**2**

votes

**1**answer

278 views

### Prime Number Theorem on APs under various conjectures

I'm trying to find the best asymptotic expansions for $\pi(x; a, q)$ in various states:
Unconditionally we have
\begin{equation}
\pi(x; a, q) = \frac{\operatorname{li(x)}}{\phi(q)} + O\left(x ...

**3**

votes

**0**answers

456 views

### A “Take a Square Root When You Can” conjecture related to the prime factorization

I would tend to think that the following has already been investigated.
But as implied from the title, I have no idea how to even start looking for it.
Let $P_n$ denote the sum of the squares of ...

**1**

vote

**0**answers

64 views

### closed form for a series with binomials and primes

does the series $\sum_{n=0}^\infty p^n \binom{x}{p^n}$ have a closed form ? ($p$ prime)
this is a special case of $\sum_{n=0}^\infty p^n \left(\sum_{k=p^n}^{p^{n+1}-1}a_k\binom{x}{k}\right)$ with the ...

**2**

votes

**1**answer

804 views

### Conjecture on the square root of the sum of the squares of the prime factors of a number

Let $A_{n}$ denote the square root of the sum of the squares of the prime factors of $n$.
For example, $A_{60}=\sqrt{2^2+2^2+3^2+5^2}\approx6.48$.
I have recently made the following observations:
...

**10**

votes

**1**answer

677 views

### Divergence of a series similar to $\sum\frac{1}{p}$

Suppose we start with $k$ primes $p_1,p_2,\ldots ,p_k$ (not necessarily consecutive) and a residue class for each prime $r_1,r_2,\ldots ,r_k$.
We denote the least integer not covered by the arithmetic ...

**8**

votes

**3**answers

645 views

### Is $n = p-q$ equivalent to Goldbach's Conjecture?

One open conjecture is that every even integer greater than two is the difference of two primes. (Some superficial discussion here.)
Goldbach's conjecture states that every even integer greater than ...

**5**

votes

**1**answer

644 views

### Any way to prove Prime Number Theorem using Hyperbolic Geometry? [closed]

The prime number theorem says that the density of prime numbers is inverse as the number of digits of $n$:
$$\displaystyle \frac{\{1 \leq k \leq n : \text{ prime } \}}{n} \approx \frac{1}{\log n}$$
...

**2**

votes

**2**answers

474 views

### Prime divisors of $p^n+1$

Let $p$ be a rational prime and $n$ be a positive integer.
It can be easily deduced from Zsigmondy's theorem that $p^n+1$ has a prime divisor greater than $2n$ except when $(p,n)=(2,3)$ or ...

**5**

votes

**1**answer

394 views

### Ruth-Aaron triples, etc

A Ruth-Aaron pair is two numbers $(n,n+1)$ such that
their sum of prime factors is equal, counting repeated prime factors.
(The name refers to Hank Aaron's 715 homeruns surpassing Babe Ruth's 714!)
So
...

**3**

votes

**0**answers

470 views

### Two (strictly related) proofs by induction of inequalities

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...

**3**

votes

**1**answer

318 views

### Primes $p=x^2+27y^2$ and Ramanujan's $x_1^{1/3} + x_2^{1/3} + x_3^{1/3}$

I was trying to generalize,
...

**6**

votes

**1**answer

632 views

### Prime labelling of graphs

A prime labeling of a graph is an injective function $f: V(G) \to \{1, 2, ..., |V(G)|\}$
such that for every pair of adjacent vertices $u$ and $v$, $\text{gcd}(f(u), f(v)) = 1$ (labels of any two ...

**2**

votes

**1**answer

151 views

### Cardinality of the prime divisor set of a k-power sum

Let $a_{1},\dots,a_{n}$ be positive natural numbers ($n>2$) such that $a_{i}\neq a_{j}$ if $i\neq j$. I want to prove that
$$ \left\lvert \left\{ p \text{ prime} \; : \; p \mid \sum_{i=1}^n ...

**12**

votes

**1**answer

597 views

### Is there a known primitive recursive upper bound on the nth “Zhang prime”

(This question is pure curiosity. Feel free to close it if you feel it is not appropriate for mathoverflow.)
In 2013 Zhang showed that there are infinitely many pairs of primes which are less that ...

**2**

votes

**1**answer

504 views

### Have there been any new developments in the Firoozbakht conjecture? [duplicate]

Let $(p_{n})_{n∈ℕ}$ be the sequence of consecutive primes. In P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1995, page 185, the author says:
A new conjecture by F. ...

**4**

votes

**0**answers

142 views

### The divisors of $p-1$ and high-degree residues modulo $p$

Here is a somewhat more explicit version of a question that I asked a while ago.
Suppose that $p$ is a prime of the form $p=2n(n+1)+1$, with a positive integer $n$. Can every odd prime divisor of ...

**2**

votes

**0**answers

110 views

### counting irreducible factors

In How hard is it to compute the number of prime factors of a given integer? a question was asked on computing number of prime factors of an integer.
Suppose we have a polynomial $f(X)\in \Bbb Z[X]$ ...

**4**

votes

**1**answer

215 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 ...

**1**

vote

**0**answers

145 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.

**0**

votes

**0**answers

125 views

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

(EDIT from scratch).
Let $\ \mathbf a := (a_1\ \ldots\ a_n)\ $ be an increasing non-constant arithmetic progression of odd positive numbers. The goal here is to resolve efficiently one of the two ...

**3**

votes

**1**answer

435 views

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

Bunyakovsky's 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), ...

**1**

vote

**1**answer

240 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

413 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 ...

**6**

votes

**0**answers

168 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

114 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:
...

**6**

votes

**0**answers

222 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} ...

**1**

vote

**0**answers

137 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

473 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

233 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

462 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 $n^{th}$ ...

**3**

votes

**0**answers

192 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?

**7**

votes

**1**answer

1k 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

593 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

311 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 ...

**7**

votes

**1**answer

411 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} + ...

**3**

votes

**0**answers

118 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

239 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 $0<d_i \le i$. ...

**2**

votes

**1**answer

166 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

598 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

294 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

333 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 ...

**18**

votes

**0**answers

580 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

278 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

212 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

190 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:
$\ ...

**3**

votes

**2**answers

393 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 ...

**5**

votes

**1**answer

222 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

203 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 ...

**3**

votes

**2**answers

528 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: ...

**33**

votes

**2**answers

1k 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 ...