**53**

votes

**0**answers

3k views

### The topology of Arithmetic Progressions of primes

The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner ...

**16**

votes

**0**answers

453 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
$$ \binom{2n}{n} \equiv 2\pmod p ? $$
...

**13**

votes

**0**answers

864 views

### Small primes attract large primes

I posted a version of this to stackexchange and got 12 up-votes and no answers in somewhat more than a day. Someone in a comment construed it as asking for a lot of novel research including figuring ...

**11**

votes

**0**answers

188 views

### Small quotients of smooth numbers

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...

**11**

votes

**0**answers

302 views

### Word complexity of primes mod 4

For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...

**11**

votes

**0**answers

221 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, ...

**11**

votes

**0**answers

637 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 ...

**9**

votes

**0**answers

372 views

### Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system,
$$x_1^2+x_2^2+\dots+x_n^2 = y^2$$
$$x_1^3+x_2^3+\dots+x_n^3 = z^3$$
and define the function,
$$F(s_m) = x_1+x_2+\dots+x_n$$
For $n\geq3$, using an ...

**9**

votes

**0**answers

381 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, ...

**8**

votes

**0**answers

128 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

**0**answers

718 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, ...

**8**

votes

**0**answers

548 views

### Two different ways to count Mersenne Primes

Hi there, the motivation for this question is to better understand the heuristics of Mersenne primes, and I was motivated by the recent questions (Mersenne quasi-primes) and (Primes in generalized ...

**7**

votes

**0**answers

226 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 $0<d_i \le i$. ...

**7**

votes

**0**answers

197 views

### In which orders can the numbers of prime factors of consecutive integers be?

Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur.
Given ...

**7**

votes

**0**answers

309 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 ...

**7**

votes

**0**answers

621 views

### “probabilistic” density of primes?

A certain set $\cal P$ of primes is defined by two assumedly independent conditions:
The first condition on a prime $p$ can be characterized in terms of the type of splitting of $p$ in certain Galois ...

**6**

votes

**0**answers

231 views

### Can integers be distorted to make primes more regular?

Given a set $P$ of real numbers $\ge 1$, define the gap among different products in $P$ as
$$g(P) = \inf \big\{\prod_{i=1}^n p_i^{a_i} - \prod_{i=1}^n p_i^{b_i} \mid p_i\in P;\,\, p_i\ne p_j \,\text{ ...

**6**

votes

**0**answers

160 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 ...

**6**

votes

**0**answers

208 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} ...

**6**

votes

**0**answers

258 views

### On the sum of consecutive primes and product of first and last

Lets consider the sequence of consecutive prime numbers $p_1=2 , p_2=3 ,p_4=5 , \cdots$
. $(p_n,p_{j})$ is to be called good prime pair if $$\sum_{i =n }^{j}p_i= p_n p_{j}$$
Meaning the sum of set of ...

**6**

votes

**0**answers

310 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? ...

**6**

votes

**0**answers

235 views

### Generalizing prime numbers to product-indecomposable objects in toposes

Any object $A$ in any topos decomposes as $A\times1$ and $1\times A$, and in $FinSet$ objects with no other product decompositions are "prime numbers". Is there any extension of the theory of ...

**6**

votes

**0**answers

613 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, ...

**6**

votes

**0**answers

476 views

### Any published references for this $O(n)$ time, $O(n^\epsilon)$ space identity for the count of primes?

I would like to know if this identity (or trivial equivalents) for $\pi(n)$, the count of primes, is currently published anywhere.
$D_{0,a}(n) = 1$
$D_{1,a}(n) = \lfloor n\rfloor-a-1 \ \ \ \ \ \ \ \ ...

**6**

votes

**0**answers

352 views

### Primes of the form $x^2+ny^2$ such that when you swap x and y you get another prime

Hello, Im looking at primes of the form $x^2+ny^2$ for $n>1$ where we can swap $x$ and $y$ and get another prime, I have found many such pairs for many values of n, and i wanted to know if there ...

**5**

votes

**0**answers

251 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 ...

**5**

votes

**0**answers

250 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 ...

**5**

votes

**0**answers

255 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 ...

**5**

votes

**0**answers

253 views

### $n\varphi(n)\equiv 2\pmod{\sigma(n)}$ as a primality test

It is known from Subbarao, "On two congruences for primality" that $n>22$ is a prime iff $$n\sigma(n)\equiv 2\pmod{\varphi(n)},$$ where $\varphi(n)$ is Euler's function and $\sigma(n)$ is sum of ...

**5**

votes

**0**answers

308 views

### How many ways can a number be Fortunate?

An integer $m$ is Fortunate if it can be written as $q-P$, where $P$ is a primorial and $q$ is the smallest prime greater than $P+1$. It is conjectured that Fortunate numbers are always prime.
It ...

**5**

votes

**0**answers

245 views

### Picking out an odd subset

I want to know whether the following property holds:
There exists a constant C such that for any big positive integer N and any nonempty subset M of {1,...,N} there exists a positive integer ...

**5**

votes

**0**answers

516 views

### Are there an infinite number of prime Euclid numbers?

A number defined as the product of first $n$ prime numbers $+1$ is called $n$th Euclid number. Are there any survey on the progress for answering the following question: are there an infinite number ...

**4**

votes

**0**answers

271 views

### Is there a hidden symmetry in the prime numbers distribution?

Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime.
...

**4**

votes

**0**answers

134 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 ...

**4**

votes

**0**answers

168 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 + ...

**4**

votes

**0**answers

315 views

### About sign changes of Li(x)-π(x)

Given a constant $C$, which are the best known upper bounds for the number of sign changes
of the function
$$
f: \mathbb{N} \rightarrow \mathbb{R}, \ \ x \mapsto {\rm Li}(x)-\pi(x)
$$
in the range ...

**4**

votes

**0**answers

90 views

### On a weighted sum in a lemma for sieve methods

I'm reading James Maynard's paper "Small gaps between primes".
Lemma 6.1 (p.14) in this paper confused me. This lemma was taken from
Goldston-Graham-Pintz-Yildirim's paper "Small gaps between ...

**4**

votes

**0**answers

414 views

### AKS Algorithm Pseudoprimes

The AKS algorithm is based on the following fully deterministic primality check:
Let input $n>1$ and $a \in \mathbb{N}$ such that $(a,n)=1$. Then $n$ is prime if and only if
$$\tag{1}(x+a)^n ...

**4**

votes

**0**answers

319 views

### Estimate on radical of $2^n \pm 1$

Not sure if this belongs to MO or not.
Are there any lower bound on radical of $2^n \pm 1$ (Recall that radical of integer $rad(k)$ is a product of primes which divide integer $k$)?
As an example If ...

**4**

votes

**0**answers

148 views

### smallest k such that highest prime factor of m(m+1)…(m+k-1) is > n if m > n.

I am fascinated by this entry OEIS A213253 which lists the smallest $k$ such that highest prime factor of $m(m+1)\dots(m+k-1)$ is $> n$ if $m > n$.
The article has references to the algorithm ...

**4**

votes

**0**answers

287 views

### Generalization of Tamarkin’s ARO 1993, final round, problem 10/8: part II

Let us use the notations of my previous question about Tamarkin's problem.
Let $\ell\in\left\lbrace 0,1,...,p\right\rbrace$.
An element $f\in \mathbb Z^{\mathbb Z}$ is said to be ...

**4**

votes

**0**answers

363 views

### Number of $k$-partitions of $n$ into odd prime parts

Browsing through OESIS I have found that if $p_p(n)$ denotes the number of partitions of $n$ into prime parts then $p_p(n) = O(e^{\frac{2 \Pi}{\sqrt{3}}\sqrt{n/\log n}})$.
I am interested in the ...

**3**

votes

**0**answers

427 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

**0**answers

171 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?

**3**

votes

**0**answers

545 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 ...

**3**

votes

**0**answers

322 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 ...

**3**

votes

**0**answers

198 views

### Arguments for the second Hardy–Littlewood conjecture being false?

Assume that $x,y > 2$, and that $x<y$. Then the second Hardy–Littlewood conjecture states that
$$\pi(x + y) - \pi(y) \leq \pi(x).$$
We can easily justify this heuristically, since
$$
...

**3**

votes

**0**answers

106 views

### Numbers expressible as sums of prime powers larger than n

Given a fixed $n \in \mathbb{N}$ larger than $1$, let $G(n)$ denote the largest number that is not expressible as a sum of prime powers larger than $n$ (the 'base' prime of the prime power need not be ...

**3**

votes

**0**answers

190 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 ...

**3**

votes

**0**answers

235 views

### Generating function for the characteristic function of prime numbers

What do we know about the generating function of $\chi(n)$ (A010051)
$$
f(x) = \sum_{n=0}^\infty \chi(n)x^n = \sum_{p\text{ prime}} x^p
$$
for $\chi(n)$ the characteristic function of the primes:
...