**3**

votes

**2**answers

311 views

### Statements going against the grain of Riemann Hypothesis (R.H.)

Let $M(N) := \sum_{n=1}^N \mu(n)$
It is known that bounding $M(N)$ by $N^{1/2+\epsilon}$ implies R.H.
A bound of $M(N)$ by $K\sqrt N$ for say $ K\ge2 $ is also sufficient. $K = 1$ is excluded as ...

**-1**

votes

**0**answers

66 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 ...

**-1**

votes

**0**answers

40 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 ...

**13**

votes

**1**answer

320 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

75 views

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

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,.......

**-4**

votes

**0**answers

79 views

### A basic Query regarding Riemann Zeta function [closed]

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 ...

**1**

vote

**0**answers

75 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 ...

**-1**

votes

**0**answers

71 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

90 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 ...

**-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 ...

**5**

votes

**1**answer

246 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 = ...

**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 ...

**8**

votes

**0**answers

265 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 ...

**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

468 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 ...

**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 ...

**6**

votes

**1**answer

144 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

273 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.

**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 ...

**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 ...

**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}...

**8**

votes

**1**answer

235 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

428 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 ...

**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:...

**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 ...

**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)} \...

**-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

242 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 $i$-...

**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 ...

**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 $...

**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? ...

**6**

votes

**1**answer

230 views

### Can $b^4+1$ be a pseudoprime to base 2 (except for Fermat numbers)?

Cross-post: This very elementary question was first posted to Mathematics Stack Exchange but the response I got there (even after offering a bounty) was not useful.
For the purpose of this question, ...

**35**

votes

**3**answers

3k views

### Why such an interest in studying prime gaps?

Prime gaps studies seems to be one of the most fertile topics in analytic number theory, for long and in lots of directions :
lower bounds (recent works by Maynard, Tao et al. [1])
upper bounds (...

**4**

votes

**3**answers

394 views

### On a theorem of Hensel about congruence of binomial coefficient

In the paper Binomial coefficients modulo prime powers, Andrew Granville stated the following theorem:
Let $n, m$ and $r=n-m$ be three given positive integer and $p^k$ is the exact power of $p$ ...

**7**

votes

**0**answers

194 views

### Can primes be (almost) random sequence in von Mises sense?

Random models for primes (such as Cramer's model) have been extensively used for informal justification of various conjectures involving primes. It is crucial to understand in what sense sequence of ...

**0**

votes

**1**answer

138 views

### An upper bound on $\sum_{n^{1/3}<p,q\leq n^{1/2}} \frac{n}{pq}-\lfloor \frac{n}{pq}\rfloor$

I would like to ask if there is a good upper bound on the difference $$D_2(n)=\sum_{n^{1/3}<p,q\leq n^{1/2}} \left(\frac{n}{pq}-\left\lfloor \frac{n}{pq}\right\rfloor\right)\quad (1) $$where $p$ ...

**15**

votes

**2**answers

2k views

### Could this unexpected bias in the distribution of consecutive primes have any impact on the security of encryption algorithms?

In a recent paper a quite unexpected result about a new pattern in prime numbers emerged:
Unexpected biases in the distribution of consecutive primesby Oliver, R. J. L.; Soundararajan, K. (Submitted ...

**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 ...

**4**

votes

**0**answers

83 views

### Large Gaps Between Almost Primes

What is the best lower bound for the longest interval contained in $[1,x]$ free of primes and products of two primes? In other words I am asking for the best lower bounds in a variant of the ...

**1**

vote

**0**answers

63 views

### Behavior of partial Euler product in the critical strip (with Dirichlet Character)

Consider a primitive Dirichlet Character $\chi$ (non principal) and the partial Euler product attached to the L-function $L(\chi,s)$ ($p_i$ are the prime numbers) :
$$P(\chi,N)=\prod_{i=1}^{N} \frac{...

**1**

vote

**0**answers

158 views

### Is the following claim true about systems of quadratic congruences modulo consecutive prime numbers

Is the following true?
Choose any value for $y : y \in \mathbb{N}$
If
$N(y)$ is the smallest natural number that satisfies the following system of quadratic congruences:
$N(y)^2 \not\equiv 1$ ...

**4**

votes

**2**answers

154 views

### Evolution of partial sum of a sequence of induced Dirichlet characters

Let's consider the Dirichlet Character $\chi_3(n)$ modulo 3 given by $\chi_3(1)=1$, $\chi_3(2)=-1$ and $\chi_3(3)=\chi_3(0)=0$.
Lets consider the sequence of induced characters $\chi^{P_N} $ obtained ...