Related papers: Properties of counterexample to Robin hypothesis

A graph $G$ is defined encapsulating the number theoretic notion of the Fundamental Theorem of Arithmetic. We then provide a graph theoretic approach to the fundamental results on the coprimality of two natural numbers, through the use of…

The abc conjecture is one of the most famous unsolved problems in number theory. The conjecture claims for each real $\epsilon > 0$ that there are only a finite number of coprime positive integer solutions to the equation $a+b = c$ with $c…

Let $\sigma(n)$ be the sum of the divisors of $n$. Kalita and Saikia defined a number $n$ to be near superperfect if $2n+d=\sigma(\sigma(n))$ for some positive divisor $d$ of $n$. We extend some of their results about near superperfect…

Wright proved that there exists a number $c$ such that if $g_0 = c$ and $g_{n+1} = 2^{g_n}$, then $\lfloor g_n \rfloor$ is prime for all $n > 0$. Wright gave $c = 1.9287800$ as an example. This value of $c$ produces three primes, $\lfloor…

We show that if $\gamma \in \omega \cup \{\aleph_{0}\}$ and $\mathcal{A}$ is a counterexample to the analytic Vaught conjecture having exactly $\gamma$ many models of Scott rank $\omega_{1}$, then there exists a club $C \subseteq…

We construct a family of countexamples to a conjecture of Galvin [5], which stated that for any $n$-vertex, $d$-regular graph $G$ and any graph $H$ (possibly with loops), \[\hom(G,H) \leq \max\left\lbrace\hom(K_{d,d}, H)^{\frac{n}{2d}},…

The inequality is computed through the so-called Gini index. The population is assumed to have the variable of interest distributed according to the Gamma probability distribution. The results show that the Gini index is reduced when the…

Shelah Spencer [ShSp:304] proved the 0-1 law for the random graphs G(n,p_n), p_n=n^{- alpha}, alpha in (0,1) irrational (set of nodes in [n]= {1, ...,n}, the edges are drawn independently, probability of edge is p_n). One may wonder what…

We prove that there exists a constant $\gamma_{\mathrm{crit}}\approx .17566$ such that if $G\sim \mathbb{G}(n,1/2)$ then for any $\varepsilon > 0$ with high probability $G$ has a equipartition such that each vertex has…

We discuss many surprising implications of a positive answer to a question raised in some cases by Greenberg in the $`70$s and more generally by Shalom in the early $2000$s. We refer to this positive answer as the Greenberg-Shalom…

Assuming the Riemann Hypothesis we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Omega-term, we prove that our result is essentially the best…

Let $q_n$ denote the $n^{th}$ number that is a product of exactly two distinct primes. We prove that $$\liminf_{n\to \infty} (q_{n+1}-q_n) \le 6.$$ This sharpens an earlier result of the authors (arXivMath NT/0506067), which had 26 in place…

We prove that if \Gamma is subgroup of Diff_{+}^{1+\epsilon}(I) and N is a natural number such that every non-identity element of \Gamma has at most N fixed points then \Gamma is solvable. If in addition \Gamma is a subgroup of…

Let $P$ be the set of all prime numbers, ${q_1},{q_2}, \cdots ,{q_m} \in P$, $P_k$ be the k-th $(k = 1,2, \cdots m)$ element of $P$ in ascending order of size, ${\alpha _1},{\alpha _2}, \cdots ,{\alpha _m}$ be positive integers, and ${\beta…

Let Omega(n) be the volume of the unit ball in R^n. We formulate as an infinite product the gamma function ratio gamma(x+1/2)/gamma(x),x>0, which allows us to reproduce and /or produce a variety of formulas and inequalities, some of them…

Exact and asymptotic formulae are displayed for the coefficients $\lambda_n$ used in Li's criterion for the Riemann Hypothesis. In particular, we argue that if (and only if) the Hypothesis is true, $\lambda_n \sim n(A \log n +B)$ for $n \to…

For a positive integer $n$, we denote by $F(n)$ the distance from $n$ to the nearest prime number. We prove that every sufficiently large positive integer $N$ can be represented as the sum $N=n_1+n_2$, where $$ F(n_i) \geqslant (\log…

For a finite group $G$ denote by $\gamma(L(G))$ the genus of the subgroup graph of $G.$ We prove that $\gamma(L(G))$ tends to infinity as either the rank of $G$ or the number of prime divisors of $|G|$ tends to infinity.

Let F(G) be a fixed linear combination of the k extremal eigenvalues of a graph G and of its complement. The problem of finding max{F(G):v(G)=n} generalizes a number of problems raised previously in the literature. We show that the limit…

The covering number of a group $G$, denoted by $\sigma(G)$, is the size of a minimal collection of proper subgroups of $G$ whose union is $G$. We investigate which integers are covering numbers of groups. We determine which integers $129$…