Questions tagged [prime-gaps]
The difference of two prime consecutive prime numbers is the prime gap. $g_i := p_{i+1} - p_i$.
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Making some sort of Logical Error in a Twin Prime Analysis [closed]
I'm trying to do some analysis on prime numbers, but I'm running into what has to be some sort of error on my part, but I can't figure it out. I'm using the concept of a wheel from wheel factorization,...
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About prime gaps ; $\sum_{m = 2}^{\ln_2^{*}(n)} \pi_{m}(n) \leq \pi(n)$?
Let $\pi(n)$ be the number of primes between $1$ and $n$.
Let $\pi_2(n)$ be the number of prime twins (gap $2$) between $1$ and $n$.
Let $\pi_3(n)$ be the number of prime cousins (gap $4$) between $1$ ...
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Devising an upper bound for $r_{0}(n)$ through Stefan-Boltzmann law
Define the fundamental primality radius of an integer $n>1$ as $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$.
Can we intepret the average prime gap around $n$ as an absolute temperature $T$...
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An interesting finding on twin primes
I was doing some research on prime gaps including twin primes and this led me to this finding, which is:
$$ \lim\limits_{n\to \infty} \frac{\pi^2(n)}{n\pi_2(n)} = 0.7550363087870907 \cdots\cdots (1) $$...
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All twin prime averages in the range $[9, 119]$ are of the form $6(5[3(z-x)]_{\pmod 7} + x)$ for some $x \in \{0,2,3\}, z \in \{0,2,3,4,5\}$.
Question. Can we come up with a general formula $f(x_5, x_7, x_{11}, \dots, x_{p_n})$ such that each twin prime average $a \in [p_n + 2, p_{n+1}^2 - 2]$ is expressible as $f(x_5, \dots, x_{p_n})$ for ...
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Prime gap size of order square root p(n)
I have the below construction.
Take a standard parabola $y=x^2$ with points on the x-axis at $(-p_n,0)$ and $(p_{n+1},0)$ and corresponding points on the parabola of $(-p_n,p_n^2)$ and $(p_{n+1},p_{n+...
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Does the merit of the prime gap size measure how far off the gap number is from its expected number based on the natural log?
I was watching the Stand-up Maths video *Exploring the mysteries of the Prime (gaps!) Line. and had some questions.
First, just to make sure I have everything straight, as I understand it, a "...
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Generating a random prime
How can I generate a random prime of the form $2^ab+1$ for small $b$ value without actually creating a list of such primes, and then choose from the list at random?
For example: I can generate a ...
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Does a prime surrounded by arbitrary large prime gaps always exists?
A question which could naturally arise when studying or thinking on prime number is: does gaps between prime numbers of arbitrary length always exists? In other words, given a positive integer $m$, ...
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Does anyone know how to prove or disprove that for all even integers $k$ that there exists a prime number $n$ such that $n+k$ & $n-k$ are both prime?
This problem came into my head while working on something similar. There was a similar question asked, but it was something slightly different. I’ve been stuck on this,unable to get any progress since ...
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Large gaps between small primes
Does there exist a positive integer $n>5$ such that the sum of the two largest primes less than $n$ equals $n$? If yes, lovely! If not, what is the largest prime gap possible between the two ...
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Stirling's approximation and gaps between primes
Let $G(n)$ denote the largest prime gap $p_{k+1}-p_k$ occuring between $1$ and $n$.
By considering the $n-1$ composite consecutive integers $n!+2$,...,$n!+n$ we can conclude that $$G(n!+n) \geq n-1 $$
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Is some twin prime average the sum of two twin prime averages, two ways?
Accoring to this question and a linked duplicate, it's been verified empirically up to some number that all twin prime averages greater than six, are the sum of two smaller twin prime averages.
I was ...
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An alien gives you a sequence, claiming it is the $1000$ prime gaps starting from the $10^{100}$th prime gap. How to check if they are likely lying?
I read an article that describes how to distinguish between real and fake sequences of coin tosses, with good reliability: we should check the longest run of heads (in real sequences of length $n$, ...
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$ C_{MRB}=\sum _{x=1}^{\infty } (-1)^x\left(e^{\frac{\log x}{x}}-1\right)$ Is its absolutely convergent arrangement prime number theorem related?
$ C_{MRB}=\sum _{x=1}^{\infty } (-1)^x\left(e^{\frac{\log x}{x}}-1\right)$ Is its absolutely convergent neighbor prime number theorem related?
$ C_{MRB}=\sum _{x=1}^{\infty } (-1)^x\left(e^{\frac{\...
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Difference of prime powers as linear combination of jumping champions
Reading the preprint Bounded gaps between
primes in short intervals by Ryan Alweiss and Sammy Luo (https://arxiv.org/abs/1707.05437), I came up with the following question:
Can the difference $\Delta$ ...
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Every twin prime average $x \gt 6$ is the sum of two twin prime averages (Code checked up to $x \leq 1,000,000$).
If $p,q$ are a pair of twin primes, then $x = \dfrac{p+ q}{2} = q-1 = p+1$ is their twin prime average.
Conjecture. Every twin prime average $x \gt 6$ is the sum of two smaller twin prime averages, $...
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Best algorithm to tell if an odd semi prime exists between a given pair of even semi primes.
Problem:
Let two even semi primes be $2q_1$ and $2q_2$:
you are to find if any $n$ exists such that $n$ is odd , $n$ is semi-prime and $2q_1 < n < 2q_2$. We don't need to know the $n$ , we just ...
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Existence of an integer $k$ whose square $k^2$ is the average of two consecutive primes with a gap of size $8$ between them.
It can be seen that the average of two consecutive primes with a gap of size $8$ between them is odd. If this average is $k^2$, then we may assume it ends with a digit $1, 5,$ or $9$. But $1$ and $9$ ...
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Asymptote of the prime gap
A famous conjecture in number theory states that $p_{n+1}-p_n=O((\log p_n)^2)$, where $p_n$ is the $n$-th prime number. However, it is well-know that $p_n \sim n\log n$. So why is the leading ...
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Valid Elementary Proof of the Bertrand-Chebyshev Theorem/Bertrand's Postulate? [closed]
$\textbf{Theorem}$ (Bertrand-Chebyshev theorem/Bertrand's postulate): For all integers $n\geq 2$, there exists an odd prime number $p\geq 3$ satisfying $n<p<2n$.
$\textit{Proof }$: For $n=2$, we ...
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Prime that can be expressed as the sum of other two distinct primes
Let us say that p1, p2, p3 are distinct primes such that p1+p2=p3. Now, since p1+p2>=2+3=5, so p3>=5 and p3 is therefore an odd prime. So either p1 or p2 must be even and the other must be odd. ...
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Which is the lowest or most accurate upper bound formula for the maximum gap in this modified sieve of Eratosthenes after a particular iteration?
As I discuss a similar question regarding the best upper bound for maximum gap after $n^{th}$ iteration of sieve of Eratosthenes here, I'm interested to know whether such a thing is possible for a ...
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Finite number of consecutive prime numbers
Suppose we have $n$ consecutive prime numbers $p_k$ with $n \in\mathbb{N}$ such that $p_k-p_{k-1}=m$ and $k=2..n$. Is it possible to find $m$ and $n$ in order to have a finite number of these ...
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Product of first $k$ primes compared to $p_{k+1}^2$
Let $p_i$ ($i \in \mathbb{N}^+$) be the $i^\text{th}$ prime. Is the product of the first $k$ primes always strictly greater than $p_{k+1}^2$ when $k > 3\text{?}$ (For instance, for $k=4,$ $2\cdot 3\...
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Which is the lowest or most accurate upper bound formula for the maximum gap in sieve of Eratosthenes after a particular iteration?
Consider a function $a(x)$ which gives the larget gap in the sieve of Eratosthenes after the $x^{th}$ iteration.
So,
$a(0) = 0$
$a(1) = 1$, After removing the multiples of 2.
$a(2) = 3$, After ...
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Coman's Last Conjecture stating that every prime $q \geq 11$ can be written as $3 \cdot (p_1-1) + p_2$, where both $p_1$ and $p_2$ are prime numbers.
Today I was taking a look at Coman's book entitled Conjectures on Primes and Fermat Pseudoprimes, many based on Smarandache function (starting from the end, as I often do) and his last conjecture, the ...
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Find all numbers n such that n+1, n+5, n+7, n+11, n+13, n+17, n+23 are all prime
Find all natural numbers $n$ such that $n+1$, $n+5$, $n+7$, $n+11$, $n+13$, $n+17$, $n+23$ are all prime.
So far I've made the following progress on this problem:
a) n must be even. Otherwise some of ...
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Is my proof for the fact that there are arbitrarily long strips of numbers between successive primes correct?
My attempt at a proof by induction:
Let two successive primes be $a$ and $b$.
$a - b = c$ gives the number of numbers between them.
Base case: The first five values of $c$ are $0,1,1,3$ and $1$.
There ...
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A question on primes larger than a bound in an arithmetic progression
Let $n \in \mathbb{Z}$ be a semi-prime with unknown factorization, $n = pq$, where $p, q \in \mathbb{P}$, the set of primes. Without loss of generality, let $p \lt q$.
Say we have done trial division ...
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How many are some special gaps between primes?
Gap primes are certainly even numbers. (starting from the consecutive primes $3$ and $5$)
Let $n$ be however an odd number and define by $n^2$ the range of the examined primes ( i.e. the range is ...
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Can the sum of $n$ consecutive primes be $n$ times a prime?
I recently saw a "coffin problem" asking to prove that the sum of two consecutive primes is not twice a prime. This got me wondering if three consecutive primes can sum to be three times a ...
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Find a prime before or after some $n$ consecutive composite numbers.
Goal
I'd like a method to find a prime before or after some $n$ consecutive composites.
One method would be to brute force every prime and check for $n$ consecutive composites before or after it. ...
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Are there infinitely many primes of the form [X]? We probably don't know.
Are there infinitely many primes of the form [expression]?
(We probably don't know. Sorry.)
This question appears pretty often, with any number of various expressions. The sad reality is that the ...
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Doubts in the $k -$tuple of primes $\mathcal{H}=\{0,2,6,12,20,26,30,32\}$
While studying some $k-$tuples of primes I found the following $\mathcal{H}=\{0,2,6,12,20,26,30,32\}$ which seems admissible to me.
using the primes package in Rstudio I get the following $p_1$ ...
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For every odd prime number $p \gt 3$ there exists another prime number $q \lt p$ such that $p - q = 2^n$ for some $n \geq 1$. Can you prove it? [closed]
Conjecture. Let $p$ be an odd prime number greater than $3$. Then there exists another odd prime number $q \lt p$ such that $p - q = 2^n$ for some positive exponent $n$.
Can we prove this or is it ...
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Confusion on the proof of Erdös's prime gap inequality
I am currently reading Erdös's paper "The difference of consecutive primes" published in 1940, in which he shows that there exists $\delta>0$ such that
$$
A=\liminf_{n\to\infty}{p_{n+1}-...
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Multiple of $15$ between quadruples of primes
Given a quadruple of primes (distinct from $5,7,11,13$) note that:
The quadruplet $(11,13,17,19)$ I can write it as $(15n-2^2, 15n-2^1, 15n+2^1, 15n+2^2)$, $n=1$.
For the quadruplet $(101, 103, 107, ...
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On Erdös's inequality regarding prime gaps
By prime number theorem, it is possible to shown that
$$
\liminf_{n\to\infty}{p_{n+1}-p_n\over\log p_n}\le1,
$$
and I learned from an exercise in Cojocaru & Murty that there exists $0<\delta<...
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Divisibility into primes quadruples
Reading about quadruples of primes, it is clear that these must have the form $(p,p+2,p+6,p+8)$, ($p>5$).
Consider the first three quadruples:
$\{11, 13, 17, 19\}, \{101, 103, 107, 109\}, \{191, ...
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Why does the ninth successive difference of primes appear to have two distinct groups?
Was exploring successive differences of primes and noticed an interesting pattern of the histogram of counts for the sixth and ninth difference. The ninth is more pronounced, code and image below.
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Full derivation inside of twin prime statement in terms of multiplicative arithmetic functions. How can the last formula be rearranged?
Let $(\cdot\mid\cdot) : \Bbb{N}\times\Bbb{N} \to \Bbb{Z}_2$ be the divisibility function which takes on the value $(x|y) = 1$ whenever $x$ divides $y$ and the value $(x|y) = 0$ whenever it does not ...
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No infinite arithmetic progression exists with prime numbers
I am trying to prove there is no infinite arithmetic progression involving only prime numbers. (In other words, I want to prove that if $a, b \in \mathbb{N}$, then there exists some $n$ such that $a + ...
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Does the following hold as a conjecture for maximum gaps between prime numbers? and can it be proved?
Even though I used matrix related mathjax on the backend, the frontend is intended to be just a regular table.
$$\begin{matrix}
a&X&X:explanation
\\1&1
\\2&3
\\3&5
\\4&(6)&(...
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Proof for strict inequality $\pi(ab) > \pi(a)\pi(b)$?
I asked about the very similar $\pi(ab)\geq \pi(a)\pi(b)$ a while ago, and this is indeed a proven result for $a,b\geq \sqrt{53}$.
Empirically, the stricter $\pi(ab)>\pi(a)\pi(b)$ looks true so ...
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Conjecture on ordering the first $p^2$ naturals by prime factor count
Let $\text{bump}(n)$ for $n\in\mathbb N$ be a function that increases the prime index of each prime factor of $n$ (with multiplicity) by $1$. I'll also use the notation $\text{bump}^k(n)$ to signify $\...
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What's so hard about contradiction proofs w.r.t. primes?
I've always felt like there should be some relatively straightforward proofs by contradiction establishing theorems about prime distribution. Ideally they would exist for things like twin prime ...
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Conjecture: there are more than $\pi(p)$ primes between consecutive prime squares
For any $p_i$ (being the $i$th prime), it seems empirically certain that
$$\pi(p_{i+1}^2)-\pi(p_i^2)>i.$$
Equivalently: there are more than $\pi(p)$ primes between any $p^2$ and the next higher ...
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The sets $\mathcal{F}_d(m,n,k) = \Big\{ x \in (m, n) : x^2 = k^2 \pmod d\Big\}$ seem to have relationships with each other. What is their structure?
For $m,n,k,d\in \Bbb{Z}, m\leq n$, define for interval of integers $(m,n)$:
$$
\mathcal{F}_d(m,n,k) := \Big\{ x \in (m,n): x^2 = k^2 \pmod d\Big\}
$$
Then $\mathcal{F}_d(m,n, i)\cdot\mathcal{F}_d(m', ...
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