Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [prime-gaps]

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

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Sorting of prime gaps

Let $g_i $ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we re-arrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence $(\hat{g}_{n,i})_{i=1}^n.$ For example, for $n =…
daniel
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A question about divisibility of sum of two consecutive primes

I was curious about the sum of two consecutive primes and after proving that the sum for the odd primes always has at least 3 prime divisors, I came up with this question: Find the least natural number $k$ so that there will be only a finite number…
CODE
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Are there arbitrarily large gaps between consecutive primes?

I made a program to find out the number of primes within a certain range, for example between $1$ and $10000$ I found $1229$ primes, I then increased my range to $20000$ and then I found $2262$ primes, after doing it for $1$ to $30000$, I found…
Nikunj
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A trivial proof of Bertrand's postulate

Write the integers from any $n$ through $0$ descending in a column, where $n \geq 2$, and begin a second column with the value $2n$. For each entry after that, if the two numbers on that line share a factor, copy the the entry unchanged, but if…
Trevor
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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: $P_n \neq P_m$ for $m\neq n$. While working on…
Basanta Raj Pahari
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Is every finite list of integers coprime to $n$ congruent $\pmod n$ to a list of consecutive primes?

For example the list $(2, 1, 2, 1)$ is congruent $\pmod 3$ to the consecutive primes $(5, 7, 11, 13)$. But how about the list $(1,1,1,1,1,1,1,1,2,3,4,3,2,3,1) \mod 5$? More generally, we are given some integer $n \geq 2$ and a finite list of…
Ahmad
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The significance and acceptance of Helfgott’s proof of the weak Goldbach Conjecture

Recently I was browsing math Wikipedia, and found that Harald Helfgott announced the complete proof of the weak Goldbach Conjecture in 2013, a proof which has been accepted widely by the math community, but according to Wikipedia hasn’t been…
D.R.
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The Gaussian moat problem and its extension to other rings in $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$

One of my favourite open problems in number theory, an area in which I enjoy only as a hobbyist, is the Gaussian moat problem, namely "Is it possible to walk to infinity in $\mathbb{C}$, taking steps of bounded length, using the Gaussian primes as…
Bennett Gardiner
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Geometric mean of prime gaps?

The arithmetic mean of prime gaps around $x$ is $\ln(x)$. What is the geometric mean of prime gaps around $x$ ? Does that strongly depend on the conjectures about the smallest and largest gap such as Cramer's conjecture or the twin prime conjecture…
mick
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On the difference between consecutive primes

Let $(p_n)$ be the sequence of prime numbers and $g_n = p_{n+1} - p_n$ Question: Is it known that $g_n \le n$? Remark: it's known that $g_n < p_n^{\theta}$ with $\theta = 0.525$ for $n$ sufficiently large (see here), and that $p_n < n(\ln n +…
Sebastien Palcoux
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is 1001 the only sum of two positive cubes that is the product of three consecutive odd primes?

That is $\ 10^3+1^3=7.11.13$. I could find no other examples. So I am looking to see if there are any more solutions to $ x^3+y^3=p.q.r$, where $ x, y$ are positive integers and $ p
pauldjackson
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Did Landau prove that there is a prime on $\bigl(x,\frac65x\bigr)$?

Was Landau the first to prove that there is a prime on $\bigl(x,\frac65x\bigr)$? In his Handbuch $\!^1$ discussing the limit $$\lim_{n\to\infty} \bigl(\pi\bigl((1+\epsilon)x\bigr)-\pi(x)\bigr)=\infty $$ he seems to say that in the next chapter he…
daniel
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The largest possible prime gap?

What is the largest possible prime gap if we observe only 1000-digits numbers? There are few conjectures about this question but is there something that we can say and be absolutely sure of it?
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Prime chains with large gaps

It is well known that the gap between consecutive primes is unbounded. Is this still true for a chain of consecutive primes ? More Formally : Is the following statement true for all natural numbers m and n ? There are m consecutive primes…
Peter
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A way to solve the gap between prime numbers

Some time ago I found this sum of prime numbers converge(*) $$\sum_{k=1}^\infty \frac{p(k+1)-2p(k+2)+p(k+3)}{p(k)-p(k+1)+p(k+2)}\ \approx \frac{5}{7}\zeta(3)^{-2} $$ where: $\zeta(s)$ is the Riemann zeta function $p(n)$ is the $n^{th}$ prime…
Patrick Danzi
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