Mathematics

Questions tagged [prime-gaps]

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

446 questions
204
votes
0 answers

Sorting of prime gaps

Let $g_i$ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we rearrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$ , if the gaps are arranged from smallest to largest, we have a new sequence $(\hat{g}_{n,i})_{i=1}^n.$ For example, for…
daniel
  • 10,601
126
votes
0 answers

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
  • 5,197
50
votes
2 answers

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.
  • 11,217
43
votes
0 answers

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 modulo $3$ 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…
user411780
37
votes
3 answers

Are there infinitely many primes of the form [X]? We probably don't know.

Are there infinitely many primes of the form [expression]? (We probably don't know. Sorry.) This question appears pretty often, with any number of various expressions. The sad reality is that the answer, more likely than not, is that we don't know.…
Eric Snyder
  • 3,177
35
votes
7 answers

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
  • 6,334
  • 1
  • 20
  • 37
33
votes
2 answers

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 prime…
Basanta Pahari
  • 2,774
32
votes
3 answers

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
  • 6,198
22
votes
2 answers

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
  • 4,273
20
votes
3 answers

On the regularity of the alternating sum of prime numbers

Let's define $(p_n)_{n\in \mathbb N}$ the ordered list of prime numbers ($p_0=2$, $p_1=3$, $p_2=5$...). I am interested in the following sum: $$S_n:=\sum_{k=1}^n (-1)^kp_k$$ Since the sequence $(S_n)$ is related to the gaps between prime numbers, I…
E. Joseph
  • 15,149
  • 8
  • 45
  • 79
18
votes
2 answers

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
  • 18,413
14
votes
2 answers

What do we know about the first occurrences of prime gaps?

Are there any conjectures from which we can infer something about the first occurrences of prime gaps length $n$ and their distribution? I've made an interesting graph of these values to make this problem easier, with gauss's approximate of their…
jg mr chapb
  • 1,660
13
votes
3 answers

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
  • 6,167
13
votes
1 answer

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
  • 87,158
12
votes
2 answers

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<q<r$ are consecutive odd primes. Now $ x^3 + y^3 =LQ$ where $…
pauldjackson
  • 141
1
2 3
…
29 30