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Questions tagged [prime-gaps]

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

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Easy proof of falsehood of $\pi(n) \leq C \cdot \text{ln}(n)$ for the prime counting function $\pi$

Let $\pi(n)$ be the number of primes in the range $1,\dotsc,n$. The following statement is true: There is no $C>0$ such that $\pi(n) \leq C \cdot \text{ln}(n)$ for all $n\geq 1$. It follows ...
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How do you compute the singular series?

Terence Tao gives at his blog the following formula for something called the singular series: $$\large\mathfrak{S}(h)=2\Pi_{2}\prod\limits_{p|h;p>2}\frac{p-2}{p-1}$$ where $\Pi_{2}=0.66016...$ is ...
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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 ...
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Convergence of series related to prime gaps and Gilbreath's conjecture

Gilbreath's conjecture states that when we take the absolute values of the consecutive differences of prime numbers, as so : 2, 3, 5, 7,11,13,17,19,23,... 1, 2, 2, 4, 2, 4, 2, 4,... 1, 0, 2, 2, 2, 2, ...
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Bounds for the $n$-th record gap between primes in a residue class

(Following question 2269073.) Let $q$ and $r$ be coprime integers, $1\le r < q$, and consider the arithmetic progression $$ r, \ r+q, \ r+2q, \ r+3q, \ldots \tag{P} $$ Dirichlet proved that there ...
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Which other unsolved problems, have necessary restrictions on the prime gaps?

We all know of Unsolved problems, like Goldbach,Legendre, and Grimm's conjectures. Goldbach has the necessary condition of: There exists a prime between $n$ and $2n-2$, which means prime gaps are ...
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Can we prove that infinite many primes begin with any given digitstring?

With Dirichlet's theorem, we can easily prove that infinite many primes end with a given digitstring with final digit $1,3,7$ or $9$. Can we also prove that infinite many primes begin with a given ...
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Why are all primes squared less one a multiple of 24, but not also a multiple of 12?

So I can't wrap my head around this problem from numberphile (https://www.youtube.com/watch?v=ZMkIiFs35HQ&t=28s). Why does this work for multiples of 24 but not 12?
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243 views

Fastest way to find all the prime factors of a very large number without calculator? [closed]

Largest possible factor of a very large number n would be number itself. The largest would be n/2 (if prime) if n/2 is not prime then it would be less than n/2 . The smallest factor would be 1. Is ...
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Constantes in asymptotique formulas of consecutif primes

Let $q$ be a prime number, and $m$ an even number. Let $\displaystyle\mathcal{B}_q=\{b \in \mathbb{N}^{*} \, | \, b \wedge {\small \left( \prod_{\substack{a \leq q \\ \text{a prime}}} {\normalsize a} ...
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Method of Proof concerning Prime Gap congruence Relation over two integer variables

$$p_n-p_{\lfloor n^{\frac{1}{m}}\rfloor^m}=0 \Rightarrow m=2$$ $$\quad\quad(\operatorname{tooth1})$$ $$p_n-p_{\lfloor n^{\frac{1}{m}}\rfloor^m}\not=0 \Rightarrow m\not=2$$ $$\quad\quad(\...
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Prime gap around x expressed as $w(x)\log^{w(x)}x$

For $ x $ a positive real number greater than $3 $ and not equal to a prime, there exists a unique pair of consecutive primes $ (p,q) $ such that $ p<x<q $. Let $ g(x) : =q-p $. Expressing ...
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Maximum runs of composites in arithmetic progressions

Is there a proof that every arithmetic progression of gap $p$ has a prime in the interval $[p, p^2)$? Put another way, can you prove the following: For all primes $p$, and all integers $0 \le m <...
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Is there a way to determine exactly the difference between $N$th & $(N+1)$th prime number?

So I was trying to find the time complexity of an algorithm to find the $N$th prime number (where $N$ could be any positive integer). So is there any way to exactly determine how far $(N+1)$th prime ...
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Goldbach conjecture: Every integer $n>3$ is halfway between $2$ primes.

Prove that the following conjecture is equivalent to the strong Goldbach conjecture: Every integer $n>3$ is halfway between $2$ primes. I'm able to prove it, but i don't have much experience in ...
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For any positive integer $n>3$, there exists at least $1$ integer $k$ such that $n+k$ and $n-k$ are primes.

How to prove the following conjecture: For any positive integer $n>3$, there exists at least $1$ integer $k$ such that $n+k$ and $n-k$ are both primes. Any hint, idea or reference would be ...
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71 views

Number of primes between $n$ and $2n$

What is a good lower bound on $\pi(2n)-\pi(n)$? Bertrand's postulate gives $1$. It is expected to be as I understand of form $\frac{c\cdot n}{\log n}$ from Prime Number Theorem. Does the ratio ...
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Expected size of the maximal prime gap below x under Hardy-Littlewood conjecture

Let $ \pi_{n}(x) $ denote the number of prime gaps of size $ n $ below $ x $ for even $ n $. The Hardy-Littlewood conjecture predicts that $ \pi_{n}(x)\sim C_{n}\frac{x}{\log^{2}x} $ with $ C_{n}=...
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60 views

Twin prime conjecture and gaps between primes

This is just a thought: if gaps between prime numbers can be arbitrarily large then it should be possible to find infinitely many gaps, such that the product $m=\prod_{n=1}^{N}Pn<P_{N+1}^{2}$, ...
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Fortune's conjecture solved for limited cases?

I am not a mathematician, but while doing other work, I came across the Fortune conjecture. According to Wikipedia and other research, it seems that it has not yet been proven. I thought about the ...
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Finding the n-th prime number [closed]

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, ...
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consecutive prime gaps and explicit bounds

I am aware of the theorem that $p_{n+1} - p_n \leq n^{0.535}$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit for all sufficiently large number to ...
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Do we know of more than one occurance where Prime Gap=1?

A prime gap $g_n$ is the difference between two prime numbers, and as we know, the first two primes are 2 and 3, thus their prime gap is 1; $$ g_n = p_{n+1}-p_n=\big\{ n=1 \big\}=3 -2=1. $$ But have ...
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Asymptoptic density of prime k-tuples

The first Hardy-Littlewood conjecture concerns the asymptotic density of prime k-tuples. Assuming that the tuple $\{p, p+2m_1, \ldots, p+2m_k\}$, where all the elements are primes and $m_i$ for all $i$...
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How many prime numbers in a given interval?

Is there any algorithm or a technique to calculate how many prime numbers lie in a given closed interval [a1, an], knowing the values of a1 and an, with a1,an ∈ ℕ? Example: [2, 10] --> 4 prime ...
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Is there a pair of prime $(p,q)$ such that $p >q>7$ and $2q-p=3$ and $p, q$ are successive?

I'm failed to find at a least one pair of primes$(p,q)$ $p >q>7$ and $2q-p=3$ with $p, q$ are successive , I think tha's impossible because we do not know more about Gaps between prime and if ...
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GCD of $p+q$ for all pairs of primes $p,q$ of the form $p-q=12n+2x$

I came across this result, and i'm having trouble explaining it. I use it as an argument in a proof, so i need to explain this behavior, in the shortest possible way, and most importantly prove that ...
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The definition of a prime constellation

On Mathworld it is first stated that a prime constellation is a sequence of $k$ prime numbers, for which the gap between the last and the first minimizes. But later they show a table with prime ...
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Is it possible to find EXACTLY $101$ consecutive composite numbers

Here is a similar question that asks for $101$ numbers none of them are prime and it is well known that $101!+1,101!+2,101!+3,\cdots,101!+101$ are those numbers. I am interested to know, how we can ...
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distributions of prime numbers - theorem of Chebyshev

I was thinking: let $a\in(0,1]$, $1<b$ be given, and let $c$ be given as a positive integer. Can we find $N$ with the property that if $N$ is large enough, than the interval $(N^a,(b N)^a)$ always ...
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Prime gaps and gaps between successive critical zeros of zeta

Assuming RH, the sequence of critical zeros of the Riemann zeta function can be viewed as the Fourier transform of the sequence of primes. From a physicist point of view, the average gap between the ...
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Quality of prime seeking methods

I am working on prime numbers with emphasis of prime search heuristics, and found the probabilistic methods for primes seeking, I am looking for a review of those methods quality in terms of machine ...
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Is there any algorithm to identify the smallest example of gap size $n-1$ between consecutive prime numbers

In contemplating Goldbach's conjecture, I became interested in gaps between successive primes. If $n<a<b<2n$ and the range $a$ to $b$ is a primeless gap, then one could ignore any primes in ...
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Brun’s constant and irrational numbers

It is trivial that if there are finitely many twin primes then Brun’s constant must be a rational number. And GammaTester (below) has offered an example of an infinite series that converges to a ...
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131 views

Generalization of Opperman's Conjecture

Does this conjecture have a name? What about a counterexample?: $$ \forall n,k \in \mathbb{N}, k \gt 1, \exists d \in (kn-n,kn] \text{ s.t. } d \perp n! $$ An equivalent statement is this: Take a ...
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Lower bound for $p_{n^3}-p_{(n-1)^3}$?

The difference between two primes is at least $2$ so $p_{n^3}-p_{(n-1)^3} \geq 6n^2$. Is there any known sharper bound?
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Sequences of consecutive numbers

If you pick a natural number n, for example 3, and you take the string from 1,..,n and shift it via 2,...,n+1; 3, ... n+2 etc, then it looks like you will first find some strings where you can do the ...
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On some conjectures about an inequality involving different arithmetic functions related to prime numbers

I've written the following puzzle about prime numbers. This exercise is thus a curiosity/miscellany about the distribution of prime number $p_k$, that I wondered when I was playing with different ...
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gaps between square roots of primes

It is well known that the difference $p_{n+1}-p_n$ can be arbitrarily large. What about $\sqrt{p_{n+1}}-\sqrt{p_n}$, or in general, $p_{n+1}^t-p_n^t$ for $t<1$? Has this problem been investigated? ...
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a prime is a midpoint of two primes

Take three consecutive primes $p_1,p_2,p_3$: What is the opinion on the question that $p_3-p_2=p_2-p_1$ occurs endlessly and is harder to find as the primes increase? Has anyone examined the ...
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35 views

Difference between prime numbers

Let $p_n$ denote the $n$th prime number and $\phi:\mathbb{N}\to\mathbb{N}$ an increasing function, with $\phi(n)>n$. Is it true that $p_{\phi(n+1)}-p_{\phi(n)}<p_{\phi(n)}$ for infinitely many $...
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On miscellaneous questions about perfect numbers I

Let $\varphi(m)$ the Euler's totient function and $\sigma(m)$ the sum of divisors function. If $n$ is an odd perfect number then $n$ satisfies $$\varphi(n)=\varphi(\sigma(n)).\tag{1}$$ The sequence ...
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Show that for any open subset of $\Bbb R$, there is a fraction with prime terms that belongs to it [duplicate]

Be $\Bbb P\Bbb Q$ the set of all fractions $f_{m,n}=\frac{p_m}{p_n}$ whose numerador and denominator are both prime numbers. i) Show that for any open set $A\subset \Bbb R^+$, there is at least one $...
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Ratios of prime gaps $(p_{n+1}-p_n)/(p_{2n+1}-p_{2n})$

This is a question about prime gaps $g_n = p_{n+1}-p_n$ that started with a look at the average of ratios $$r_n=\frac{p_{n+1}-p_n}{p_{2n+1}-p_{2n}}$$ and of the inverse, $$ s_n=\frac{p_{2n+1}-p_{2n}}{...
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Can we prove that $p_N + 3 \leq 2 p_{N-1}$ for sufficiently large $N$?

Question in the title. It intuitively seems absurd that $p_N - p_{N-1} \gt p_{N-1} - 3 = $ the largest gap formable from all $p_i = $ odd primes $3, \dots, p_{N-1}$. Was wondering how difficult the ...
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Let $p_{k} $be the $k^{\text{th}}$ prime number. Show that there are infinitely many $k$ such that $p_{k+1} − p_{k} > 2$. [closed]

Let $p_{k} $be the $k^{\text{th}}$ prime number. Show that there are infinitely many $k$ such that $p_{k+1} − p_{k} > 2$. I was thinking about Dirichlet's theorem as i don't know to prove its ...
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Bounding Maximal gaps with Ramanujan primes

Gaps and Maximal Gaps We define terms used in this article. A prime gap as $g_n := p_{n+1} - p_n$, and we define $g_n$ as a maximal gap, if $g_i < g_n$ for all $i < n$. Define $M_{m,g}$ to be ...
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142 views

Question about prime gap records

Let $g_n$ be the $n$ th prime gap. Let $f_n = max( g_1,g_2,...,g_n)$ Now take the sequence $f_n$ and remove the duplicates. Also sort from small to large. Then $f_n $ gives the sequence $$ 1,2,4,...
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107 views

A remarkable(?) condition on sequences of natural numbers

There is a remarkable condition on increasing sequences of natural numbers $(a_n)_n$: $$\bigg\lfloor\frac{a_n^2}{a_{n+1}}\bigg\rfloor=2a_n-a_{n+1}\tag 1$$ that - when $n$ is big enough - seems to hold ...
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1answer
108 views

The distribution of powers of primes [closed]

How often do we see two or more powers of primes between two consecutive primes $p_k$ and $p_{k+1}$? One example is $p_4=7$ and $p_5=11$; we have $$ 7 < 2^3 < 3^2 < 11. $$ Are there any other ...