On the Erdős-Turán additive base conjecture
In this paper, we formulate and prove several variants of the Erdős-Turán additive bases conjecture.
math.GM↗
arXiv subjects
Theophilus Agama
Publications and source records attributed to Theophilus Agama.
Distribution of boundary points of expansion and application to the lonely runner conjecture
In this paper, we study the distribution of the boundary points of expansion. As an application, we say something about the lonely runner problem. We show that given $k$ runners $\mathcal{S}_i$ round a unit circular track with the condition that at some time $||\mathcal{S}_i-\mathcal{S}_{i+1}||=||\mathcal{S}_{i+1}-\mathcal{S}_{i+2}||$ for all $i=1,2\ldots,k-2$, then at that time we have $$ ||\mathcal{S}_{i+1}-\mathcal{S}_i||>\frac{\mathcal{D}(n)π}{k-1} $$ for all $i=1,\ldots,k-1$ and where $1>\mathcal{D}(n)>0$ is a constant depending on the degree of a certain polynomial of degree $n$. In particular, we show that given at most eight $\mathcal{S}_i$~($i=1,2,\ldots, 8$) runners running around a unit circular track with distinct constant speed and the additional condition $||\mathcal{S}_i-\mathcal{S}_{i+1}||=||\mathcal{S}_{i+1}-\mathcal{S}_{i+2}||$ for all $1\leq i\leq 6$ at some time $s>1$, then at that time their mutual distance must satisfy the lower bound $$ ||\mathcal{S}_{i}-\mathcal{S}_{i+1}||>\frac{Cπ}{7} $$ for some constant $1>C>0$ for all $1\leq i\leq 7$.
math.CO↗
Balanced matrices
In this paper, we introduce a particular class of matrices. We study the concept of a matrix to be \emph{balanced}. We study some properties of this concept in the context of matrix operations. We examine the behaviour of various matrix statistics in this setting. The crux will be to understand the determinants and the eigenvalues of balanced matrices. It turns out that there exist a direct communication among the leading entry, the trace, determinants and, hence, the eigenvalues of these matrices of order $2\times 2$. These matrices have an interesting property that allows us to predict their quadratic forms using their spectrum, without an information about their entries.
math.RA↗
Limited polynomials and sendov's conjecture
In this paper we study a particular class of polynomials. We study the distribution of their zeros, including the zeros of their derivatives as well as the interaction between this two. We prove a weak variant of the sendov conjecture in the case the zeros are real and are of the same sign.
math.CA↗
The spanning method and the Lehmer totient problem
In this paper, we introduce and develop the notion of spanning of integers along functions $f:\mathbb{N}\longrightarrow \mathbb{R}$. We apply this method to a class of problems that requires to determine if the equations of the form $tf(n)=n-k$ has a solution $n\in \mathbb{N}$ for a fixed $k\in \mathbb{N}$ and some $t\in \mathbb{N}$. In particular, we show that \begin{align} \# \{n\leq s~|~tφ(n)+1=n,~\mathbf{for~some}~t\in \mathbb{N}\}\geq \frac{s}{2\log s}\prod \limits_{p | \lfloor s\rfloor }(1-\frac{1}{p})^{-1}-\frac{3}{2}e^γ\nonumber \end{align} as $s\longrightarrow \infty$, where $φ$ is the Euler totient function and $γ=0.5772\cdots$ is the Euler-Macheroni constant.
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Ulam numbers have zero density
In this paper, we show that the natural density $\mathcal{D}[(U_m)]$ of Ulam numbers $(U_m)$ satisfies $\mathcal{D}[(U_m)]=0$. That is, we show that for $(U_m)\subset [1,k]$ \begin{align} \lim \limits_{k\longrightarrow \infty}\frac{\left |(U_m)\cap [1,k]\right |}{k}=0.\nonumber \end{align}
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A proof of the union-close set conjecture
In this paper, we introduce the notion of the universe, induced communities, and cells with their corresponding spots. Using this language, we formulate and prove the union close set conjecture by showing that for any finite universe $\mathbb{U}$ and any induced community $\mathcal{M}_{\mathbb{U}}$ there exist some spot $a\in \mathbb{U}$ such that the density \begin{align} \mathcal{D}_{\mathcal{M}_{\mathbb{U}}}(a)\geq \frac{1}{2}.\nonumber \end{align}
math.GM↗
The diagonalization method and Brocard's problem
In this paper, we introduce and develop the method of diagonalization of functions $f:\mathbb{N}\longrightarrow \mathbb{R}$. We apply this method to show that the equations of the form $Γ_r(n)+k=m^2$ has a finite number of solutions $n\in \mathbb{N}$ with $n>r$ for any fixed $k,r\in \mathbb{N}$, where $Γ_r(n)=n(n-1)\cdots (n-r)$ denotes the $r^{th}$ truncated Gamma function.
math.GM↗
The theory of the Collatz process and the method of dynamical balls
In this paper, we introduce and develop the theory of the Collatz process and the method of dynamical balls. We leverage this theory to study the Collatz conjecture. This theory also has a subtle connection with the infamous problem of the distribution of Sophie Germain primes. We provide several formulations of the Collatz conjecture in this language. Furthermore, we introduce and develop the notion of dynamical systems induced by fixed $a\in \mathbb{N}$ and their associated induced dynamical balls. We develop tools to study problems that require determining the convergence of certain sequences generated by iterating on a fixed integer.
math.GM↗
A proof of the twin prime conjecture
In this paper, we prove the twin prime conjecture showing that \begin{align} \sum \limits_{\substack{p\leq x\\p,p+2\in \mathbb{P}}}1\geq (1+o(1))\frac{x}{2\mathcal{C}\log^2 x}\nonumber \end{align} where $\mathcal{C}:=\mathcal{C}(2)>0$ fixed and $\mathbb{P}$ is the set of all prime numbers. In particular, it implies \begin{align} \sum \limits_{p,p+2\in \mathbb{P}}1=\infty\nonumber \end{align} when we take $x\longrightarrow \infty$ on both sides of the inequality. We start by developing a general method for estimating correlations of the form \begin{align} \sum \limits_{n\leq x}G(n)G(n+l)\nonumber \end{align} for a fixed $1\leq l\leq x$ and where $G:\mathbb{N}\longrightarrow \mathbb{R}^{+}$.
math.GM↗
The Theory of ramification
In this paper, we introduce and develop the concept of \emph{ramification} in a given modulus. We study some properties in relation to this concept and it's connection to some important problems in mathematics, particularly the Goldbach conjecture.
math.NT↗
An extension of the Erdős-Turán additive base conjecture via generalized circles of partition
This paper is an extension program of the notion of circle of partition developed in our first paper \cite{CoP}. As an application we prove the Erdős-Turán additive base conjecture.
math.NT↗
A new proof of some matrix inequalities
In this paper we give alternate proofs of some well-known matrix inequalities. In particular, we show that under certain conditions the inequality holds \begin{align}\sum \limits_{λ_i\in \mathrm{Spec}(ab^{T})}\mathrm{min}\{\log |t-λ_i|\}_{[||a||,||b||]}&\leq \# \mathrm{Spec}(ab^T)\log\bigg(\frac{||b||+||a||}{2}\bigg)\nonumber \\&+\frac{1}{||b||-||a||}\sum \limits_{λ_i\in \mathrm{Spec}(ab^T)}\log \bigg(1-\frac{2λ_i}{||b||+||a||}\bigg).\nonumber \end{align}Also under the same condition, the inequality also holds\begin{align}\int \limits_{||a||}^{||b||}\log|\mathrm{det}(ab^{T}-tI)|dt&\leq \# \mathrm{Spec}(ab^T)(||b||-||a||)\log\bigg(\frac{||b||+||a||}{2}\bigg)\nonumber \\&+\sum \limits_{λ_i\in \mathrm{Spec}(ab^T)}\log \bigg(1-\frac{2λ_i}{||b||+||a||}\bigg).\nonumber \end{align}
math.FA↗
The prime pairs are equidistributed among the coset lattice congruence classes
In this paper we show that for some constant $c>0$ and for any $A>0$ there exist some $x(A)>0$ such that, If $q\leq (\log x)^{A}$ then we have \begin{align} Ψ_z(x;\mathcal{N}_q(a,b),q) &= \frac{Θ(z)}{2ϕ(q)}x + O\bigg(\frac{x}{e^{c\sqrt{\log x}}}\bigg)\nonumber \end{align}for $x\geq x(A)$ for some $Θ(z)>0$. In particular for $q\leq (\log x)^{A}$ for any $A>0$\begin{align}Ψ_z(x;\mathcal{N}_q(a,b),q)\sim \frac{x\mathcal{D}(z)}{2ϕ(q)}\nonumber \end{align}for some constant $\mathcal{D}(z)>0$ and where $ϕ(q)= \# \{(a,b):(p_i,p_{i+z})\in \mathcal{N}_q(a,b)\}$.
math.NT↗
The prime index function
In this paper we introduce the prime index function \begin{align}ι(n)=(-1)^{π(n)},\nonumber \end{align} where $π(n)$ is the prime counting function. We study some elementary properties and theories associated with the partial sums of this function given by\begin{align}ξ(x):=\sum \limits_{n\leq x}ι(n).\nonumber \end{align}We show that a prime $p>2$ is a twin prime if and only if $ξ(p)=ξ(p+2)$. We also relate the prime index function to Cramer's conjecture by showing that \begin{align}|ξ(p_{n+1})-ξ(p_n)|+2=p_{n+1}-p_n.\nonumber \end{align}That is, Cramer's conjecture can be stated as \begin{align}ξ(p_{n+1})-ξ(p_n)\ll (\log p_n)^2.\nonumber \end{align}This reduces the problem to obtaining very good estimates of the second prime index function.
math.GM↗
The theta splitting function
In this paper we study the Theta splitting function $Θ(s+1)$, a function defined on the positive integers. We study the distribution of this function for sufficiently large values of the integers. As an application we show that \begin{align}\sum \limits_{m=0}^{s}\prod \limits_{\substack{0\leq j \leq m\\σ:[0,m]\rightarrow [0,m]\\σ(j)\neq σ(i)}}(s-σ(j))\sim s^s\sqrt{s}e^{-s}\sum \limits_{m=1}^{\infty}\frac{e^m}{m^{m+\frac{1}{2}}}.\nonumber \end{align} and that \begin{align}\sum \limits_{j=0}^{s-1}e^{-γj}\prod \limits_{m=1}^{\infty}\bigg(1+\frac{s-j}{m}\bigg)e^{\frac{-(s-j)}{m}}\sim \frac{e^{-γs}}{\sqrt{2π}}\sum \limits_{m=1}^{\infty}\frac{e^m}{m^{m+\frac{1}{2}}}.\nonumber \end{align}
math.GM↗
On the number of monic admissible polynomials in the ring $\mathbb{Z}[x]$
In this paper we study admissible polynomials. We establish an estimate for the number of admissible polynomials of degree $n$ with coeffients $a_i$ satisfying $0\leq a_i\leq H$ for a fixed $H$, for $i=0,1,2, \ldots, n-1$. In particular, letting $\mathcal{N}(H)$ denotes the number of monic admissible polynomials of degree $n\geq 3$ with coefficients satisfying the inequality $0\leq a_i\leq H$, we show that \begin{align}\frac{H^{n-1}}{(n-1)!}+O(H^{n-2})\leq \mathcal{N}(H) \leq \frac{n^{n-1}H^{n-1}}{(n-1)!}+O(H^{n-2}).\nonumber \end{align} Also letting $\mathcal{A}(H)$ denotes the number of monic irreducible admissible polynomials, with coefficients satisfying the same condition , we show that \begin{align}\mathcal{A}(H)\geq \frac{H^{n-1}}{(n-1)!}+O\bigg( H^{n-4/3}(\log H)^{2/3}\bigg).\nonumber \end{align}
math.NT↗
Improved estimate for the prime counting function $π(x)$
Using some simple combinatorial arguments, we establish some new estimates for the prime counting function and its allied functions. In particular we show that \begin{align}π(x)=Θ(x)+O\bigg(\frac{1}{\log x}\bigg), \nonumber \end{align}where \begin{align}Θ(x)=\frac{θ(x)}{\log x}+\frac{x}{2\log x}-\frac{1}{4}-\frac{\log 2}{\log x}\sum \limits_{\substack{n\leq x\\Ω(n)=k\\k\geq 2\\2\not| n}} \frac{\log (\frac{x}{n})}{\log 2}.\nonumber \end{align}This is an improvement to the estimate \begin{align}π(x)=\frac{θ(x)}{\log x}+O\bigg(\frac{x}{\log^2 x}\bigg)\nonumber \end{align}found in the literature.
math.NT↗