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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Smoothed and truncated Von Mangoldt function
The Von Mangoldt function $\Lambda : \mathbb{N} \to \mathbb{R}$ is defined as
$$\Lambda (n)={\begin{cases}\log p&{\text{if }}n=p^{k}{\text{ for some prime }}p{\text{ and integer }}k\geq 1,\\0&{...
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Is De Polignac's conjecture equivalent with the the statement that for any positive even $n$, there are infinitely prime pairs with difference $n$?
Suppose $n$ is a positive even integer. I wonder if the following two statements are equivalent:
(1) There are infinitely many pairs of consecutive primes with difference $n$.
(2) There are infinitely ...
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Does a set with no divisibility pairs necessarily have arbitrarily large gaps? [duplicate]
The set of prime numbers has the following properties:
No element is divisible by any other element.
We can find arbitrarily large gaps between consecutive elements.
Does (1) imply (2) for arbitrary ...
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How do we measure the "holes" (zeros) of the set of $\Bbb{Z}$-linear combinations of $f_i : G \to G$ where $G$ is an abelian group?
Question:
Let $G$ be a normed abelian group. Namely the triangle inequality holds $|g + h|\leq |g| + |h|$. An example would be $G = \Bbb{Z}$ together with $|\cdot| =$ absolute value.
Now suppose ...
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Non-trivial prime gaps
A simple proof that there are prime gaps of size at least $n+1$ for every $n$ can be seen in the first answer to this question. I consider prime gaps of the form $n!, n!+1, \ldots, n!+n$ of length at ...
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Minimum $k$ for which every positive integer of the interval $(kn,(k+1)n)$ is composite
I am looking for references containing results on the minimum $k$ for which every positive integer of the interval $(kn,(k+1)n)$ is composite.
If we denote as $k(n)$ this minimum $k$ for some $n$, $k$ ...
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The Growth of Primes
I've been experimenting with the function $π(x)$ such that $π(x)$ counts the number of primes from $1$ to $x$.
I found that after $10$, $π(x^2) - π(\lfloor \frac{x^2}{2} \rfloor)$ is always larger ...
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Mirror symmetry in distances of remaining numbers of Eratosthenes-sieve
Trying my first steps in Python I found an interesting phenomenon when I tried Eratosthenes sieve. Especially I looked for the distances between the remaining numbers.
For example, if you already have ...
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Are there contiguous sequences of prime numbers of length $k$ which are convex (similarly, concave) for every $k\in\mathbb{N}?$
Does the sequence of prime numbers contain contiguous subsequences of length $k$ which are strictly convex (similarly, strictly concave), for every $k\in\mathbb{N}?$
For example,
$$ 17, 19, 23, 29 $$
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Consecutive composite numbers using the Chinese Remainder Theorem. [duplicate]
Consecutive Composite Numbers
Define a list of the first $n$ prime numbers $p_1, p_2, \ldots, p_n$.
Create a set of $n$ congruences
\begin{align*}
x + 1 &\equiv 0 \pmod{p_1} \\
x + 2 &\equiv 0 ...
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question on estimator for $\frac{\pi(n)}{n}$ and $\frac{\pi_2(n)}{\pi(n)}$
$\pi(n)$ and $\pi_2(n)$ represent the count of primes and count of twin primes $\leq n$ respectively.
Suppose we want to estimate $\frac{\pi(n)}{n}$. One way which obviously is not error-free is to ...
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broader meaning of twin prime constant?
It appears that the twin prime constant has meaning outside of the strict twin prime constant. I attempted to keep this post as short as possible.
Definitions:
Let $p,q$ represent primes and let $n$ ...
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A weaker version of the second Hardy-Littlewood conjecture.
Let $n$ be a positive integer and let $f(n)$ be the counting function for non-composite numbers.
So
$$f(0)=0,f(1)=1,f(2)=2,f(3)=3,f(4)=3,f(5)=4$$
Now my mentor noticed as a kid that apparantly
$$f(x+y)...
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What series approaches to $\log(\log(\log(n)))$?
Background: Harmonic series approaches to $\lim_{n->\infty}\log(n)$. This gives the Euler–Mascheroni constant. In addition, the harmonic series summed only over the primes approaches to $\lim_{n-&...
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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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