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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I'm looking for a set of prime numbers that satisfy the following property in the text below [closed]

I'm looking for a set of prime numbers that satisfy the following property in the text below: $p_{k+1}-p_k=p_k-p_{k-1}$, $p_{k+i}-p_k=p_k-p_{k-i}$, $p_{k-1},p_k,p_{k+1}$ consecutive prime numbers. ...
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For every odd prime number $p \gt 3$ there exists another prime number $q \lt p$ such that $p - q = 2^n$ for some $n \geq 1$. Can you prove it? [closed]

Conjecture. Let $p$ be an odd prime number greater than $3$. Then there exists another odd prime number $q \lt p$ such that $p - q = 2^n$ for some positive exponent $n$. Can we prove this or is it ...
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4 votes
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Confusion on the proof of Erdös's prime gap inequality

I am currently reading Erdös's paper "The difference of consecutive primes" published in 1940, in which he shows that there exists $\delta>0$ such that $$ A=\liminf_{n\to\infty}{p_{n+1}-...
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Multiple of $15$ between quadruples of primes

Given a quadruple of primes (distinct from $5,7,11,13$) note that: The quadruplet $(11,13,17,19)$ I can write it as $(15n-2^2, 15n-2^1, 15n+2^1, 15n+2^2)$, $n=1$. For the quadruplet $(101, 103, 107, ...
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3 votes
0 answers
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On Erdös's inequality regarding prime gaps

By prime number theorem, it is possible to shown that $$ \liminf_{n\to\infty}{p_{n+1}-p_n\over\log p_n}\le1, $$ and I learned from an exercise in Cojocaru & Murty that there exists $0<\delta<...
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Divisibility into primes quadruples

Reading about quadruples of primes, it is clear that these must have the form $(p,p+2,p+6,p+8)$, ($p>5$). Consider the first three quadruples: $\{11, 13, 17, 19\}, \{101, 103, 107, 109\}, \{191, ...
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Symmetric $k-$ tuples of primes in arithmetic progressions

I've been working on this for a long time but Im stuck, I hope someone here can guide me. First I define a symmetric k-tuple as $\mathcal{H}=\{h_1,h_2,...,h_{k-1},h_k\}$ such that $h_1+h_k=h_2+h_{k-1}=...
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5 votes
1 answer
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Why does the ninth successive difference of primes appear to have two distinct groups?

Was exploring successive differences of primes and noticed an interesting pattern of the histogram of counts for the sixth and ninth difference. The ninth is more pronounced, code and image below. ...
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Full derivation inside of twin prime statement in terms of multiplicative arithmetic functions. How can the last formula be rearranged?

Let $(\cdot\mid\cdot) : \Bbb{N}\times\Bbb{N} \to \Bbb{Z}_2$ be the divisibility function which takes on the value $(x|y) = 1$ whenever $x$ divides $y$ and the value $(x|y) = 0$ whenever it does not ...
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3 answers
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No infinite arithmetic progression exists with prime numbers

I am trying to prove there is no infinite arithmetic progression involving only prime numbers. (In other words, I want to prove that if $a, b \in \mathbb{N}$, then there exists some $n$ such that $a + ...
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3 votes
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Does the following hold as a conjecture for maximum gaps between prime numbers? and can it be proved?

Even though I used matrix related mathjax on the backend, the frontend is intended to be just a regular table. $$\begin{matrix} a&X&X:explanation \\1&1 \\2&3 \\3&5 \\4&(6)&(...
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7 votes
1 answer
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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 ...
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2 votes
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Conjecture on ordering the first $p^2$ naturals by prime factor count

Let $\text{bump}(n)$ for $n\in\mathbb N$ be a function that increases the prime index of each prime factor of $n$ (with multiplicity) by $1$. I'll also use the notation $\text{bump}^k(n)$ to signify $\...
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What's so hard about contradiction proofs w.r.t. primes?

I've always felt like there should be some relatively straightforward proofs by contradiction establishing theorems about prime distribution. Ideally they would exist for things like twin prime ...
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3 votes
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Conjecture: there are more than $\pi(p)$ primes between consecutive prime squares

For any $p_i$ (being the $i$th prime), it seems empirically certain that $$\pi(p_{i+1}^2)-\pi(p_i^2)>i.$$ Equivalently: there are more than $\pi(p)$ primes between any $p^2$ and the next higher ...
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The sets $\mathcal{F}_d(m,n,k) = \Big\{ x \in (m, n) : x^2 = k^2 \pmod d\Big\}$ seem to have relationships with each other. What is their structure?

For $m,n,k,d\in \Bbb{Z}, m\leq n$, define for interval of integers $(m,n)$: $$ \mathcal{F}_d(m,n,k) := \Big\{ x \in (m,n): x^2 = k^2 \pmod d\Big\} $$ Then $\mathcal{F}_d(m,n, i)\cdot\mathcal{F}_d(m', ...
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2 votes
1 answer
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For any prime $p_{n}>7$, there is at least as much odd composites before $p_{n}$ than between $p_{n}$ and $p_{n+1}$?

How to prove this conjecture: For any prime $p_{n}>7$, there is at least as much odd composites before $p_{n}$ than between $p_{n}$ and $p_{n+1}$. Let $o_{n}$ denote the number of odd integers ...
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5 votes
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Conjecture: between any two consecutive squares, there are integers matching each of $2p, 3p,$ and $4p$; also, more terms with higher degrees

This is a minor twist on Legendre's conjecture, of course. To restate: I submit that for all $n>1$, every interval $\left(n^2,(n+1)^2\right)$ contains at least one integer matching each form $p, 2p,...
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1 vote
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Definition of Prime Triplet & Prime Triple

According to the Wikipedia's Prime Triplet article, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by $6$. In particular, the sets must have the ...
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2 votes
1 answer
127 views

What does Cramer's model say?

I know that Cramer's model is stated as follows: "With a probability =1, the relation $$\displaystyle \limsup_{n\to\infty}\frac{p_{n+1}-p_n}{(\log p_n)^2}=1$$ is satisfied" Could someone ...
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1 vote
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What is the proper name for the "second ratio" of a prime number (meaning the second difference / second sum)?

I became interested in the second differences of prime numbers, and as part of my amateur investigations, I wanted simple metric that would allow me to directly compare prime number sequences across ...
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How does this inequality imply Chebyshev's statement

$\ A(x) = \sum_{n\leq x} a_n$ is slowly increasing. that is for a fixed $\theta>0$ $\lim _{x \to \infty} \frac {\ A (\theta x)}{A(x)}=1$ With the above notation we have $\lim_{x\rightarrow \infty} \...
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what sort of proof did Chebyshev use to prove Bertrand's conjecture?

what sort of proof did Tchebycheff use to prove Bertrand's conjecture? did he use proof by contradiction or induction? the reason i'm asking is because I'm looking for Chebyshev's proof that is a bit ...
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I am trying to find the maximum gap between any prime and the nearest prime (whether smaller or bigger)?

I am trying to find the maximum gap between any prime and the nearest prime number (whether smaller or bigger)? Here is what I have: Assuming: I don’t know whether any of the multiples that are ...
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3 votes
1 answer
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A formula that counts exactly the twin prime averages occuring in an interval $[a,b]$ is surprisingly succinct!

Let $p_n$ denote the $n$th prime number. Let $p_n \lt a \lt b \lt p_{n+1}^2$ be any such integers. Their oddness or divisibility does not matter as in my previous posts, which makes this formula ...
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8 votes
4 answers
186 views

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_{...
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-1 votes
1 answer
75 views

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 $(2^q⋅3^r⋅5^s⋅7^t)$? [closed]

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 ...
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1 vote
1 answer
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There must exist a prime between $P_n$ and (including) $|(P_n\#)/2-2^y|$ as long as $|(P_n\#)/2-2^y| > P_n$.

There must exist a prime between $P_n$ and (including) $|(P_n\#)/2-2^y|$ as long as $|(P_n\#)/2-2^y| > P_n$. The Primorial function is expressed as: $p_n\#=\prod_{k=1}^{n}p_k$ I will show my ...
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0 votes
1 answer
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Gaps between primes and Pigeonhole Principle

Prime number theorem: $\#\{\text{primes} \leq x\} = (1 + o(1))\frac{x}{\log(x)}.$ Pigeonhole Principle $\implies$ among the primes${}\leq x$, there's a prime gap (at least one) $p_{n+1} - p_{n} \geq (...
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3 votes
2 answers
107 views

Smallest $\epsilon > 0$ for which there's always a prime between $n$ and $(1+\epsilon)n$

One can show via the PNT that $$\lim_{n\to \infty} \frac{\pi((1+\epsilon)n) - \pi(n)}{n/\log{n}} = \epsilon,$$ for any $\epsilon > 0$, which in particular implies that for any $\epsilon > 0$ and ...
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4 votes
3 answers
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Product of primes below some number $n$

I was asked this question in an exam For an integer $n>3$ denote by $f(n)$ the product of all prime numbers less than $n$. So $f(6) = 30$, $f(5) = 6$. Which of the following are true? A. There are ...
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2 votes
1 answer
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Conjecture about the density of primes

Conjecture For any sufficiently large integer $kn$ , the sequence representing the number of primes in each block obtained by splitting $kn$ into $k$ equal blocks, is a strictly decreasing sequence, ...
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3 votes
1 answer
200 views

Is it possible to come up with a formula for upper bound for this?

Consider a sieve, where the only numbers left are $n \equiv 5 mod (6)$ So the sieve has 5, 11, 17, 23... Where the gap is uniform and is 6, initially. Now we'll continue to sieve out the multiples of ...
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1 vote
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What is the significance (if any) of being able to deterministically calculate the gap to the next prime number in this way?

(Hello, this is my first post here so I hope I do a good job of laying it out. I am happy to clarify or clean up examples if it might help out.) Consider a table of numbers (n - horizontal axis) and ...
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1 vote
1 answer
102 views

No primes for increments for factorial n

I was reading the following exercise: Prove that if $n \ge 2\space$ then among the numbers: $n! + 2, \space n! + 3,..., n! + n$ none are prime (where $n! = 1\cdot 2 \cdot 3 \cdot ... n$ My ...
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1 vote
2 answers
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Conjecture: There are infinitely many $N$ such that $0\equiv N\pmod2$, $1\equiv N\pmod3$, and $0\leq N\pmod P\leq P-4$ for primes $P$ with $5\leq P<N$

Here's a conjecture, There are infinitely many numbers $N$, such that for all prime numbers $P<N$ $0 \equiv N \pmod 2$ and $1 \equiv N \pmod 3$ and $0 \leq N \pmod P \leq (P-4)$, for $P \geq 5$ ...
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3 votes
0 answers
133 views

Spiral's from (prime) squares $p^2$ and gaps [closed]

This is recreational math. I created some colorful spirals from natural numbers and prime numbers. I do not understand observations I made. If possible I would like a more indepth answer and some ...
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2 votes
0 answers
58 views

A quite efficient way to get large gaps.

I know that the method of 'factorial' can guarantee that there are arbitrarily large gaps between consecutive primes, but I have a question about this method, consider the number $$g=p_1\cdot p_2......
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4 votes
1 answer
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Do there exists infinitely many primes that satisfy $p_a-p_b=k$

I have read that Terence Tao proved that there exists infinitely many primes that satisfy $p_n-p_{n-1}\le246$ ($p_n$ denotes the $n^{th}$ prime) I want to know whether it has been proven that there ...
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1 vote
1 answer
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Is there an elementary proof that next prime number is guaranteed to be relatively nearby, that does not involve prime number theorem-related maths?

I asked this question yesterday, perhaps a bit too hastily: Does the prime number theorem tell us that the next prime number is guaranteed to be relatively nearby? I think I bit off more than I can ...
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3 votes
0 answers
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Does any odd prime number belong to at least one set of a minimum of $3$ prime numbers, that are separated from each other by the same gap?

I have tried searching for the following and I mostly got results dealing with: the rarity of different gaps, twin gaps, cousin gaps... As a self learner, there is a chance that I didn't know of a ...
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3 votes
2 answers
143 views

Does the prime number theorem tell us that the next prime number is guaranteed to be relatively nearby?

Let $\ p_n\ $ be the $\ n$-th prime number. Does the prime number theorem , $\Large{\lim_{x\to\infty}\frac{\pi(x)}{\left[ \frac{x}{\log(x)}\right]} = 1},$ imply that: $ \displaystyle\lim_{n\to\infty}\...
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0 votes
0 answers
41 views

Weaken Dickson's conjecture

I'm interesting in Dickson's conjecture. But it is hard to consider. I want some `weaken' Dickson's conjecture. It is the following statement. Let $a$ and $b$ be constant integers. $S$ is a set of ...
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  • 655
3 votes
1 answer
135 views

Distribution of prime gaps - is it an unsolved problem?

Numerical experiments show the distribution of prime gaps conforms to some quite firm constraints. The following plot visualises these constraints - it shows the log of the count of prime gaps against ...
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1 vote
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What can we derive from an experimental plot of prime gap counts?

The below chart shows the counts of prime gaps in the number range up to $5\times10^8$. It is clearly an interesting shape and some key features are: an approximate linear relationship between the ...
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3 votes
0 answers
124 views

On the inequality $p_{n+1}-p_n\leq n$

Not a very well known inequality is $$p_{n+1}-p_n\leq n$$ where $p_i$ is the $i^{th}$ prime number. I know this can be proven using the following inequality: (B. Rosser, L. Schoenfeld) $$\forall x\geq ...
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1 vote
1 answer
129 views

Frequency of gaps between consecutive prime numbers

This plot, from Odlyzko, Rubinstein & Wolf, 1999, shows the frequency of the gaps between consecutive primes of a given size. Where $N(x,d)$ is the number of primes $p \leq x$ such that $p+2d$ ...
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  • 1,750
1 vote
1 answer
128 views

Density of Primes in an interval

Is it possible to characterize all or at least some $k \in \mathbb{N}$ such that there are more than $k$ primes between $k^2$ and $\lfloor k^2/2\rfloor$? Or is it possible to characterize all or at ...
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0 votes
1 answer
43 views

is this prime probability function that generated from Eratosthenes Sieve can predict numbers of prime in closed interval?

I have two question regarding this prime probability $P(p)$ for $p$ that exists for $[p_{k-1}^2 , x]$ $P(p)=\prod^k_{i=1}\big{(} 1 -\frac{1}{p_i}\big{)}$ Where $x<p_k^2$ and $k$ was index such that ...
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3 votes
2 answers
65 views

$p_{i + k} - p_i \neq \text{const}$ for any $k \geq 1$ where $p_i = i$th prime number.

Let $p_i$ be the $i$th prime number. This should be simple to prove: $$ \forall k \geq 1, c \in \Bbb{Z}, \\ p_{i + k} - p_i \neq c, \\ $$ for some $i \geq 2$. But for example: $$ 11 - 5 = 6 \\ 13 - ...
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