I'm looking at the family of step functions $\Phi_p(x) = \pi(2x - p)$, where $\pi$ is the prime-counting function and $p$ ranges over the primes. When I superimpose these functions for many values of $p$ (see attached figure, $n$ up to $5 \times 10^7$), the result is a dense region with internal...

Let $m_1,\ldots,m_s$ be pairwise coprime positive integers, and let $$ M=\prod_{i=1}^s m_i. $$ For each $i$, let $A_i$ be a non-empty proper subset of residue classes modulo $m_i$. Define the CRT-allowed residue set $S \pmod M$ by $$ S=\{x \pmod M : x \pmod {m_i} \in A_i \text{ for every } i\}....

I found the following infinite continued fraction: $$ \operatorname{C_{B4CC}}=\cfrac{2}{1+\cfrac{2}{1+\cfrac{6}{1+\cfrac{3}{1+\cfrac{10}{1+\cfrac{4}{\ddots}}}}}} $$ where $\operatorname{C_{B4CC}}$ denotes Baxter's four-coloring constant and the partial numerators are defined by interweaving...

I am studying a deterministic construction based on the consecutive gaps between primes. The phenomenon I am interested in is NOT that arithmetic on primes sometimes produces another prime. The interesting point is that the same future value can be generated by different rows of a cumulative...

Let $x,n$ be positive integers. Consider the equation $$e^x+x^e=e^n$$ I would like to know whether this equation can have any positive integer solutions. Since $x^e \gt 0$, any solution must have $n \gt x$. Dividing by $e^x$, $$1+\left(\frac{x}{e}\right)^e=e^{n-x},$$ so $$n-x=...

I was playing around with Goldbach representations and came up with the following class of primes. Consider an even integer $n$ that can be written as a sum of two odd primes, $$ n=p+q,\qquad p\le q. $$ For a fixed $n$, define $GGT(n)$ to be the largest possible value of $q$ among all such pairs...

P.412 of volume 3 of Gauss's collected works contains two unpublished remarks of Gauss that apparently have not been discussed yet. The first one is of number-theoretic significance, while the second relates the value of $Q([a+bi]\varphi)$ at a point $\varphi=\text{arcsinlemn} (x)$ such that...

Let $p_n$ be the $n$-th prime and let $$ g_n = p_n-p_{n-1}. $$ Define $$ S_n=\sum_{i=2}^{n} g_i=p_n-2 $$ and $$ V_n=S_n+g_n=p_n+g_n-2. $$ I call a convergence the occurrence of the same value $V$ at two or more distinct positions. For example, for $V=103$: $$ 95+8=103,\qquad 99+4=103,\qquad...

We consider the odd-step block transitions of the Collatz mapping by collapsing intermediate even divisions: $$3n_i + 1 = 2^{k_i} n_{i+1}, \quad \text{where } k_i = v_2(3n_i + 1)$$ We model potential upward expansion chains ($1^r$, where $k_i = 1$ for $r$ consecutive steps) followed by...

Does the proposition "if f(n)=an+b has two distinct primes f(c),f(d), with a>0 then it has infinitely many primes" imply Dirichlet's Theorem?

Let $n$ be a positive integer and let $p>3$ be a prime number such that $p\mid n^2+n+1$. Prove that $(n+1)^p\equiv n^p+1\pmod{p^3}$. Set $P(x)=\dfrac{(x+1)^p-x^p-1}{p}$. First use $p\mid n^2+n+1$ to show that $n^3\equiv1\pmod p$ and hence $p\equiv1\pmod3$. Now let $\omega$ be a primitive third...

I know of Conway's chained arrow notation. I have read the Wikipedia article on it, but it still didn't give me a rigorous proof that it exists and is unique. So, to make my question precise, suppose we are given a finite nonempty sequence $S$ of positive integers. How does one rigorously define...

I came up with the following Diophantine equation: $$ \frac{z}{x+y}+\frac{y}{x+z} +\frac{z}{x-y}+\frac{y}{x-z}=9, $$ where $x,y,z\in\mathbb Z,$ $xyz\ne0,$ and $x+y\ne0,$ $x-y\ne0,$ $x+z\ne0,$ $x-z\ne0.$ Does this equation have a nonzero integer solution? If a solution exists, I would be...

I developed this theory. Is it correct? What do you think? Formal definition of the space $L$ as a local ring: $$\mathcal{L} \cong \mathbb{C}[\varepsilon]/(\varepsilon^2)$$ Fundamental axioms of the basis units $\{1, c, b\}$: $$c^2 = -1, \quad b^2 = 0, \quad cb = 0$$ General representation of an...

I am investigating a novel arithmetic function $f: \mathbb{Z}^+ \to \{0, 1, \dots, 9\}$ that maps an integer to a terminal state on a $\mathbb{Z}/10\mathbb{Z}$ cycle based on its distinct prime signature. Definition Let $n \in \mathbb{Z}^+$ have the unique prime factorization $n = p_1^{a_1}...

I have constructed an arithmetic function that exhibits a chaotic behavior reminiscent of Collatz-like dynamical systems, but with a feedback loop driven by the partitions of parity-shifted divisor geometries. The system appears to inherently embed the Halting Problem within standard...

Definition 1: An element of $\operatorname{PSL}_2(q)$ is semisimple if it is diagonalisable in $\operatorname{PSL}_2(\overline{\Bbb F_q})$, where $\overline{\Bbb F_q}$ is the algebraic closure of $\Bbb F_q$. Definition 2: Let $q$ be a power of a prime. Then we say $n\in \Bbb N$ is $q$-good if:...

So I am studying the arithmetic derivative and I stumbled upon following approach: We are considering here a $\Bbb{Q}$-sub-vector-space of $\Bbb{R}$. Define the set $$\log(\Bbb{P}):=\{\log(p) : p\in\Bbb{P}\}$$ and the $\Bbb{Q}$-vector-space generated by $\log(\Bbb{P})$...

Let \chi be a Dirichlet character of prime order r, and define [ A_\chi(n)=\sum_{d\mid n}\chi(d). ] For a prime p, one trivially has [ A_\chi(p)=1+\chi(p). ] I have been looking at composites n that satisfy the same identity [ A_\chi(n)=1+\chi(n). \tag{1} ] I would like to know whether...

Let $T_n = \frac{n(n+1)}{2}$ denote the $n$-th triangular number, and let $S(m)$ denote the sum of the digits of $m$ in base 10. I am investigating the conjecture that $S(T_n)$ is a prime number for infinitely many $n$. Modular constraints We know that $T_n \pmod 9$ is periodic with a period of...