Questions tagged [prime-numbers]

Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.

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0answers
74 views

Primality test for specific class of $N=8kp^n-1$

My following question is related to my question here Can you provide a proof or a counterexample for the following claim : Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\...
5
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1answer
114 views

Asymptotic density of sums of consecutive primes

Call a positive integer respectable if it is a sum of consecutive prime numbers. For example, every prime numbers is respectable. So are $3+5=8$, $2+3+5=10$, $5+7=12$, $3+5+7=15$, $2+3+5+7=17$, $7+11=...
5
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1answer
206 views

Conjectured primality test for specific class of $N=k \cdot 6^n+1$

Can you provide a proof or a counterexample for the claim given below? Inspired by Theorem 5 in this paper I have formulated the following claim: Let $N=k \cdot 6^n+1$ , $k<6^n$ and $\...
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1answer
124 views

Arithmetic progressions, given a prime

I have recently become interested in reading a little more on certain directions regarding primes in arithmetic progressions (AP). I would appreciate specific paper references (with the journal and ...
5
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2answers
238 views

Pairs of integers whose product is one more or less than a prime

Given a positive integer N it is often possible to pair each of the integers 1, 2, 3, ..., N with a different integer between N + 1 and 2 N so that the product of each pair is one less or more than a ...
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64 views

How we can characterize all positive integers, multiples of 4, that cannot be expressed as $(p_1-1)(p_2-1),\;\;p_1,\,p_2$ distinct primes

I ask how we can characterize all positive integers multiples of 4 that cannot be expressed as $(p_1-1)(p_2-1),\;\;p_1,\,p_2$ distinct primes The first multiples of 4 that cannot be expressed ...
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2answers
90 views

A variant of Turán–Kubilius inequality

Let $\omega(n)$ the number of distinct prime factors of $n$ (counted without multiplicity). A famous consequence of Turán–Kubilius inequality is $$ \sum_{n\leq x}(\omega(n)-\log\log x)^2=O(x\log \log ...
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4answers
387 views

Can one show combinatorially how $\operatorname{lcm}(1, \dotsc, n)$ grows?

Let us write $M(n)$ for $\operatorname{lcm}(1,\dotsc,n)$ for $n$ a positive integer. Asymptotically $M(n)$ tends toward $e^n$. This result uses analytic number theory. (Lcm is least common multiple, ...
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143 views

A conjecture involving $P_n=\prod_{k=1}^np_k$

For each positive integer $n$ let $P_n=\prod_{k=1}^n p_k$, where $p_k$ is the $k$th prime. Question. Is my following conjecture true? Conjecture. For any integer $n>1$, there are $k,m\in\{1,\...
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1answer
161 views

The existence of rational points [closed]

Solving some problem parametrically, I got the following answer: $$ \dfrac{5x}{4} + \sqrt{\dfrac{y^2}{4} - \dfrac{x^2}{16}} + \dfrac{1}{10} \sqrt{10x^2 + 9y^2} + \dfrac{1}{5} \sqrt{5x^2 + 16y^2} $$ ...
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72 views

Improved upper bound for second moment of reduced residues modulo $q$?

The question asked here is a follow up to this question, which was answered by user GH from MO. [EDIT: The question is edited for clarification after receiving a comment on the original posting.] As ...
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107 views

Write $2n+1=p+q$ with $p$ prime and $q$ practical

Recall that a positive integer $n$ is callled practical if every $m=1,\ldots,n$ can be written as the sum of some distinct divisors of $n$. The only odd practical number is $1$. In 1996 G. Melfi [J. ...
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203 views

Are there infinitely many primes of the form $x^2+(x+y)y^2$?

Heath-Brown [Acta Math. 186(2001), 1-84] proved in 2001 that there are infinitely many primes of the form $x^3+2y^3$ with $x,y\in\mathbb Z^+=\{1,2,3,\ldots\}$. In contrast, here I ask the following ...
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1answer
113 views

p2 - p1 = 2n for every 2n [closed]

Quote: "Chen's work mentioned in the discussion of the Goldbach conjecture also showed that every even number is the difference between a prime and a P2." from: link However I can't get this ...
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2answers
419 views

Are twin primes the only solution to this equation?

Let $p,q_i, i \ge 1$ be primes, $m$ a positive integer. The equation $$ p.\prod_{i=1}^m(q_i-1)-(p-1).\prod_{i=1}^mq_i=2 $$ for $m=1$ has all twin primes $p,q_1=p+2$ as solution. Are there solutions ...
2
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1answer
161 views

Do prime pairs inbetween and equidistant from adjacent integer powers cover all the prime numbers?

Is it true that every odd prime number is a member of one or more pairs of primes $(p_1, p_2)$ such that $p_1$ and $p_2$ are inbetween and equidistant from two adjacent powers of some number $n$? i.e. ...
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1answer
544 views

Prime-like numbers that avoid Green-Tao? [duplicate]

I would like to understand the conditions that support the Green-Tao Theorem, which established that the primes contain arbitrarily long arithmetic progressions. I am wondering: Q. Is it difficult ...
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48 views

Erdos multiplication table problem avoiding integers with too many distinct prime factors

Consider multiplication operation $$f(x_1,x_2)=x_1x_2$$ where $x_i\in\{1,\dots, n_i\}\backslash T_i$ with $n_1, n_2\in\{1,\dots,\infty\} $ where $T_i$ is set of positive integers less than $n_i$ which ...
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97 views

Possible rearrangments of double products containing sine function : [closed]

I know that the following question is not a fit (at all ) for this site , So , apologies ; but it interests me in very unusual way ; so I'm asking here . If not appropriate to post here tell me I'll ...
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2answers
453 views

A tree with prime vertices

Let us construct a simple (undirected) graph $T$ as follows: $\quad$ Let the set of all primes be the vertex set of $T$. For each prime $p$, take the least prime $q>p$ such that $2(p+1)-q$ is ...
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63 views

Construction of weight function to satisfy condition on given functional

(Sorry for similar and trivial looking question ; But could have potential application in prime number theory ) Consider the following function : $$F(z) = \omega(z)\frac{\sin^2\left(\frac{c\Gamma^2(...
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1answer
108 views

The equivalent proposition of Legendre's conjecture [closed]

Legendre's conjecture, proposed by Adrien-Marie Legendre, state that there is a prime number between $n^2$ and $(n+1)^2$ for every positive integer $n$. My conjecture: Let $n$ be a positive integer, ...
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1answer
204 views

“On the distribution of reduced residues” by Montgomery and Vaughan – missing careful argument wanted

In their paper, On the distribution of reduced residues, Montgomery and Vaughan state early on that With a more careful argument from (2) it is easily seen that $$\tag{*} qhP - qhPQ + O(qhP^2) \leq ...
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1answer
181 views

A conjecture stronger than the Legendre conjecture about prime numbers [closed]

I want to draw attention on my own question (cross posted from Mse) Hi I want to evaluate the following sum : $$S_n=\frac{1}{\log(2)}+\frac{2}{\log(3)}+\frac{3}{\log(4)}+\frac{4}{\log(5)}+\cdots+\...
4
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1answer
75 views

Divisors of shifted geometric progressions

For integers $a,b,k$ with $a \geq 1$ and $k\geq 2$, consider the shifted geometric progression $n_i = ak^i + b$. I would like to understand the set of integers (prime or otherwise) that divide at ...
4
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0answers
216 views

How to prove there is infinite prime numbers of form $5n+3$ without Dirichlet theorem? [closed]

Is there a nice elementary way to prove there is infinite prime numbers of form $5n+3$ (also for $5n+2$) with $n\in \mathbb{N}$? I know how to do it for primes of form $pn+1$ for any prime $p\geq 3$ ...
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1answer
89 views

On existence of conjecture relating prime zeta function:

There is an article on Wikipedia about prime zeta function (PZF): https://en.m.wikipedia.org/wiki/Prime_zeta_function In that article , there is table of fairly accurate values of PZF for different ...
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74 views

On the asymptotics of the Chebyshev psi function

Denote by $\psi(x)$ the Chebyshev psi function over prime powers. Assuming the RH, it can be shown via the Riemann explicit formula that $$\psi(x)-x \ll √x |\sum_{|\gamma| < T} \frac{1}{\rho} | +...
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1answer
123 views

What are the exceptional properties of Mersenne exponent for known largest prime? [closed]

It is a clear that largest known primes are Mersenne prime. It is well known that $2^p - 1$ is prime only if $p$ is prime; however, the converse is not true - take $p = 11$. My question is: is there ...
2
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0answers
141 views

A conjecture/algorithm about prime numbers [closed]

I am not a mathematician, but I used to love reading Euclid's elements. In 2013 I discovered a very inefficient way of generating prime numbers exactly using a statement I deduced while reading BOOK ...
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1answer
238 views

Is there a connection of prime numbers and Extreme Value Theory?

As most others are, so am I fascinated by primes. By the theorem of Euclid and the sieve of Eratosthenes the $ k \ge 2$ - th prime is given by: $$ p_k = \min_{x>1,\gcd(x,p_1 \cdots p_{k-1})=1} x ...
2
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0answers
64 views

The number of admissible tuples with last element equal to $h_{k-1}$?

Let $k \geq 2$ and $(h_1, h_2,\cdots,h_{k-1}) \in \mathbb{N}^{k-1}$. Consider the $k$-tuple : $\mathcal{H}_k=(0,h_1,\cdots,h_{k-1})$ with $0<h_1<\cdots<h_{k-1}$. The $k$-tuple $\mathcal{H}...
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155 views

A conditional approach to twin prime conjecture

Disclaimer: this question was first asked on a French forum (here comes the link for those of you who read French: http://www.les-mathematiques.net/phorum/read.php?5,1758830,1758922), but no ...
2
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0answers
71 views

Iteration of a primeness-measuring function

Question For $n \in \mathbb{N}$ let $\delta(n)$ denote the cardinality of the set $$\left\{(a,b) \in \mathbb{N}^2 \::\: 1 < a < n,\ 1 < b < n,\ n|ab\!\: \right\}.$$ Let $D(n)$ denote ...
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166 views

Explicit Formula for $n$th prime in terms of Riemann zeros?

We all know there exists a explicit Formula for prime counting function in terms of Riemann zeros. I'm wondering if similar formula exists for $n$th prime in terms of Riemann zeros?
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126 views

Is Brun's constant a period?

Brun proved that the sum of reciprocals of the twin primes is finite. This sum, Brun's constant, is not known to be irrational, (otherwise this would immediately entail the veracity of the twin prime ...
4
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2answers
487 views

Error term in Mertens' third theorem

Mertens' third theorem states that: $$\prod_{\substack{ p \leq x \\ \text{p prime} }} \left( 1 - \dfrac{1}{p} \right) \sim \dfrac{e^{-\gamma}}{\log(x)}$$ Question: what is the best functions (...
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0answers
63 views

Are numbers which are the product of n primes more common than numbers which are the product of n-1 primes? [duplicate]

In a recent video (https://www.facebook.com/188916357807416/videos/519169035700435/) Stephen Wolfram wonders whether, for every integer n>2, eventually the number of integers which are precisely the ...
4
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1answer
194 views

Reference / Survey for finite field analog number theory

It is folklore that many number theoretic results on prime numbers have a simpler-to-prove finite field analog. For example, on the one hand, the proof of the Prime Number Theorem $$\#\{\text{prime ...
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1answer
504 views

How to explain this prime gap bias around last digits?

My question is related to this article by Oliver and Soundararajan (article about a bias in the distribution of the last digits of consecutive prime numbers). After trying some python experimental ...
1
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1answer
119 views

Another kind of primality related to tessellations by polygons

You can define a number $p$ to be prime by "no tessellation of $p$ identical squares forms a convex figure". This suggests what I'll call a t-prime $p$, defined by "no tessellation of $p$ identical ...
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71 views

Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^{\frac{{p_1}-3}{p_2}}-1)\equiv 0 \bmod p_1$ with $p_1,p_2\equiv 3\bmod 4$?

let $p_1$ and $p_2$ be positive primes such that $p_1,p_2 \equiv 3\bmod 4$ and $\phi$ is the Euler totiont function , I want to find the Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^...
2
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1answer
189 views

How does one prove that the density of unusual numbers is $\ln 2$?

The Wikipedia page for unusual number states that the density of that set is $\ln 2$, and that this was proven by Schroeppel in 1972. The only source that I found for that is the HAKMEM document, and ...
3
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1answer
130 views

Primes in arithmetic progressions above a given threshold

Given co-prime $a,b$, Dirichlet's theorem states that there are infinitely many primes in the arithmetic progression $M = \{ a + bn : n \in \mathbb N\}$. Linnik's theorem asserts that the first such ...
1
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1answer
218 views

Equation of the Chebyshev $\psi$ function

Consider $\Psi(x)$ to be the Chebyshev function given by $$\Psi(x)=\sum_{n\leq x} \Lambda(n)$$ where $\Lambda(n)$ is the Mangoldt function which is equal 0 unless $n $ is prime power, and let $(E)$ ...
2
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0answers
91 views

Write $p^2$ as $x^2+2y^2+3\times 2^z$ with $x,y,z$ nonnegative integers

In April 2018, I noted that the first integer $n>1$ with $n^2\not\in\{x^2+2y^2+3\times 2^z:\ x,y,z=0,1,2,\ldots\}$ is $$5884015571=7\times17\times49445509.$$ Question. Is it true that for each ...
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1answer
111 views

Sufficient conditions on $ a_i,b_i$ for $a_1\phi(n)+b_1, \cdots, a_k\phi(n)+b_k$ to be simultaneously prime infinitely often?

I am really interested in sufficient conditions on $a_i, b_i$ guaranteeing that the linear forms $a_1\phi(n)+b_1,\dots, a_k\phi(n)+b_k$ become simultaneously prime for infinitely many positive ...
2
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0answers
63 views

quadratic residues and cubic polynomials [closed]

I'm really not sure about this, but I've heard somewhere that for any prime $p$, $|\sum_{x=0}^{p-1} (\frac{ax^3 +bx^2 +cx +d}{p} ) |\le \sqrt{2p}$ holds. Does anyone know a proof for this inequality ...
7
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1answer
867 views

Some interesting experimental results about the distribution of primes

Let's consider the following metric of the gap between consecutive primes $$m(k)=\frac {p_k^2-p_{k-1}^2} {24}\;\;\;\;\;(k\ge4)$$ Now, let's define the function $\delta(k)=m(k)\;\;\;\;$ if $\,m(k)\,$ ...
2
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0answers
75 views

Sequence of least prime-multiples with smallest Hamming weight

It is known that Almost all primes have a multiple of small Hamming weight, which makes me wonder what is known about the least multiples of primes that have least Hamming weight. Questions: what ...

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