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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Proof of the fact that $\Delta (\Bbb{P}_{\geq 3} \sqcup \{\pm 1\}) = 2\Bbb{Z} =$ close to the open problem of $\Delta\Bbb{P}_{\geq 3}=2\Bbb{Z}$
Define $R_i = \Bbb{Z}/p_i\#$ where $p_i$ is the $i$th prime number and $p_i\# = p_i p_{i-1} \cdots p_1$. Let the inverse-system maps be given by all possible compositions of $f_{i,i+1}:R_{i+1} \...
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Is it possible to prove that iterating this hailstone-resembling algorithm will always end in $1$?
I've been messing with a rule I came up with recently, that resembles the Collatz function in formulation but shares very few practical similarities. Using $p_n$ to represent the $n^{\text{th}}$ prime ...
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Reasoning about reduced residue systems as a generalization from prime gaps
I have been thinking a lot about prime gaps.
It seemed to me that it is much easier to reason about reduced residue systems which can then be used to reason back to prime gaps.
Below is an example of ...
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Smoothed and truncated Von Mangoldt function
The Von Mangoldt function $\Lambda : \mathbb{N} \to \mathbb{R}$ is defined as
$$\Lambda (n)={\begin{cases}\log p&{\text{if }}n=p^{k}{\text{ for some prime }}p{\text{ and integer }}k\geq 1,\\0&{...
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Is De Polignac's conjecture equivalent with the the statement that for any positive even $n$, there are infinitely prime pairs with difference $n$?
Suppose $n$ is a positive even integer. I wonder if the following two statements are equivalent:
(1) There are infinitely many pairs of consecutive primes with difference $n$.
(2) There are infinitely ...
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Does a set with no divisibility pairs necessarily have arbitrarily large gaps? [duplicate]
The set of prime numbers has the following properties:
No element is divisible by any other element.
We can find arbitrarily large gaps between consecutive elements.
Does (1) imply (2) for arbitrary ...
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How do we measure the "holes" (zeros) of the set of $\Bbb{Z}$-linear combinations of $f_i : G \to G$ where $G$ is an abelian group?
Question:
Let $G$ be a normed abelian group. Namely the triangle inequality holds $|g + h|\leq |g| + |h|$. An example would be $G = \Bbb{Z}$ together with $|\cdot| =$ absolute value.
Now suppose ...
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Non-trivial prime gaps
A simple proof that there are prime gaps of size at least $n+1$ for every $n$ can be seen in the first answer to this question. I consider prime gaps of the form $n!, n!+1, \ldots, n!+n$ of length at ...
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Minimum $k$ for which every positive integer of the interval $(kn,(k+1)n)$ is composite
I am looking for references containing results on the minimum $k$ for which every positive integer of the interval $(kn,(k+1)n)$ is composite.
If we denote as $k(n)$ this minimum $k$ for some $n$, $k$ ...
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The Growth of Primes
I've been experimenting with the function $π(x)$ such that $π(x)$ counts the number of primes from $1$ to $x$.
I found that after $10$, $π(x^2) - π(\lfloor \frac{x^2}{2} \rfloor)$ is always larger ...
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Mirror symmetry in distances of remaining numbers of Eratosthenes-sieve
Trying my first steps in Python I found an interesting phenomenon when I tried Eratosthenes sieve. Especially I looked for the distances between the remaining numbers.
For example, if you already have ...
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Are there contiguous sequences of prime numbers of length $k$ which are convex (similarly, concave) for every $k\in\mathbb{N}?$
Does the sequence of prime numbers contain contiguous subsequences of length $k$ which are strictly convex (similarly, strictly concave), for every $k\in\mathbb{N}?$
For example,
$$ 17, 19, 23, 29 $$
...
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Consecutive composite numbers using the Chinese Remainder Theorem. [duplicate]
Consecutive Composite Numbers
Define a list of the first $n$ prime numbers $p_1, p_2, \ldots, p_n$.
Create a set of $n$ congruences
\begin{align*}
x + 1 &\equiv 0 \pmod{p_1} \\
x + 2 &\equiv 0 ...
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question on estimator for $\frac{\pi(n)}{n}$ and $\frac{\pi_2(n)}{\pi(n)}$
$\pi(n)$ and $\pi_2(n)$ represent the count of primes and count of twin primes $\leq n$ respectively.
Suppose we want to estimate $\frac{\pi(n)}{n}$. One way which obviously is not error-free is to ...
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broader meaning of twin prime constant?
It appears that the twin prime constant has meaning outside of the strict twin prime constant. I attempted to keep this post as short as possible.
Definitions:
Let $p,q$ represent primes and let $n$ ...
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A weaker version of the second Hardy-Littlewood conjecture.
Let $n$ be a positive integer and let $f(n)$ be the counting function for non-composite numbers.
So
$$f(0)=0,f(1)=1,f(2)=2,f(3)=3,f(4)=3,f(5)=4$$
Now my mentor noticed as a kid that apparantly
$$f(x+y)...
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What series approaches to $\log(\log(\log(n)))$?
Background: Harmonic series approaches to $\lim_{n->\infty}\log(n)$. This gives the Euler–Mascheroni constant. In addition, the harmonic series summed only over the primes approaches to $\lim_{n-&...
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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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Conjecture. If $n \in \Bbb{N}\setminus 1$ is not a twin prime average, then $n^2 - 1$ is not square-free.
From the evidence:
...
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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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