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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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 $C$ exist such that for every prime $p_n$ the inequality…
Peter
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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 generating a given gap up to 1998. Things get…
Ed Pegg
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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?
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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 argument: Lemma 1: For any integers $x,w,n$, there are at…
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Is $(\ln p)^\alpha$ a limit for gaps proceeding $p$ for some $\alpha$?

By computational experiments I found that there are no primes $p_n$ with $7(\ln p_n)^\alpha$, if $\alpha\ge 1.8932$. Are there primes $p_n>7$ such that $p_{n+1}-p_n>(\ln p_n)^2$?
Lehs
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Gaps between numbers of the form $pq$, part 2

I asked this question: Gaps between numbers of the form $pq$ , and received a very satisfactory answer. Now I'm curious about a related one. We know that we have have arbitrarily long stretches of numbers with no primes, so there is no upper bound…
G Tony Jacobs
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Is this a valid partial refinement of Ingham's upper bound for prime gaps?

This is a follow-up question from this one that was kindly answered by @JordanPayette. The corrections were applied for this solution. Let $p_n$ denote the $n^{th}$ prime number. Ingham showed that: $$p_{n+1} - p_n \lt K p_n^{\frac{5}{8}}$$ where…
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Is it possible to prove that this function $F(X)$ evaluates to $1$ for any positive integer $X$?

Let $P_{1} = 2$, $P_{2} = 3$, $P_{3} = 5$, $P_{4} = 7$, ... (the list of all primes). Consider the function $F(x)$. It evaluates to $1$ if there exists some positive integer $N$ ( $N \ge 4$ ) such that $$\left\{ \begin{array}{l} {P_{N - 2}} -…
lyrically wicked
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Longest prime $k$-tuplet such that each prime $p$, $\gcd(p-1, 420) > 2$?

What is the longest possible length $k$ of a prime $k$-tuplet (smallest groupings of $k$ primes) such that each prime $p$ in the $k$-tuplet,$\gcd(p-1, 420) > 2$. In other words, each prime $p$ in this $k$-tuplet is either $1$ $\pmod 3$, $1$ $\pmod…
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Calculate the max gap size out of the Firoozbakht's conjecture

You can find the following image at the Wikipedia page for the Firoozbakht's conjecture. The conjecture states that $p_n^{1/n}$ is a strictly decreasing function. How can one calculate the gap size out of the conjecture? Or how is the Firoozbakht's…
JoCa
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Does there exist an explicit formula that expresses $p_{n+1}-p_{n}$ in terms of $p_n$, the $n$th prime number?

Given the $n$th prime number, is it possible to express explicitly a function $\phi:P \mapsto \mathbb{N}$, where $P$ is the set of all prime numbers, such that $\phi(p_n)$ gives the prime gap after $p_n$?
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Maximum number of primes $>5$ in a sequence separated by the same gap

Start with prime $p$ then find the maximum sequence of primes $p+gap, p+2\cdot gap, p+3\cdot gap$ etc. Looking at the first gaps I found: \begin{array}{|c|c|} \hline 2 & 1 \\ \hline 4 & 2 \\ \hline 6 & 5 \\ \hline 8 & 2 \\ \hline 10 & 2 \\ …
pietfermat
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How can I show that: $\pi(n)\simeq\dfrac{n}{10}+\dfrac{5}{\sqrt{n}}+\dfrac{\log{n}+n}{\log_2{n}}$, where $\pi(n)$ is the prime counting function

Progress: I observed this result: $\pi(n)\simeq\Bigg\lfloor\dfrac{n}{10}+\dfrac{5}{\sqrt{n}}+\dfrac{\log{n}+n}{\log_2{n}}\Bigg\rfloor$ while studying several inequalities and relations concerning prime gaps and PNT and it tends to be extremely…
user18724
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Does Zhang's work imply that the next greatest prime number is necessarily less than $2^{57,885,161}-1+ 70000000$?

Actually the greatest known prime is $2^{57,885,161}-1$. Zhang's work proved that the interval between pairs of prime numbers is limited to $70,000,000$. Does it imply that the next greatest prime number is necessarily less than $2^{57,885,161}-1 +…
MFornari
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Enhancements of Bertrand's postulate?

Is there a smallest real number $a$ such that there exist a natural number $N$ so that: $n>N\implies p_{n+1}\leq a\cdot p_n$? I believe it can be proved that $n>7\implies p_{n+1}\leq \sqrt 2\cdot p_n$.
Lehs
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