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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### Renyi entropy of prime gaps

Denote with $p_n$ the $n$-th prime number and let
$$
h_N(d) = |\{ n : p_{n+1} < N, p_{n+1} - p_n = d \}|
$$
be the number of times that prime gap $d$ happens for primes less than $N$.
Let $H = \sum_{d \in {\mathbb Z}} h_N(d)$ and let
$$
f_N(d) =…

Riccardo B.

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### Proof for strict inequality $\pi(ab) > \pi(a)\pi(b)$?

I asked about the very similar $\pi(ab)\geq \pi(a)\pi(b)$ a while ago, and this is indeed a proven result for $a,b\geq \sqrt{53}$.
Empirically, the stricter $\pi(ab)>\pi(a)\pi(b)$ looks true so long as either $a$ or $b$ is $\geq 23$.
It also seems…

Trevor

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### Does there exist such a sequence $B$ when $p>5$?

Let $A = (a_1, a_2, \ldots, a_n)$ be the sequence of odd primes are less than or equal to a prime number $p$.
Let $C$ be the infinite ascending sequence of composite numbers that their factors are all in $A$.
Let $B$ be a sequence of $n$ consecutive…

user49311

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### Conjecture: smallest missing mod value always yields previous prime

I've come up with a conjecture that seems similar in strength to Legendre's or Oppermann's, but maybe subtly different.
Let $a_n$ be the smallest nonnegative value such that there is no $m$ in $1

Trevor

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### Analog of Cramer's conjecture for primes in a residue class

Let $q$ and $r$ be fixed coprime positive integers,
$$
1 \le r < q, \qquad \gcd(q,r)=1.
$$
Suppose that two prime numbers $p$ and $p'$, with $p

Alex

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### Combinations of four consecutive primes in the form $10n+1,10n+3,10n+7,10n+9$

Here $n$ is some natural number. For example, among the primes $< 1000$ I found four such combinations:
\begin{array}( 11 & 13 & 17 & 19 \\ 101 & 103 & 107 & 109 \\ 191 & 193 & 197 & 199 \\ 821 & 823 & 827 & 829 \end{array}
Using Mathematica I was…

Yuriy S

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### How to see that the prime gaps functions isn't eventually monotonic?

Let $g(n)$ be the distance between the $n$th prime and the next.
By elementary means we can see that $g(n)$ is not eventually constant and that $g(n)$ is not strictly monotonic.
Further we know that it isn't eventually monotonic (meaning $g(k) \le…

Brennan.Tobias

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### Does the following equation have only 1 solution of $n=2$?

Conjecture:
If $$I=\frac{1}{1+p_{n+1}}+\sum_{k=1}^{n}\frac{1}{p_k}$$
(where $p_n$ denotes the $n$'th prime)
then $n=2$ is the only natural number for $n$ that makes $I$ an integer.
All I really understand to do is to input numbers in for $n$. Beyond…

user266519

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### What is the smallest prime $p$ such that the next prime is greater than $p+2000\ $?

I studied this site
https://en.wikipedia.org/wiki/Prime_gap
and wondered if the smallest prime gap greater than $2000$ can still be determined, in other words :
Which is the smallest prime $p$, such that
$q-p>2000$, where $p$ and $q$ are…

Peter

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### How to find this number, which is probably a very big prime or a product of big primes?

Let $\mathcal{N}(n)$ be the next prime greater than $n$.
Which is the smallest natural number $n>0\;$ such that:
$\mathcal N(2\cdot 3\cdot 5\cdot 7\cdot 11\cdot n)−2\cdot 3\cdot 5\cdot 7\cdot 11\cdot n\;\;\;$ is neither a prime nor $1$?
I'm…

Lehs

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### Elementary proofs of prime gap theorems?

"Obviously" it is thrue that $p_{n+1}<2p_n$. Testing for $n<10$ shows it is true for small $n$ and no mathematician or wannabe has ever doubt that it is true for big $n$. But there is no real simple arithmetic proof, so far, not using the prime…

Lehs

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### Polynomial equations in $p$ and $q$ with $p,q$ primes

Is there a non zero polynomial $R \in \mathbb{Z}[X,Y]$ such that there exists an infinite number of pair $(p,q)$ with $p$ and $q$ primes, $p \neq q$ and $R(p,q)=0$ ?
I know the curve must be of genus $0$ (Faltings-Mordell).
My question is related…

francis-jamet

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### The sequence of prime gaps is never strictly monotonic

I have an assignment question that asks me to show that the sequence of prime gaps is never strictly monotonic. I'm also allowed to assume the Prime Number Theorem.
I've managed to show that it cannot be strictly decreasing by considering the…

Haikal Yeo

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### Techniques on finding consecutive primes with large gaps

I found two consecutive prime numbers $401!-3463$ and $401!+4021$. These have a difference of $7664$. Is there some kind of technique that is known in order to find consecutive prime numbers with sufficiently large gaps?
I just used the fact that…

Mr Pie

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### The crosshatch conjecture, on primes in $(p,p^2)$

If the first $p^2$ integers are laid out in a $p\times p$ square, every row and column will have at least one prime. Easily visualized as so:
I recognize this should maybe be packaged as two conjectures, but hey, I like the aesthetics.…

Trevor

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