**1**

vote

**1**answer

252 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

181 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

182 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

337 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 ...

**2**

votes

**0**answers

79 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

192 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

404 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

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 ...

**5**

votes

**0**answers

236 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

596 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 ...

**3**

votes

**1**answer

248 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

318 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

261 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

352 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

398 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

231 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 ...

**8**

votes

**1**answer

259 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

256 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 ...

**0**

votes

**1**answer

180 views

### Conjectured Primality Test for Numbers of the Form $k \cdot 2^n+1$ 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

327 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

312 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

668 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 ...

**0**

votes

**0**answers

128 views

### Does $\pi(n+r)+\pi(n-r)$ decrease as $r$ increases?

Assume Goldbach's conjecture. Then for every large enough positive integer $n$ there exists a non negative integer $r$ such that both $n+r$ and $n-r$ are primes. Such an integer $r$ will be called a ...

**6**

votes

**1**answer

326 views

### is there any heuristics suggesting that the number of Fibonacci primes below $x$ is equivalent to $\log_{\phi}\log_{\phi}x$?

The question of knowing whether there are infinitely many Fibonacci primes is an open question. As $F_p$ is prime only if $p$ is prime, one has $\pi_{FP}(x)\le \pi(\log_{\phi} x+0.5\log 5)$, but ...

**14**

votes

**3**answers

1k views

### Prime factorization “demoted” leads to function whose fixed points are primes?

Let $n$ be a natural number whose prime factorization is
$$n=\prod_{i=1}^{k}p_i^{\alpha_i} \; .$$
Define a function $g(n)$ as follows
$$g(n)=\sum_{i=1}^{k}p_i {\alpha_i} \,$$
i.e., exponentiation is ...

**3**

votes

**1**answer

321 views

### Lower bound for a prime gap occurring infinitely often

In his striking paper of may 2013, Zhang showed the existence of an even integer $g\lt 70,000,000$ such that $g$ is a prime gap occurring infinitely often. What is the best unconditional lower bound ...

**13**

votes

**1**answer

693 views

### Tight prime bounds

This is a cross-post of this question on MSE. I would not usually do this, but have decided to in this case since it has had no responses having been posted as a bounty question. I did not delete the ...

**0**

votes

**0**answers

118 views

### When is the earliest large prime gap also the latest large prime gap?

Suppose one finds the earliest prime gap of at least a certain size $g$, so that $p_{n+1}-p_n=g$ and $n$ is the smallest index for which the gap is as big as $g$.
Now consider the relative size of ...

**1**

vote

**2**answers

230 views

### overlap quadratic residues

Let $p$ be a prime number of form $4k+1$ and $M$ is its quadratic residue set.
Let $M_i=\{i+x|\forall x\in M\}$ $\forall 0<i<p$.
Does there exist a positive constant $\varepsilon$ such that ...

**4**

votes

**1**answer

216 views

### How frequently is 3 a cubic residue mod primes in an arithmetic progression?

Suppose $(a,3q)=1$ and $a\equiv 1\pmod 3$. Are there infinitely many primes $p\equiv a\pmod {3q}$ such that $3$ is a cubic nonresidue modulo $p$?
Or, an equivalent formulation using quadratic forms: ...

**6**

votes

**1**answer

513 views

### Is this weak asymptotic Goldbach's conjecture open?

Let $\tau(x)$ be the number of even numbers $2<2n<x$ which can't be written as a sum of two primes.
Goldbach's conjecture: $\tau(x) = 0$
Asymptotic Goldbach's conjecture: $\tau(x) = O(1) $
...

**0**

votes

**2**answers

242 views

### Goldbach-type problem: the valuation of irreducible elements of a subgroup of $\mathbb{Q}_{> 0}$

Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number.
It follows that $\phi(\prod_i p_i^{n_i}) = \sum_i n_i p_i$, with $p_i$ a prime ...

**2**

votes

**0**answers

179 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
$$
...

**17**

votes

**1**answer

725 views

### The conjecture of Montgomery and Soundararajan on primes in short intervals: Empirical inconsistencies?

Assume that $y/ \log x \rightarrow \infty$ and that $y/x \rightarrow 0$. Then, from a conjecture by Montgomery and Soundararajan, we expect the number of primes in the interval $[x,x+y]$ to be ...

**0**

votes

**0**answers

286 views

### Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by:
$$
\ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1.
$$
For all terms of $A$ greater than $\ ...

**2**

votes

**0**answers

106 views

### Primality Criterion for Specific Class of Numbers of the Form $k\cdot b^n-1$

Let $N=k\cdot b^n-1$ where $b$ is an even integer , $3\nmid b$ , $3\nmid N$ ,
$k \equiv 1,5 \pmod{6}$ , $k< b^n $ and $n>2$ .
Let $S_i=P_b(S_{i-1})$ with $S_0=P_{k\cdot b/2}(P_{b/2}(4))$ , ...

**2**

votes

**0**answers

97 views

### Primality Criterion for Specific Classes of Generalized Fermat Numbers

Let $F_n(b)= b^{2^n}+1 $ where $b$ is an even integer , $ 3\nmid b , 5\nmid b $ and $n\ge2$
Let $S_i=P_b(S_{i-1})$ with $S_0=P_{b/2}(P_{b/2}(8))$ where
...

**3**

votes

**2**answers

417 views

### Primes from a Dirichlet sequence and an irrational number

From Dirichlet's theorem on arithmetic progressions, if $\text{gcd}(a,b)=1$ we know $\{ak+b\}_{k\ge 0}$ contains infinitely many primes. Let those primes be $p_1,p_2,\cdots$. Then the real
...

**0**

votes

**0**answers

59 views

### Prime Hadamard Matrices

Assume that $n$ is a sufficiently large number. Is there a Hadamaed matrix $H_{4n \times 4n}=(h_{ij})$ with the last row and the last cloumn $J$ (thet is, for every $k$, $h_{k,4n}=1$ and $h_{4n, ...

**10**

votes

**1**answer

374 views

### Primes dividing $2^a+2^b-1$

From Fermat's little theorem we know that every odd prime $p$ divides $2^a-1$ with $a=p-1$.
Is it possible to prove that there are infinitely many primes not
dividing $2^a+2^b-1$?
(With ...

**7**

votes

**4**answers

1k views

### Arbitrarily long arithmetic progressions

Are there arbitrarily long arithmetic progressions in which all the
prime factors of all the terms are at most $N$, for some $N$? Assume
all the terms are positive and the sequence of terms is ...

**5**

votes

**2**answers

199 views

### Relationship of Euler product to coprimality densities for arbitrary sets of primes

Continuing the curiosity of my last couple questions: Is it the case that for every set of primes $F$, the asymptotic density of the integers coprime to all of $F$ is $\displaystyle \prod_{p \in F} (1 ...

**2**

votes

**1**answer

312 views

### Finding a suitable number

Let $n,m$ be two positive integers. By $r_n$ we denote the largest prime not exceeding $n$. If $r_n\leq m\leq n$ and $q$ is the largest prime factor of $n!/m!$ such that $q\geq 17$ and $q\geq n-m+3$, ...

**1**

vote

**2**answers

237 views

### Consecutive primes versus prime twins

First a warm-up. Let $\ V\ $ be an arbitrary set of odd natural numbers. Let $\ S(V)\ $ be the generated multiplicative semi-group. What are the necessary and/or sufficient conditions on $\ V\ $ for ...

**6**

votes

**5**answers

2k views

### Optical methods for number theory?

I found a paper: 'A New Method of Finding the Distribution of Prime Number', saying
We stack discs and annuluses with certain rules then turn on the light to illuminate. The projection of ...

**4**

votes

**1**answer

157 views

### Log weight removal in general (weaker) prime number theorem

Let $a_n$ be a sequence of non-negative numbers.
Assume that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} a_p\log p}{X}\leq 1.$$
Can we prove that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} ...

**4**

votes

**1**answer

170 views

### Prime residua races and two views on primes

Let $\ a>1\ \ r\ \ k\ $ be arbitrary natural numbers such that $\ a\ r\ $ are relatively prime. The natural conjecture below, is it known?, is probably true in full generality:
Q1. There exists a ...

**7**

votes

**0**answers

191 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 ...

**6**

votes

**0**answers

248 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 ...