \beta(n)\ $ and $\ \gamma(n),\ $ which I define as follows:
$\ ...

**2**

votes

**2**answers

317 views

### Conjectured relation between alternating Prime zeta series and Riemann zeta

Let $P(s)$ be the Prime zeta function.
Numerical evidence suggests these identities:
$$ \sum_{k=1}^\infty \frac{(-1)^{k}P(3k)}{k}=\log{\bigg(\frac{1}{945}\frac{\pi^6}{\zeta(3)}\bigg)}\qquad\quad ...

**1**

vote

**0**answers

73 views

### Bounds re Asymptotic Formula for the Sum of Largest Prime Factors

I have a reference request related to the result :
$\sum_{n=2}^{x} P(n)$ ~ $\frac{\pi^2}{12}\frac{x^{2}}{log(x)}$ as $x \rightarrow \infty$
where $P(n)$ is the largest prime factor of the positive ...

**0**

votes

**0**answers

184 views

### Approximating the number of primes

Let $\pi(x)$ be the number of primes le than $x$. Is there any approach or attempt to find a nice function $f$ that does a better job in approximating $\pi$ than the logarithmic integral ...

**2**

votes

**2**answers

376 views

### Is there a von Koch-type theorem for the generalized Riemann hypothesis?

Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the error term in the prime number theorem having the bound
$$
\mid\pi(x)-\textrm{li}(x)\mid=O(\sqrt{x} \log x).
$$
Q1: ...

**31**

votes

**2**answers

978 views

### Does iterating a certain function related to the sums of divisors eventually always result in a prime value?

Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$.
For example $f(6)=6+3+2=11$, $f(5)=5$.
Note that $x$ is a fixed point for ...

**4**

votes

**0**answers

223 views

### $x^2+1$ attaining almost prime values

Iwaniec, using the linear sieve, proved that $n^2+1$ can be a product of at most two primes infinitely often and furthermore a lower bound of the correct order of magnitude for the number of such ...

**0**

votes

**0**answers

66 views

### Metric defined over Galois extensions of the rationals [duplicate]

I don't know if this of interest, but I'd be curious to know if the following idea has been pursued.
In this question (Metric on the set of subsets of the rational primes) I proposed a metric, d, ...

**8**

votes

**2**answers

585 views

### Is a Galois extension of the rationals determined by its set of completely split primes?

apologies if this is a naive question. Consider two Galois extensions, K and L, of the rational numbers. For each extension, consider the set of rational primes that split completely in the ...

**2**

votes

**1**answer

234 views

### Is it possible to sum the divergent series with prime coefficients?

It is known that the series $$ P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime} $$ cannot be summed by means of (prime) zeta function regularization. (The result was ...

**3**

votes

**0**answers

311 views

### Metric on the set of subsets of the rational primes

Note: this is a revision of an earlier post. It was kindly pointed out that my initial proposed metric was in fact not a metric, so this is a revised version.
I was thinking how to say that two sets ...

**9**

votes

**1**answer

254 views

### Repetend digit graphs for $1/n$ in base $b$

Here is a decimal expansion of $\frac{1}{34}$:
$$(1/34)_{10}=0.02941176470588235\overline{2941176470588235}\ldots$$
And here is a graphical representation of the 16-digit
"repetend," as a directed ...

**3**

votes

**1**answer

342 views

### Lower bounds on the error term of the prime number theorem

Are there any lower bounds on the error term for the prime number theorem, or in other words, is there a nontrivial $f$ s.t.
$$f(x)\ll |\psi(x) - x|$$
where $\psi$ is the Chebyshev function.

**3**

votes

**1**answer

376 views

### Smallest prime in an arithmetic progression

Let $\{a_n\}_{n\in\mathbb{N}}$ be defined as $a_n = a + bn$ for some $a, b >0,(a, b) = 1$. Are there good bounds on the minimal $k$ s.t. $a_k$ is prime. It is well known that there are infinitely ...

**8**

votes

**1**answer

228 views

### Integers with a large prime divisor in short intervals

For an integer $n$, denote by $P^+(n)$ the largest prime divisor of $n$. Then we have the following:
There exists some $c>0$, such that for all $x$ sufficiently large the number of integers ...

**7**

votes

**1**answer

237 views

### lower and upper bound for $\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k$?

Are there known any lower and upper bounds for
$$
\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k,
$$
where $\Omega(n)$ is the number of prime factors counting multiplicities of $n$?
Or at least is it known ...

**6**

votes

**2**answers

242 views

### Divisor sums over values of binary forms of primes

Let $\tau$ be the divisor function, that is
$$
\tau(n)=\sharp\{d \in \mathbb{N}, d|n\}.
$$
I was wondering if anyone has ever proved an asymptotic estimate
for the sum
$$S(x):=\sum_{p,q\leq ...

**-1**

votes

**1**answer

139 views

### Conjectured Primality Test for Numbers of the Form k2^n+1 with n>2 [closed]

Definition : Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^m+\left(x+\sqrt{x^2-4}\right)^m\right) $
where $m$ and $x$ are positive integers .
Conjecture : Let $N=k\cdot 2^n+1$ with ...

**10**

votes

**1**answer

319 views

### Squarefree numbers $n$ such that $432n+1$ is also squarefree

This is a second attempt (see Primes $p$ such that $432 p +1$ is prime)
Is the set of squarefree numbers $n$ such that $n(432 n+1)$ is also squarefree known to be infinite?
Fact: the number of such ...

**-1**

votes

**1**answer

307 views

### Primes $p$ such that $432 p +1$ is prime [closed]

Is the set of prime numbers $p$ such that $432 p + 1$ is also prime infinite?
It doesn't follow from Dirichlet's theorem as far as I can tell.

**5**

votes

**2**answers

640 views

### The shortest interval for which the prime number theorem holds [closed]

It is well known that the prime number theorem on the form
\begin{align*}
\pi(x+y) - \pi(x) \sim \frac{y}{\log (x+y)}
\end{align*}
breaks down for short enough intervals, e.g. taking $y=(\log ...