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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.

16 votes
2 answers
2k 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 …
mathoverflowUser's user avatar
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, …
mathoverflowUser's user avatar
6 votes
1 answer
474 views

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_ …
mathoverflowUser's user avatar
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 …
mathoverflowUser's user avatar
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, …
mathoverflowUser's user avatar
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 …
mathoverflowUser's user avatar
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 …
mathoverflowUser's user avatar
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 …
mathoverflowUser's user avatar
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- …
mathoverflowUser's user avatar
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 …
mathoverflowUser's user avatar
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 …
mathoverflowUser's user avatar
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 …
mathoverflowUser's user avatar
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 …
mathoverflowUser's user avatar
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) …
mathoverflowUser's user avatar
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 …
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