# 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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### About prime gaps ; $\sum_{m = 2}^{\ln_2^{*}(n)} \pi_{m}(n) \leq \pi(n)$?

Let $\pi(n)$ be the number of primes between $1$ and $n$. Let $\pi_2(n)$ be the number of prime twins (gap $2$) between $1$ and $n$. Let $\pi_3(n)$ be the number of prime cousins (gap $4$) between $1$ ...
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### Devising an upper bound for $r_{0}(n)$ through Stefan-Boltzmann law

Define the fundamental primality radius of an integer $n>1$ as $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$. Can we intepret the average prime gap around $n$ as an absolute temperature $T$...
• 2,590
568 views

### An interesting finding on twin primes

I was doing some research on prime gaps including twin primes and this led me to this finding, which is: $$\lim\limits_{n\to \infty} \frac{\pi^2(n)}{n\pi_2(n)} = 0.7550363087870907 \cdots\cdots (1)$$...
• 2,705
1 vote
57 views

### All twin prime averages in the range $[9, 119]$ are of the form $6(5[3(z-x)]_{\pmod 7} + x)$ for some $x \in \{0,2,3\}, z \in \{0,2,3,4,5\}$.

Question. Can we come up with a general formula $f(x_5, x_7, x_{11}, \dots, x_{p_n})$ such that each twin prime average $a \in [p_n + 2, p_{n+1}^2 - 2]$ is expressible as $f(x_5, \dots, x_{p_n})$ for ...
• 21.9k
68 views

284 views

### Difference of prime powers as linear combination of jumping champions

Reading the preprint Bounded gaps between primes in short intervals by Ryan Alweiss and Sammy Luo (https://arxiv.org/abs/1707.05437), I came up with the following question: Can the difference $\Delta$ ...
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• 645
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### Which is the lowest or most accurate upper bound formula for the maximum gap in sieve of Eratosthenes after a particular iteration?

Consider a function $a(x)$ which gives the larget gap in the sieve of Eratosthenes after the $x^{th}$ iteration. So, $a(0) = 0$ $a(1) = 1$, After removing the multiples of 2. $a(2) = 3$, After ...
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### Coman's Last Conjecture stating that every prime $q \geq 11$ can be written as $3 \cdot (p_1-1) + p_2$, where both $p_1$ and $p_2$ are prime numbers.

Today I was taking a look at Coman's book entitled Conjectures on Primes and Fermat Pseudoprimes, many based on Smarandache function (starting from the end, as I often do) and his last conjecture, the ...
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### Find all numbers n such that n+1, n+5, n+7, n+11, n+13, n+17, n+23 are all prime

Find all natural numbers $n$ such that $n+1$, $n+5$, $n+7$, $n+11$, $n+13$, $n+17$, $n+23$ are all prime. So far I've made the following progress on this problem: a) n must be even. Otherwise some of ...
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### Is my proof for the fact that there are arbitrarily long strips of numbers between successive primes correct?

My attempt at a proof by induction: Let two successive primes be $a$ and $b$. $a - b = c$ gives the number of numbers between them. Base case: The first five values of $c$ are $0,1,1,3$ and $1$. There ...
1 vote
59 views

### A question on primes larger than a bound in an arithmetic progression

Let $n \in \mathbb{Z}$ be a semi-prime with unknown factorization, $n = pq$, where $p, q \in \mathbb{P}$, the set of primes. Without loss of generality, let $p \lt q$. Say we have done trial division ...
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### How many are some special gaps between primes?

Gap primes are certainly even numbers. (starting from the consecutive primes $3$ and $5$) Let $n$ be however an odd number and define by $n^2$ the range of the examined primes ( i.e. the range is ...
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### Can the sum of $n$ consecutive primes be $n$ times a prime?

I recently saw a "coffin problem" asking to prove that the sum of two consecutive primes is not twice a prime. This got me wondering if three consecutive primes can sum to be three times a ...
• 69
### Find a prime before or after some $n$ consecutive composite numbers.
Goal I'd like a method to find a prime before or after some $n$ consecutive composites. One method would be to brute force every prime and check for $n$ consecutive composites before or after it. ...