Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [prime-gaps]

The difference of two prime consecutive prime numbers is the prime gap. $g_i := p_{i+1} - p_i$.

1
vote
1answer
27 views

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 $i$...
5
votes
2answers
56 views

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 ...
1
vote
4answers
47 views

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 ...
2
votes
1answer
43 views

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 ...
0
votes
2answers
39 views

The definition of a prime constellation

On Mathworld http://mathworld.wolfram.com/PrimeConstellation.html 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 ...
6
votes
1answer
171 views

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 ...
0
votes
0answers
46 views

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 ...
1
vote
0answers
43 views

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 ...
0
votes
0answers
21 views

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 ...
3
votes
0answers
46 views

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 ...
0
votes
0answers
39 views

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 ...
2
votes
1answer
118 views

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 ...
2
votes
1answer
40 views

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?
0
votes
1answer
41 views

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 ...
3
votes
1answer
62 views

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 ...
4
votes
0answers
50 views

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? ...
1
vote
0answers
48 views

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 ...
0
votes
0answers
34 views

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 $...
2
votes
0answers
28 views

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 ...
2
votes
0answers
26 views

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 $...
9
votes
0answers
165 views

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}}{...
1
vote
2answers
57 views

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 ...
1
vote
3answers
78 views

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 ...
2
votes
0answers
124 views

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 ...
1
vote
3answers
120 views

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,...
6
votes
1answer
104 views

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 ...
1
vote
1answer
77 views

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 ...
2
votes
0answers
71 views

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\!\...
-2
votes
1answer
165 views

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 ...
0
votes
0answers
125 views

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 \...
1
vote
1answer
77 views

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 ...
6
votes
1answer
160 views

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 ...
4
votes
0answers
121 views

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! :-)
29
votes
2answers
805 views

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: $...
2
votes
1answer
68 views

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 ...
1
vote
1answer
40 views

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 (...
0
votes
0answers
37 views

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 $C$ ...
0
votes
0answers
32 views

$\pi(x + x^B) - \pi(x) \sim \frac{x^B}{\log x}$

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 ...
1
vote
0answers
74 views

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 ...
0
votes
0answers
57 views

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 ...
1
vote
1answer
28 views

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 ...
3
votes
3answers
56 views

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\...
1
vote
2answers
114 views

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 ...
1
vote
0answers
24 views

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?
0
votes
0answers
42 views

The distribution of the gaps of prime numbers: What is known?

As far as I know the expected distance between two prime-numbers is about $\log(N)$. Is there more about this distribution know, e.g. the variance of this distance (depending on $N$)? I know the ...
0
votes
1answer
30 views

How would you prove that this sum involving prime gaps has a limit?

Let $p_k$ = the $k$th prime. $$ \varphi(n) = \sum_{k=1}^{n-1} e^{i 2 \pi \frac{p_{k+1} - p_k}{p_n}} $$ seems to approach a constant point in $\Bbb{C}$ as $n \to \infty$. How can I prove it though?
1
vote
1answer
54 views

The first number larger than $N$ in prime gap sequence

Is there any way to estimate the first number in the prime gap sequence which is greater than a given number $N$? For example, for $N=3$, $g_4=11-7=4$ is the first one larger than $N$. Thanks in ...
1
vote
1answer
40 views

What is the density of consecutive primes which have gaps too small or too large from the average ?

Let $p_i$ denote the $i$-th prime number and $g_i$ the $i$-th gap $g_i=p_{i+1}-p_{i}$. By the PNT we know that the average gap is $g_i \approx \ln p_i$. A gap $g_i$ is considered "too small" iff $\...
0
votes
0answers
32 views

For a given integers $x,n$, counting the number of integers $v$ where $x < v \le x+n$ and gcd$(\frac{n}{4}\#,v)=1$

If $n < 188$, is it true for all integers $x,n$ that there exist at least $4$ integers $v$ such that $x < v \le x+n$ and gcd$(v,\frac{n}{4}\#)=1$? I believe that the answer is yes. Here's my ...
5
votes
7answers
472 views

Is there a prime $p$ whose successor is greater than $2p$?

Toying with Goldbach's Conjecture, I encountered myself in a situation where the following question arose. Is there a prime $p$ whose successor is greater than $2p$? You see. If the answer to this ...