# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ... 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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### 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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### 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 $P \geq 5$ ... 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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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...... 4 votes 1 answer 150 views ### 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 ... 1 vote 1 answer 78 views ### 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 ... 3 votes 0 answers 78 views ### 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 ... 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}\... 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 ... 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 ... 1 vote 0 answers 85 views ### 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 ... 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 ... 1 vote
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### 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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### 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 ...
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 ...
### $p_{i + k} - p_i \neq \text{const}$ 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 - ...