5 problems

Let $A=2^n abcd$ and $B=2^n efgh$ be a symmetric higher-order amicable pair, with $a,b,c,d,e,f,g,h$ denoting the relevant prime factors. Define $x$ and $y$ through the normalized e…

A dendrimer nanostar is the nanostar structure obtained by growing a dendrimer through a specified number of stages; a graph is a $G_\phi$-graph if it is realized as $G_\phi(A)$ fo…

Let $R_0(x):=\sum_{n\leq x}\frac{\varphi(n)}{n}-\frac{x}{\zeta(2)}$ and, for an interval of size $H$, let … Let $H=\Theta(x^\delta)$ for some fixed $0<\delta\leq 1$. Variance conje…

Euler's totient formulation of the ternary Goldbach theorem with a congruence. The system has a solution $(x,y)$ in non-negative integers for every odd integer $n$ greater than fiv…

Euler's totient formulation of the binary Goldbach conjecture. The system has an integral solution for every natural number $n$ distinct from unity. This is presented as an alterna…