If $$S=\sum_{b\geq 2, \ \mu^2(b)=1} \frac{\mu(b)}{b^2}\sum_{c|b, c<\sqrt{b}}\frac{1}{\left(\frac{1}{c^2}+\frac{c^2}{b^2}\right)^{3/2}}$$ Prove that $S$ is conditionally convergent. Since $\mu(b)\neq0$ only for squarefree $b$, write $b=ck$, $c<k$, $(c,k)=1$. Then $ \mu(b)=\mu(ck)=\mu(c)\mu(k), $...

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...

Let $V$ be a finite dimensional complex vector space and $G$ be a finite subgroup of $G<\operatorname{GL}(V)$. My question is, essentially, when do the orbit space $V/G$ have finitely many symplectic leaves? My interest lies, specially, in the case when $V=h \oplus h^*$, and $G$ is given by a...

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 $E \subset \mathbb{R}^n$ be a Besicovitch (Kakeya) set, i.e., a compact set containing a unit line segment in every direction $e \in \mathbb{S}^{n-1}$. The Kakeya conjecture asserts that $\dim_{\text{H}}(E) = \dim_{\text{M}}(E) = n$ for all $n \ge 4$. Given the geometric obstructions in $n...

Let $f$ be a polynomial of degree $n$ over the finite field $\mathbb{F}_p$. We aim to find its Galois group. If $f$ is irreducible and separable, then $\mathbb{F}_{p^n}$ is its splitting field, and its Galois group is $Z/nZ$, which is generated by $\sigma:x\mapsto x^p$. Now, suppose we do not...

I want to find the Galois group $G$ of $f=x^5+2$ over $\mathbb{Q}$. I will write my approach, and I would like to know if there is a faster approach or a more standard one that does not require creativity. Let $\alpha$ be such that $\alpha^5=-2$ and $\zeta$ be the fifth root of unity. Then, the...

An exercise in Lang asks us to find the Galois group of $$f=x^6+22x^5-9x^4+12x^3-37x^2-29x-15$$ over the rationals. I am going to write as far as I can. Then, I will ask how to proceed. I also welcome any other suggestions to solve this problem. Step 1: Reducing mod 2, we get...

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...

My experimental observation suggests that the sum of the reciprocals of the square roots of the products of all multiplicative-inverse pairs modulo a prime $p$ tends to $5$ as $p \to \infty$. More specifically, for each prime $p$, consider the pairs $(a,b)$ satisfying $1\le a,b\le p-1$ and...

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 a strong version of a digit-sum conjecture: Does $S(T_n)$ yield infinitely many distinct prime numbers? Modular and Growth Behavior We know that...

A while ago, @Aleksandr posed this question, about finding new solutions to $$a^8+b^8+c^8+d^8=e^8+f^8+g^8+h^8$$ (where the solutions should be non-trivial and primitive) and he stated the known result that in 2006, Nuutti Kuosa discovered...

The problem was motivated by this related MSE question, although the recurrence here is structurally different. Let $f, g : \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ satisfy $$f(n) \ge n+1, \qquad g(n) \ge (1+\varepsilon)n$$ for some fixed $\varepsilon > 0$. Consider $$a_{n+2} = a_{n+1} \bigl(1 +...

We are to prove that if $n$ is fixed natural number then exists arithmetic $a_n$ and geometric $b_n$ sequences of integers that: $$b_1 < a_1 < b_2 < a_2 < \ldots < b_n < a_n.$$ Sketch It’s equivalent to construct such sequences of fractions – we can always multiply by such large $N$ as is...

I observed experimentally that for every positive integer $a$, the interval $$ (a,11a/5] $$ seems to contain at least $\lfloor\sqrt a\rfloor$ primes. Equivalently, if $\pi(x)$ denotes the prime-counting function, the claim is $$ \pi(11a/5)-\pi(a)\ge \lfloor\sqrt a\rfloor $$ for every positive...