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

**11**

votes

**1**answer

### On the regularity of the alterning 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,…

E. Joseph

- 14,453
- 8
- 36
- 67

**10**

votes

**5**answers

### does a number that contains all primes less than it exist?

I want a number that has a prime factorization that contains all prime numbers less than that number (besides $2$), anyone with an answer please show a proof.
I have made a little progress, if this number exists, then it is one less than a prime…

spydragon

- 159
- 13

**10**

votes

**1**answer

### Asymptotic Distribution of Prime Gaps in Residue Classes

Define $\pi_{n,a}(x)$ as the number of primes $p$ less than $x$ such that $p\equiv a\bmod n$ for coprime $n,a$. This function can be asymptotically approximated by $$\pi_{n,a}(x)=\frac{\operatorname{Li}(x)}{\varphi(n)}$$
This allows for the…

Romain S

- 2,725
- 13
- 37

**10**

votes

**0**answers

### For any $x\in \mathbb{N}$ does there exist $m\in \mathbb{N}$ such that $2x+1+2m, 2x+1+4m$ are both prime?

Could someone please give me a proof (or counter example) for this (I believe it is true):
For any $x$ (Whole Number) there exists some $m$ (Also Whole) such that $2x+1+2m$ and $2x+1+4m$ are both prime.
An equivalent is for any odd $n$ there…

mtheorylord

- 4,432
- 12
- 38

**10**

votes

**2**answers

### Are there two consecutive gaps of size $4$ between prime numbers?

Are there consecutive gaps(difference) of 4 between prime numbers?
Seeing at the first few gaps
$1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6,.....$.
(for example 6 is repeated consecutively at some places).
Are there consecutive…

user132079

**10**

votes

**1**answer

### 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

- 1,462
- 10
- 37

**9**

votes

**4**answers

### Gaps between primes

I recently watched a video about the recent breakthrough involving the gaps between primes. I have an idea that I'm sure is wrong, but I don't know why.
If you take the product of all prime numbers up to a certain number and call it x, won't x-1…

ThisIsAQuestion

- 205
- 1
- 7

**9**

votes

**1**answer

### Improving Zhang's prime gap

I am referring to Zhang's paper.
Since the set $\cal{H}$ is a subset of $[3.5\times 10^6, 7\times 10^7]$, shouldn't the prime gap he obtained be less than
$ 7\times 10^7 - 3.5\times 10^6$ rather than $7\times 10^7$, as stated in his paper? Or am I…

TCL

- 13,592
- 5
- 26
- 73

**9**

votes

**3**answers

### A weaker version of the Andrica's conjecture

Andrica's conjecture states that: For every pair of consecutive prime numbers $p_{k}$ and $p_{k+1}$, we have :
$$\sqrt{p_{k+1}}-\sqrt{p_{k}}<1\quad\quad \color{#2d0}{\text{(1.)}}$$
I know that is statement is not yet proved. But I am asking on a…

Safwane

- 3,713
- 18
- 40

**9**

votes

**1**answer

### Goldbach's conjecture with negative primes

Is the Goldbach conjecture any easier if we allow primes to be negative as well? That is, every even integer is the sum or difference of two primes. The twin prime conjecture talks about the occurrence of a certain kind of prime gap, but I don't…

Mario Carneiro

- 26,030
- 4
- 59
- 119

**8**

votes

**1**answer

### A Conjecture about Maximal Prime Gaps

As it is well known that prime number is $2,3,5\cdots \cdots$, thus all these prime number are denoted by$p_{1},p_{2},\cdots \cdots ,p_{n}\cdots \cdots$. The prime maximal gap $\max_{p_{n+1}\leqslant N}(p_{n+1}-p_{n})$ means the maximum value of $…

Wenlong Du

- 81
- 1

**8**

votes

**4**answers

### How to prove than $p_{2n}-(p_{2n}\mod p_{n}) = 2p_{n}$ ? where $p_{n}$ is the $_{n}$th prime number ? (for $n$ > 1)

Let the prime function $p_n$ be the $n$th prime number.
For example $p_1$ = 2, $p_2$ = 3, $p_3$ = 5, $p_4$ = 7, $p_5$ = 11 etc.
I noticed something with the prime function : it seems than $p_{2n}-(p_{2n}\mod p_{n}) = 2p_{n}$, for $n$ > 1
For example…

kijinSeija

- 303
- 1
- 11

**8**

votes

**1**answer

### Do all elements of $[n+1,2n]$ have strictly higher gpf than elements of $[1,n]$ when sorted by gpf?

For any $n\in\mathbb N$, let $A=\{x\in\mathbb N \mid 1 \leq x \leq n\}$ and $B=\{x\in\mathbb N \mid n+1 \leq x \leq 2n\}$.
Order the sets by greatest prime factor of each element, ascending. Let $A_1$ be the element in $A$ with the smallest gpf…

Trevor

- 4,973
- 12
- 31

**8**

votes

**2**answers

### Relative sizes of prime gaps

There are no prime numbers between the two primes $113$ and $127$. That gap seems quite large by comparison to the sizes of the numbers in it.
$$
\frac{\text{size of gap}}{\text{prime just below the gap}} = \frac{14}{113} = 0.12389\ldots
$$
Is that…

Michael Hardy

- 1
- 30
- 276
- 565

**8**

votes

**2**answers

### Are there an infinite number of primes which are any multiple of $n$ apart?

Are there an infinite number of primes which are any multiple of $n$ apart? That is take $n\in \mathbb{N}$, then is there an infinite number of primes which are separated by $\textbf{any}$ of the numbers in the below set
$$ \{n,2n,3n,4n,\ldots \}.…

Monty

- 1,768
- 1
- 9
- 22