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

# Questions tagged [prime-gaps]

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### As $n$ grows sufficiently larger, $\pi(n)<\pi_{1}(n)$, where $\pi(n)$ and $\pi_{1}(n)$ is the number of prime and semiprime $\leq{n}$, respectively

From $P_{12}=37$ the number of semiprime(s) appears to be higher than the number of prime(s). Though I couldn't check for a higher $n\geq{500}$ for several limitations, I could really use any proof or insight regarding this observation.
Thanks

user18724

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### Generate prime of x decimal digits using bit-oriented prime generator

I've got a question on stackoverflow where somebody asks to generate a random 18-digit prime. Unfortunately, the only prime generator is the one from OpenSSL. This prime generator is however geared towards generating primes for RSA and DSA, which…

Maarten Bodewes

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### Is there only a finite amount of primes that differ by $2k \in \mathbb N$?

With $2k \in \mathbb N$ greater or equal to the bound in Yitang Zhang's proof about prime number gaps (I put it that way since it's constantly decreasing).
As far as I know the proof states that there are infinitely many primes which differ by an…

355durch113

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### For any integer $n>0$ there is always a prime $p$ with $q_n\leq{p}\leq{3n}$ or $3n\leq{p}\leq{q_n}$ , where $q_n$ is the $n$-th prime.

I checked it for all the prime $\leq{100}$, the results seem very obvious though, so I would love a proof or two and any correction and opinion. Thanks in advance.

user18724

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### What is Lebesgue measure of sets of inverse prime numbers in $[0,1] $?

I would like to know if it is possible to know the Lebesgue mesure of
sets of inverse prime numbers in $[0,1]$.
Note : I think I should know first whether the sets of primes are infinite countable or uncountable according to the large gaps and…

zeraoulia rafik

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### Improved Betrand's postulate

I want to show that $2p_{n-2} \geq p_{n}-1$...
Bertand's postulate shows us that $4p_{n-2}\geq p_{n}$ but can we improve on this?
any ideas?

Brad Graham

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### question about first occurring prime gaps

If a prime gap $g(p)$ is the first occurring prime gap of it's size, does this imply that it is also the largest gap below $p$? In other words, is the set of first occurring prime gaps contained completely within the set of maximum prime gaps below…

jg mr chapb

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### Max gap between $(3^x⋅5^y⋅7^z)$ or $(2^w⋅3^x⋅5^y⋅7^z)$ and the closest $(3^r⋅5^s⋅7^t)$ or $(2^q⋅3^r⋅5^s⋅7^t)$?

If we take the Primorial formula and remove $(2⋅3⋅5⋅7)$, we are left with:
$p_n\#=\prod_{k=5}^{n}p_k$ ($k=5$ because $11$ is the $5$th prime).
$p_n\#=11⋅13⋅17⋅19⋅23⋅29⋅31⋅...$.
The product is getting bigger and bigger rapidly.
In comparison the…

Isaac Brenig

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### A way to find exactly $n$ consecutive composite numbers

Inspired by this quite simple question, I tried to play with the formula.
$(n+1)!+(i+1);\;1\le i\le n$
Which gives at least $n$ consecutive composite numbers. But I've found that very often they are more than $n$.
For $n=3$ we have $3! +2=8;…

Raffaele

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### Counterexample to a given claim about prime numbers

Let $(x_{k})_{k≥2}$ and $(y_{k})_{k≥2}$ be two non constant sequences of strictly increasing positive integers such that $x_{k}>1,y_{k}>1$ for all $k≥2$.
I want to get a counterexample to the following claim:
Claim: If there exist one positive…

Safwane

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### Finding the n-th prime number

We want to uniquely map hash values to prime numbers. One way to achieve this is storing the first $l$ prime numbers into an ordered list $L$ with size $|L| =l$. When the hash value $h$ calculated, return $L(h)$. However, when the required list size…

kelalaka

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### I have a proof concerning prime numbers. Should I publish my result?

I can easily and shortly prove that given $ε$, computable using all primes less than $N$, there will be at least one prime number between $n$ and $(1+ε)n$, where $n > N$.
It proves Bertrand's Postulate for $ε = 1$. And it gives a better result for…

user1582006

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### Wouldn't the Riemann hypothesis rule out a formula to predict primes?

Prime formula: a deterministic way to predict primes.
Riemann hypothesis: implies "primes are random".
If RH is true will we never have a useful prime formula?

D J Sims

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