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

# Questions tagged [prime-gaps]

358 questions

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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?

Peđa

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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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