**0**

votes

**0**answers

57 views

### A rash guess about distribution of primes based on meager empirical evidence?

Elementary number theory is a field in which imbeciles can ask questions that experts cannot answer (and I wonder if discrete geometry is a similar subject in that respect?) and herewith I submit ...

**-4**

votes

**0**answers

69 views

### four consecutive primes ending in 1,3,7, or 9 [on hold]

Examine the last four digits of four consecutive primes to seek 1,3,7,9 in any
order. You will find that they occur more than by chance. Do the same for the
frequency of two, three, four,.......

**-1**

votes

**0**answers

26 views

### product distinct prime factors of prime(n)-1 and prime(n)+1 [on hold]

The prime 127 has 127-1=126 with distinct prime factors 2,3,7 and 127+1=128 with
distinct prime factors of only 2; hence 2*3*7=42<127. Log 127/42=q=1.296. Are
such primes common? Can a value of ...

**12**

votes

**1**answer

253 views

### Does Littlewood's bound on $\zeta(1+it)$ extend to all the partial sums?

Littlewood established that $2e^{\gamma} \geq \limsup_{t \to \infty} |\zeta(1+it)| / \log{\log{t}} \geq e^{\gamma}$, the lower bound unconditionally and the upper bound on RH. It now seems to be ...

**-4**

votes

**0**answers

77 views

### A basic Query regarding Riemann Zeta function [on hold]

The Euler Definition of Zeta is given as (extreme right):
Using only Euler's definition, how can any value of s (real/complex) lead to the function being 0. As, no matter what the denominator of ...

**9**

votes

**2**answers

1k views

### What is known about primes of the form x^2-2y^2?

David Cox's book Primes of The Form: $x^2+ny^2$ does a great job proving and motivating a lot of results for $n>0$. I was unable to find anything for negative numbers, let alone the case I am ...

**1**

vote

**0**answers

72 views

### Reference request: Bounding exponential sum $\sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x )$

Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let
$$
\mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} .
$$
We ...

**-3**

votes

**1**answer

102 views

### Is a positive integer determined by its sequence of typical primality radii?

This question is a follow-up to About Goldbach's conjecture . Assuming the truth of Goldbach's conjecture, suppose $n$ and $m$ are two positive integers such that $N_{2}(n)=N_{2}(m)=:N$ and that ...

**-1**

votes

**0**answers

70 views

### Sum-free sets of powerful numbers

For $n=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ with distinct primes $p_i$, call $\alpha= (\alpha_1,\dots,\alpha_r)$ the type of $n$ and denote by $N_\alpha$ the set of all naturals of this type.
We ...

**-1**

votes

**0**answers

88 views

### A question about arithmetic progressions and prime numbers

"I took number $3$ and observed:
$3$ is an arithmetic progression of length one.
$3,5$ is an arithmetic progression of length two.
$3,5,7$ is an arithmetic progression of length three.
Then I took ...

**6**

votes

**1**answer

662 views

### A general question about strictly non-palindromic numbers

For a definition, see the wikipedia page: http://en.wikipedia.org/wiki/Strictly_non-palindromic_number
So according to the wikipedia page, under properties, all strictly non-palindromic numbers with ...

**3**

votes

**2**answers

149 views

### $p$-simple integers from between $n$ and $n+p-1$

Let $\ p\ $ be an arbitrary prime. Then an integer $\ s\ $ is called $p$-simple $\ \Leftarrow:\Rightarrow\ s\ $ is not divisible by any prime $\ q<p.\ $
Could you prove my conjecture (or is it ...

**5**

votes

**1**answer

244 views

### Infinitely many primes coming from Euclid's proof

When teaching Euclid's classic proof of the infinitude of primes today, the following question appeared to me. Let $p_1,p_2,p_3,\ldots$ be the prime numbers, listed in increasing order. Set
$$k_n = ...

**12**

votes

**3**answers

2k views

### Is there a known bound in prime gaps?

Is there known to be an $x$ such that for all positive integers $N$ there exists some $n>N$ such that $p_{n+1}-p_n \leq x$, where $p_n$ is the $n$th prime? Or, in other words, is it known that ...

**8**

votes

**0**answers

261 views

### Is the number $\sum_{p\text{ prime}}p^{-2}$ known to be irrational?

Is the number $$\sum_{p\text{ prime}}p^{-2}$$ known to be irrational?
The limit exists, since $$\sum_{p\text{ prime}}p^{-2}<\sum_{i=1}^{\infty}i^{-2}=\frac{\pi^{2}}{6}$$.

**0**

votes

**0**answers

82 views

### Recursive construction of integers from Fermat Primes

Assume there are only finitely many Fermat primes.
Fix a Fermat prime $F_t$ say $3$.
What is in general known about numbers of form $F_{t,k}=2^kF_t +1$?
Is the number of primes of form $F_{t,k}$ ...

**0**

votes

**0**answers

8 views

### Why no prime number could appear as the length of a hypotenuse in more than one Pythagorean triangle? [migrated]

Why no prime number could appear as the length of a hypotenuse in more than one Pythagorean triangle?
In other words, could any of you give me a algebraic proof for the following?
given prime ...

**8**

votes

**0**answers

187 views

### Greedy permutation of the set $\{1,2,\dots,n\}$ and prime numbers

Answering this question, another question came to my mind. For which $n$ will the greedy algorithm work?
We define a sequence of natural numbers $x_n$ recursively:
$$x_1 =1,$$
$$x_n \mbox{ is the ...

**6**

votes

**1**answer

185 views

### Permutations of the set $\{1,2,…,n\}$ and prime numbers

Here is the version of this question that I posted on math.stackexchange a few days ago and I did not receive an answer that settles my question so I thought that maybe on this site I could get a ...

**1**

vote

**1**answer

240 views

### Density of Sophie Germain $3\bmod 4$ primes

Is there any reason to expect the density of primes $p$ such that $2p+1$ is also a prime where $p=3\bmod4$ holds would be different from case of $p=1\bmod4$?
What if $2p+1$ is replaced by $2p-1$ and ...

**7**

votes

**1**answer

175 views

### Main term in the number of sign changes of $\psi(x) - x$

Define $N_\Delta(T)$ to be the number of sign changes of $\psi(x) - x$ in the interval $[1, T]$.
Landau's Theorem says $N_\Delta(T)$ is $\Omega(\log T)$ [1].
But perhaps that estimate is too crude. ...

**1**

vote

**1**answer

466 views

### What is the best currently proven bounds on prime gaps?

I did some digging around on the internet but I found tons of different equations on both lower and upper bounds for the largest possible prime gap g(n). I was wondering what are currently the best ...

**0**

votes

**0**answers

42 views

### On a count of certain number of primes in an interval

Fix a prime $p$, $\alpha\in(0,1)$ and $\beta\in(1,2)$ and let $\mathcal U$ be primes in $[p^\alpha,\beta p^\alpha]$ such that if $b\in\mathcal U$ and if $d$ is multiplicative inverse of $b$ in $\Bbb ...

**6**

votes

**1**answer

143 views

### Density of prime divisors of $a^n + b$

Suppose $a > 1, b \neq 0$ be two rational numbers. Is it known in general that the set of prime divisors of (the numerator of) $a^n + b$ has a positive relative density?

**3**

votes

**0**answers

169 views

### Effective prime number theorem

The prime number theorem implies that for every $ϵ>0$, there is $n_\epsilon$ such that for all $n≥n_\epsilon$ the number of primes in $[n,cn]$ is at least $\frac{(c−1−\epsilon)n}{\log n}$ and at ...

**0**

votes

**0**answers

47 views

### Density of set of primes which avoid given finite set of residues modulo powers of all primes

Let $k\in\mathbb{N}$, $k\ge2$ and $S\subseteq\mathbb{Z}$ be a finite set of integers. For every prime $p$ let $c_p$ be a number of invertible residue classes mod $p^k$ that contain some element of $S$....

**4**

votes

**0**answers

272 views

### Proof for new deterministic primality test possible?

Conjecture:
Let $n \in \mathbb{N}$ and $n$ odd.
Then the number $N=2^2 + n^2$ is prime, if and only if $N$ divides $2^{(N-1)/2} + 1$.
Thanks.

**2**

votes

**3**answers

343 views

### Primes in arithmetic progressions in number fields

My general question is how does one prove equi-distribution results for primes in arithmetic progressions in number fields? I am interested in the equi-distribution of prime elements of the ring of ...

**7**

votes

**1**answer

162 views

### Do arbitrary $K$-twin primes exist?

Polignac's conjecture states that for any positive even integer $K$, there exist infinitely many pairs of primes such that their difference is $K$.
I am interested the status in a much weaker form of ...

**15**

votes

**2**answers

781 views

### A set of integers whose factorial can be written as a product of two factorials

I am trying to collect informations concerning the set
$$\mathcal{A}=\left\{n\in\mathbb{N} \mid (\exists k,l\in\{2,3,\dots,n-2\})(n!=k!l!)\right\}.$$
It seems not much is known about the set $\mathcal{...

**2**

votes

**0**answers

115 views

### $f(x)$-th largest number of prime factors

Given a sufficiently well-behaved function $1\le f(x)\le x$ and a multiset $S=\{\omega(n): 1\le n\le x\}$, what can be said about the asymptotics of the $f(x)$-th largest member of $S$? In other words,...

**1**

vote

**0**answers

135 views

### On the cardinality of the set of right-truncatable primes

We say that the (base ten) prime number $p=a_{n}a_{n-1}a_{n-2}\cdots a_{1}a_{0}$ is right-truncatable if all of the following numbers are prime:
\begin{eqnarray*}a_{n},\\a_{n}a_{n-1},\\ a_{n}a_{n-1}...

**19**

votes

**3**answers

1k views

### For consecutive primes $a\lt b\lt c$, prove that $a+b\ge c$.

For consecutive primes $a\lt b\lt c$, prove that $a+b\ge c$.
I cannot find a counter-example to this. Do we know if this inequality is true? Alternatively, is this some documented problem (solved or ...

**8**

votes

**1**answer

233 views

### Arbitrarily many primes in a Fibonacci-type sequence

It is conjectured that the standard Fibonacci sequence contains infinitely many primes. While this is perhaps too difficult, I am wondering about the following simpler version:
Question. For any $K$, ...

**3**

votes

**1**answer

427 views

### origin of analogy “primes as the atoms of number theory/ arithmetic”

a math student recently challenged me on the old comparison/ analogy of prime numbers to "the atoms of number theory or arithmetic" and then was wondering the origin of the phrase.
where does this ...

**20**

votes

**1**answer

1k 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 $2^n \pm 1$?
We recall that radical of an integer $rad(k)$ is a product of primes which divide $k$.
As an example, if ...

**20**

votes

**1**answer

1k views

### How many primes can there be in a short interval?

Given $n \in \mathbb{N}$, let $\pi(n)$ denote the number of prime numbers $\leq n$.
What is
$$
\limsup_{m \rightarrow \infty} \left( \limsup_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} \...

**3**

votes

**1**answer

115 views

### Reference Request on the existence of $k$ satisfying $P(\Phi_d(2))^k \gt \Phi_d(2)$ for all $d$

I am working my way through the literature regarding the following conjecture: There is a positive integer $k$ such that for all positive integers $d$,
$$P(\Phi_d(2))^k \gt \Phi_d(2).$$
I am ...

**0**

votes

**0**answers

155 views

### Asymptotic value of sum over Möbius function

Consider the sum
$$
S(x)=\sum_{\substack{{d\mid P(\sqrt{x})}\\{d\leq x}}}|\mu(d)|,
$$
where $P(x)$ is the product of all primes less than or equal to $x$, and $\mu(d)$ is the Möbius function.
Q:...

**12**

votes

**3**answers

898 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? ...

**18**

votes

**1**answer

312 views

### On random divisor sums modulo $2^k$

Let $k,n,\ell$ be positive integers with $k,n\ge 2$ and $0\le \ell \le k-1$. For each integer $2\le j \le n$, choose a divisor $d_j$ of $j$, uniformly at random from the divisors of $j$. We denote by $...

**12**

votes

**2**answers

667 views

### 3-7 primes in base 10

After a quick look at the sequence (of primes) A020463,
$$
3, 7, 37, 73, 337, 373, 733, 773, 3373, \dots,
$$
the following question is straighforward:
Are there infinitely many primes compiled from ...

**-1**

votes

**1**answer

129 views

### Find the values of $n$ that satisfy this inequality involving a product over prime numbers [closed]

Inequality
What values of $n$ satisfy the following inequality?
$$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$
$p$ are prime numbers and the notation $p_i$ indicates the $i$th ...

**4**

votes

**1**answer

241 views

### The values of $n$ which satisfy an inequality about prime numbers

For which values of $n$ does the following inequality hold for?
$$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-1}{p_i}\right)$$
$p$ are prime numbers and the notation $p_i$ indicates the $i$-...

**4**

votes

**0**answers

210 views

### Asymptotic estimate for a random model of primes

Question
Let
$$
\pi_{rm_c}(x) = \sum_{ \substack{ {n\leq x}\\{(n+a,P(\sqrt{n}))=1}}} 1-1,
$$
where $P(x)$ is the product of all primes less or equal to $x$ and $a$ is a random integer constrained to ...

**2**

votes

**1**answer

88 views

### Set of triple-primes satisfying a certain equation

Is there a set of triple-primes satisfying the following equation?
$p_1p_2+p_2p_3+p_3p_1+p_1+p_2+p_3=2^β,p_1p_2p_3=2^α-1,α>β.$
I have checked the first 11 numbers that no one satisfy the above ...

**7**

votes

**0**answers

138 views

### k-Almost Primes in short intervals

According to this question every interval $[x, x + x^{0.45}]$ contains a product of two primes, and this has been improved further slightly. Are there better results available for $k$-almost primes? ...

**13**

votes

**0**answers

437 views

### Intersection between the sums of the first positive integers, primes and non primes

Is the following conjecture true ?
$$\left\{\sum\limits_{\begin{array}{c}k=1\\k\in\mathbb{Z}\end{array}}^nk \ |\ n\in\Bbb Z\right\} \cap
\left\lbrace \sum\limits_{\begin{array}{c}k=1\\k\in\mathbb{Z}...