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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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Expected size of the maximal prime gap below x under Hardy-Littlewood conjecture

Denoting by $ \pi_{n}(x) $ the number of prime gaps of size $ n $ below $ x $ for even $ n $, Hardy-Littlewood conjecture predicts that $ \pi_{n}(x)\sim C_{n}\frac{x}{\log^{2}x} $ with $ C_{n}=...
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Twin prime conjecture and gaps between primes

This is just a thought: if gaps between prime numbers can be arbitrarily large then it should be possible to find infinitely many gaps, such that the product $m=\prod_{n=1}^{N}Pn<P_{N+1}^{2}$, ...
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Fortune's conjecture solved for limited cases?

I am not a mathematician, but while doing other work, I came across the Fortune conjecture. According to Wikipedia and other research, it seems that it has not yet been proven. I thought about the ...
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Finding the n-th prime number [closed]

We want to uniquely map hash values to prime numbers. One way to achieve this is storing the first $l$ prime numbers into an ordered list $L$ with size $|L| =l$. When the hash value $h$ calculated, ...
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consecutive prime gaps and explicit bounds

I am aware of the theorem that $p_{n+1} - p_n \leq n^{0.535}$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit for all sufficiently large number to ...
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Do we know of more than one occurance where Prime Gap=1?

A prime gap $g_n$ is the difference between two prime numbers, and as we know, the first two primes are 2 and 3, thus their prime gap is 1; $$ g_n = p_{n+1}-p_n=\big\{ n=1 \big\}=3 -2=1. $$ But have ...
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Asymptoptic density of prime k-tuples

The first Hardy-Littlewood conjecture concerns the asymptotic density of prime k-tuples. Assuming that the tuple $\{p, p+2m_1, \ldots, p+2m_k\}$, where all the elements are primes and $m_i$ for all $i$...
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How many prime numbers in a given interval?

Is there any algorithm or a technique to calculate how many prime numbers lie in a given closed interval [a1, an], knowing the values of a1 and an, with a1,an ∈ ℕ? Example: [2, 10] --> 4 prime ...
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Is there a pair of prime $(p,q)$ such that $p >q>7$ and $2q-p=3$ and $p, q$ are successive?

I'm failed to find at a least one pair of primes$(p,q)$ $p >q>7$ and $2q-p=3$ with $p, q$ are successive , I think tha's impossible because we do not know more about Gaps between prime and if ...
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GCD of $p+q$ for all pairs of primes $p,q$ of the form $p-q=12n+2x$

I came across this result, and i'm having trouble explaining it. I use it as an argument in a proof, so i need to explain this behavior, in the shortest possible way, and most importantly prove that ...
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The definition of a prime constellation

On Mathworld it is first stated that a prime constellation is a sequence of $k$ prime numbers, for which the gap between the last and the first minimizes. But later they show a table with prime ...
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Is it possible to find EXACTLY $101$ consecutive composite numbers

Here is a similar question that asks for $101$ numbers none of them are prime and it is well known that $101!+1,101!+2,101!+3,\cdots,101!+101$ are those numbers. I am interested to know, how we can ...
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distributions of prime numbers - theorem of Chebyshev

I was thinking: let $a\in(0,1]$, $1<b$ be given, and let $c$ be given as a positive integer. Can we find $N$ with the property that if $N$ is large enough, than the interval $(N^a,(b N)^a)$ always ...
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Prime gaps and gaps between successive critical zeros of zeta

Assuming RH, the sequence of critical zeros of the Riemann zeta function can be viewed as the Fourier transform of the sequence of primes. From a physicist point of view, the average gap between the ...
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Quality of prime seeking methods

I am working on prime numbers with emphasis of prime search heuristics, and found the probabilistic methods for primes seeking, I am looking for a review of those methods quality in terms of machine ...
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Is there any algorithm to identify the smallest example of gap size $n-1$ between consecutive prime numbers

In contemplating Goldbach's conjecture, I became interested in gaps between successive primes. If $n<a<b<2n$ and the range $a$ to $b$ is a primeless gap, then one could ignore any primes in ...
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Brun’s constant and irrational numbers

It is trivial that if there are finitely many twin primes then Brun’s constant must be a rational number. And GammaTester (below) has offered an example of an infinite series that converges to a ...
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Generalization of Opperman's Conjecture

Does this conjecture have a name? What about a counterexample?: $$ \forall n,k \in \mathbb{N}, k \gt 1, \exists d \in (kn-n,kn] \text{ s.t. } d \perp n! $$ An equivalent statement is this: Take a ...
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Lower bound for $p_{n^3}-p_{(n-1)^3}$?

The difference between two primes is at least $2$ so $p_{n^3}-p_{(n-1)^3} \geq 6n^2$. Is there any known sharper bound?
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Sequences of consecutive numbers

If you pick a natural number n, for example 3, and you take the string from 1,..,n and shift it via 2,...,n+1; 3, ... n+2 etc, then it looks like you will first find some strings where you can do the ...
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On some conjectures about an inequality involving different arithmetic functions related to prime numbers

I've written the following puzzle about prime numbers. This exercise is thus a curiosity/miscellany about the distribution of prime number $p_k$, that I wondered when I was playing with different ...
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gaps between square roots of primes

It is well known that the difference $p_{n+1}-p_n$ can be arbitrarily large. What about $\sqrt{p_{n+1}}-\sqrt{p_n}$, or in general, $p_{n+1}^t-p_n^t$ for $t<1$? Has this problem been investigated? ...
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a prime is a midpoint of two primes

Take three consecutive primes $p_1,p_2,p_3$: What is the opinion on the question that $p_3-p_2=p_2-p_1$ occurs endlessly and is harder to find as the primes increase? Has anyone examined the ...
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Difference between prime numbers

Let $p_n$ denote the $n$th prime number and $\phi:\mathbb{N}\to\mathbb{N}$ an increasing function, with $\phi(n)>n$. Is it true that $p_{\phi(n+1)}-p_{\phi(n)}<p_{\phi(n)}$ for infinitely many $...
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On miscellaneous questions about perfect numbers I

Let $\varphi(m)$ the Euler's totient function and $\sigma(m)$ the sum of divisors function. If $n$ is an odd perfect number then $n$ satisfies $$\varphi(n)=\varphi(\sigma(n)).\tag{1}$$ The sequence ...
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Show that for any open subset of $\Bbb R$, there is a fraction with prime terms that belongs to it [duplicate]

Be $\Bbb P\Bbb Q$ the set of all fractions $f_{m,n}=\frac{p_m}{p_n}$ whose numerador and denominator are both prime numbers. i) Show that for any open set $A\subset \Bbb R^+$, there is at least one $...
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Ratios of prime gaps $(p_{n+1}-p_n)/(p_{2n+1}-p_{2n})$

This is a question about prime gaps $g_n = p_{n+1}-p_n$ that started with a look at the average of ratios $$r_n=\frac{p_{n+1}-p_n}{p_{2n+1}-p_{2n}}$$ and of the inverse, $$ s_n=\frac{p_{2n+1}-p_{2n}}{...
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Can we prove that $p_N + 3 \leq 2 p_{N-1}$ for sufficiently large $N$?

Question in the title. It intuitively seems absurd that $p_N - p_{N-1} \gt p_{N-1} - 3 = $ the largest gap formable from all $p_i = $ odd primes $3, \dots, p_{N-1}$. Was wondering how difficult the ...
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Let $p_{k} $be the $k^{\text{th}}$ prime number. Show that there are infinitely many $k$ such that $p_{k+1} − p_{k} > 2$.

Let $p_{k} $be the $k^{\text{th}}$ prime number. Show that there are infinitely many $k$ such that $p_{k+1} − p_{k} > 2$. I was thinking about Dirichlet's theorem as i don't know to prove its ...
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Bounding Maximal gaps with Ramanujan primes

Gaps and Maximal Gaps We define terms used in this article. A prime gap as $g_n := p_{n+1} - p_n$, and we define $g_n$ as a maximal gap, if $g_i < g_n$ for all $i < n$. Define $M_{m,g}$ to be ...
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Question about prime gap records

Let $g_n$ be the $n$ th prime gap. Let $f_n = max( g_1,g_2,...,g_n)$ Now take the sequence $f_n$ and remove the duplicates. Also sort from small to large. Then $f_n $ gives the sequence $$ 1,2,4,...
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A remarkable(?) condition on sequences of natural numbers

There is a remarkable condition on increasing sequences of natural numbers $(a_n)_n$: $$\bigg\lfloor\frac{a_n^2}{a_{n+1}}\bigg\rfloor=2a_n-a_{n+1}\tag 1$$ that - when $n$ is big enough - seems to hold ...
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The distribution of powers of primes [closed]

How often do we see two or more powers of primes between two consecutive primes $p_k$ and $p_{k+1}$? One example is $p_4=7$ and $p_5=11$; we have $$ 7 < 2^3 < 3^2 < 11. $$ Are there any other ...
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A conjecture concerning primes and perhaps prime gaps

I found out this conjecture which is tested for $m\leq 100$: For all natural numbers $n>\!30$, for all $\alpha\in\mathbb Z$ and all $\mu=0,\pm 1,\pm 2$: $m\equiv p_n^2+\mu p_n+\alpha\!\...
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I have a proof concerning prime numbers. Should I publish my result?

I can easily and shortly prove that given $ε$, computable using all primes less than $N$, there will be at least one prime number between $n$ and $(1+ε)n$, where $n > N$. It proves Bertrand's ...
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Upper bound for the $n$th record gap between primes in an arithmetic progression

(Following question 2269073. See also Mathoverflow question 289974.) Let $q$ and $r$ be coprime integers, $1\le r < q$, and consider the arithmetic progression $$ r, \ r+q, \ r+2q, \ r+3q, \ldots \...
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Prime counts with maximal prime spacing.

I think this question is related to Maier's theorem but I am unsure. Notation: $\pi(x)$ is the prime-counting function up to $x$. $g_k := p_{k+1} - p_k$. Define $M_n$ to be the $n$th maximal gap ...
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Something strange about prime gaps and $p-q =999999999999182774421592902$

Coming across the post First 100s place without a prime, I went to the informative link "First occurrence prime gaps" suggested by Jack D'Aurizio. The main list of $999$ prime $p$ covers the smallest ...
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2017 was prime. 2018=2 x 1009 is double a prime. What does the future portend?

What is the likelihood in the future that a year will be prime or double a prime? Are these years rare? Dependent on the prime gaps? What's the best proven frequency? Happy New Year! :-)
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A conjecture regarding prime numbers

For $n,m \geq 3$, define $ P_n = \{ p : p$ is a prime such that $ p\leq n$ and $ p \nmid n \}$ . For example : $P_3= \{ 2 \}$ $P_4= \{ 3 \}$ $P_5= \{ 2, 3 \}$, $P_6= \{ 5 \}$ and so on. Claim: $...
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Can pairs of consecutive primes with large merit be found efficiently?

The merit of a prime gap between the consecutive primes $p_{n}$ and $p_{n+1}$ is defines as $$m:=\frac{p_{n+1}-p_n}{\ln p_n}$$ How can I find efficiently, lets say, a pair of consecutive random ...
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Need Help With Proof Regarding Prime Numbers

some friends and I are stuck on a proof regarding primes. It goes as follows: Let $p_1, p_2$ be primes with $p_1 < p_2$. Show that there is a $n \in > \mathbb{N}$ such that $p_1 + n \cdot (...
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Is every prime gap bounded from above by $C\cdot \ln^2(p_n)$?

It is known that the merit of a prime gap can be arbitary large : The merit is defined by $$\frac{p_{n+1}-p_n}{\ln(p_n)}$$ , where $p_n$ and $p_{n+1}$ are consecutive primes. Does a constant $C$ ...
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$\pi(x + x^B) - \pi(x) \sim \frac{x^B}{\log x}$

Does anyone have any reference where I can explore the proof or a outline of this theorem by Hoheisel (https://en.wikipedia.org/wiki/Prime_gap) and the further work of others that tried to get to a ...
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How to improve Chebyshev bound on the prime counting inequality?

So, I've understood the proof of A*x/logx < pi(x) < B*x/logx for (A,B) = (0.5,2), but how can I make this difference smaller? Does any one know the methods ...
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Prime Gap number runs

Take a random base 10 number of 32 digits. The odds of a run of 4 or more identical digits is about 1 in 40. At First occurrence prime gaps by Dr. Thomas R. Nicely, you can see the minimal primes ...
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Prime numbers in factorizations of natural numbers

I would like to know if the following affirmation is true or not: If M is a set of consecutive natural numbers there is a prime number in the factorization of one of M's elements that doesn't divide ...
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For $N,M\in \mathbb{N}\gt\gt 1$ can we have $N$ consecutive natural numbers of which $M$ are prime?

We know that for an arbitrarily large $N \in \mathbb{N}$ we can have $N$ consecutive natural numbers of which none is prime. A construction that verifies this is the set $B(n)=\{n!+2, n!+3,\cdots,n!+n\...
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Two kinds of prime gaps

$$1361 - 1327 = 34$$ Between these two prime numbers there are no others. No prime gaps this big come before this one; i.e. this one is "maximal". The largest prime not exceeding the square roots of ...
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Proportion of Prime Gaps Divisible by n (expected)?

We seem to know that the proportion of prime gaps divisible by 6 (as the gaps tend toward infinity) ≈ 1/2. Are we able to find the expected proportion of prime gaps divisible by 4, or 8, or 10?