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Results tagged with prime-numbers
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user 165920
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
6
votes
1
answer
474
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How to define a fractal from the lexicographic sorting on the prime factorization of natural...
Consider on the natural number the lexicographic ordering on the prime factorization:
If $m = p_1^{a_1}\cdots p_r^{a_r},n = q_1^{b_1}\cdots q_s^{b_s}$ then we define:
$$m \vartriangleleft n :\iff [(p_ …
- 3,054
2
votes
0
answers
268
views
A relation of the prime counting function $\pi$ to counting the ordered ways of a number $n$...
The definitions are from these two questions:
https://math.stackexchange.com/questions/3164216/a-series-related-to-prime-numbers
https://math.stackexchange.com/questions/4349186/trying-to-understand- …
- 3,054
1
vote
2
answers
384
views
Solving a recurrence relation for the prime counting function?
I have found some number sequence $c_n = 1+b_n$ for $n \ge 0$, where $b_n = $ A307977(n).
I am trying to solve the following recurrence relation for the prime counting function:
$$\forall n \ge 3: \pi …
- 3,054
1
vote
0
answers
153
views
A transformation game for natural numbers?
Consider the completely additive function $\eta(n) := \sum_{p\mid n} v_p(n)p$ defined on natural numbers, with values in natural numbers. For literature, on this function, see the corresponding OEIS s …
- 3,054
3
votes
1
answer
735
views
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , where $\pi$ denotes the prim...
Is
$$1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n},$$
where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?
Context:
This question came out as a result in tryi …
- 3,054
3
votes
1
answer
952
views
A geometric proof that there are infinitely many primes?
Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{N})$.
Let $h(n) = J_2(n)$ be the second Jordan totient function, defined by:
$$J_2(n) = n^2 \prod_{p|n}(1-1/p^2)$$
De …
- 3,054
5
votes
1
answer
790
views
A consequence of Firoozbakht's conjecture?
This is a question out of curiosity, while looking at the Firoozbakht's conjecture. It might not be research related, but as usual, I am not really sure if a question ever is research related or not, …
- 3,054
16
votes
2
answers
1k
views
Polynomials for natural numbers and irreducible polynomials for prime numbers?
Let $p$ be a prime and $n$ be a natural number.
Define inductively for prime numbers: $f_1(x) := 1$, $f_2(x):=x$, $f_p(x) := 1+\prod_{q\mid p-1} f_q(x)^{v_q(p-1)}$.
Is $f_p(x)$ always irreducible for …
- 3,054
2
votes
0
answers
96
views
Primes as expected values?
This is a follow-up question, which is related to the answer of this quesiton: Is there a connection of prime numbers and extreme value theory?
I will duplicate the answer here, so this question is se …
- 3,054
2
votes
0
answers
358
views
Largest eigenvalue of a Laplacian matrix to lower bound the prime counting function? (Recurs...
Let $L_n$ be the Laplacian matrix of the undirected graph $G_n = (V_n, E_n)$ (which is defined here: Why is this bipartite graph a partial cube, if it is? ) with sorted spectrum:
$$\lambda_1 (G_n) \ge …
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5
votes
1
answer
609
views
Why does this convolution of the prime counting function $\pi$ look like a parabola?
In this previous question it is shown that the convolution of the prime counting function $\pi$ with itself, is related to the Goldbach conjecture:
$$\pi^*(n):=\sum_{k=0}^n \pi(k) \pi(n-k)$$
The Goldb …
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0
votes
1
answer
283
views
Factorization trees and (continued) fractions?
This question is inspired by trying to understand the lexicographic sorting of the natural numbers in the fractal at this question:
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , wher …
- 3,054
4
votes
0
answers
532
views
Is the integer factorization into prime numbers normally distributed?
Edit:
Sorry, for the inconvenience: I have edited the question, since there was a misconception in my thinking.
Let $P_1(n) := 1$ if $n=1$ and $\max_{q\mid n, \text{ } q\text{ prime}} q$ otherwise, de …
- 3,054
1
vote
0
answers
75
views
Partial sums of Möbius function and Euler characteristic of a simplicial complex for closed ...
In A cell complex in number theory by Anders Björner, 2011 a number theoretic cell complex is described which has the property that the Euler characteristic is related to the Mertens function:
$$M(n) …
- 3,054
9
votes
2
answers
776
views
Why do these finite group Dedekind matrices seem to have integer spectrum when specialized t...
Let $p$ be a prime and let $f_p$ be the permutation on the set $\{1,2,\cdots,p-1\}$ which is given by taking inverses in $\mathbb{Z}/(p)$:
$$x \bmod(p) \mapsto \frac{1}{x} \bmod (p)$$
So for instance, …
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