**3**

votes

**1**answer

302 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.

**0**

votes

**0**answers

235 views

### Reverse to Chinese Remainder Theorem [on hold]

Theorem. Let $m_1,m_2,\ldots,m_r$ be positive integers that are relatively prime in pairs, that is, $(m_i,m_j) = 1$ if $i\neq j$. Then for any integers $a_1, a_2,\ldots a_r$. the $r$ congruences
$$
...

**3**

votes

**1**answer

338 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 ...

**-2**

votes

**0**answers

85 views

### Double prime condition $p^{p-1} \equiv 1\!\pmod{q}$ and $q^{q-1} \equiv 1\!\pmod{p}$

This is a follow-up question to
Does the congruence $a^p \equiv 1 \pmod{b^p}$ with prime $p \ge 5$ force $b \le p$?
Let $p,q$ be two odd primes such that $$p^{p-1} \equiv 1\!\pmod{q}$$ and $$q^{q-1} ...

**8**

votes

**1**answer

205 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 ...

**6**

votes

**1**answer

203 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

209 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 ...

**-1**

votes

**0**answers

55 views

### Property of set of prime numbers [migrated]

let $\{p_1,p_2,p_2,\cdots ,p_r\}$ be the set of $r$($\ge2$) pair wise distinct prime numbers
i.e.., $(i\ne j \implies p_i \ne p_j)$ for all $1\le i,j\le r$
${Statement}$ :
For any such ...

**-1**

votes

**1**answer

131 views

### Conjectured Primality Test for Numbers of the Form k2^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 ...

**9**

votes

**1**answer

300 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

291 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

567 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

112 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

232 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

279 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 ...

**10**

votes

**1**answer

519 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

99 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

206 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

194 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

495 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

227 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

140 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

648 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

265 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

81 views

### Primality Criterion for Specific Class of Numbers of the Form kb^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

86 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

388 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

48 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

351 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

178 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

310 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

220 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

1k 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

150 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

166 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

157 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 ...

**5**

votes

**0**answers

213 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 ...

**3**

votes

**3**answers

436 views

### Conjecture about a sequence of natural numbers, such that, $\forall n : A_n<P_n<A_{n+1}$

Conjecture - no natural number $k$ exists such that:
$P$ is the sequence of all primes starting from the $k$th prime
$A$ is a sequence of natural numbers such that:
$\forall n : ...

**18**

votes

**2**answers

914 views

### The prime numbers modulo $k$, are not periodic

Consider the sequence of prime numbers: $2,3,5,7, \cdots$. Now reduce this sequence modulo $k$ for some integer $k > 2$. Show the resulting sequence is not periodic. :
EDIT: As noted in the ...

**2**

votes

**0**answers

338 views

### New proofs of Euclid's theorem of the infinitude of primes?

Playing around with elementary inclusion-exclusion, I arrived at two simple variations of proofs of Euclid's theorem that I thought would be long known in the literature. So far I haven't been able to ...

**0**

votes

**0**answers

194 views

### Relationship between this conjecture and Lehmer's Theorem?

Let A be:
n such that $\ \frac{n-1}{ord_n 2}=2^x\ $ and $n$ with the conditions of the conjecture in OEIS A226014,$\ n \in \mathbb{Z^+} ,\ x \in \mathbb{Z}_{\geq 0},\ $then $n$ is prime ...

**4**

votes

**1**answer

191 views

### Uniformity of the distribution of the prime numbers on the prime residue classes (mod $m$)

Given positive integers $m$, $r$ and $n$, let $\pi(m,r,n)$ denote the number
of prime numbers $p \leq n$ in the residue class $r$ (mod $m$).
Further let $1 = r_1 < r_2 < \dots < ...

**3**

votes

**2**answers

353 views

### Are all counterexamples of OEIS A226181 both Poulet numbers and Proth numbers?

OEIS A226181:
3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79,
83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, ...
Primes $p$ ...

**10**

votes

**2**answers

1k views

### Natural numbers that cannot be expressed as a difference between a square and a prime?

We wish to find the set of natural numbers that cannot be expressed as a difference between a square and a prime.
e.g.
$1 = 2^2 - 3$
$2 = 3^2 - 7$
$3 = 4^2 - 13$
and so on.
The smallest such ...

**2**

votes

**0**answers

211 views

### Relation between Maier's theorem and a conjecture of Montgomery and Soundararajan

Let us consider the number of primes in the interval $[N,N+h]$, with $h\leq N$. According to the answer given by Lucia to a previous question on the distribution of primes, it is natural to consider ...

**0**

votes

**1**answer

312 views

### Is a certain sumset derived from primes of a certain form the set of all naturals?

OEIS A167055 Numbers n such that $12n + 5$ is prime.
$0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21,...$ are items of OEIS $A167055$.
I conjecture that the set of the sum of every two items of this ...

**2**

votes

**0**answers

122 views

### n-ary quadratic forms with $S$-integer values

Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form.
Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to ...

**1**

vote

**0**answers

100 views

### Prime counting function with a form of finite product using perron's formula

There's a form of complex integral what Riemann obtained to finding $\pi (x)$,
$$ \pi^{*}(x)=\int_{L}\frac{\log \zeta (s)}{s}x^{s}ds, (1)$$
we already know that it lead us to the Prime Number ...