arXiv subjects
Publications and source records attributed to N. A. Carella.
A method of constructing specific polynomial representations $f(x)$ over the finite field $\mathbb{F}_p$ of the square roots function modulo a prime $p = 2^kn + 1$, $n$ odd, is presented. The formulas for the cases $k = 2$, $3$ and $4$ are given.
Let $x\geq 1$ be a large integer, and let $μ:\mathbb{N}\longrightarrow\{-1,0,1\}$ be the Mobius function. This article proposes an effective asymptotic result for the autocorrelation function $\sum_{n \leq x} μ(n) μ(n+t) =O\left( e^{-c\sqrt{\log x}}\right) $, where $t\ne 0$ be a small fixed integer, and $c>0$ is a constant.
This note presents new results for the squarefree value sets of quartic polynomials over the integers.
This short note provides a sharper upper bound of a well known inequality for the sum of divisors function. This is a problem in pure mathematics related to the distribution of prime numbers. Furthermore, the technique is completely elementary.
Let $x\geq 1$ be a large integer, and let $a_0 0$ is an arbitrary constant.
Let $E$ be a nonsingular elliptic curve over the rational numbers, and let $τ^n=p^n+1-\#E(\mathbb{F}_{p^n})$. A result in the current literature claims that the normalized traces of Frobenius $α_n=(τ^n+\overlineτ^n)/2p^{n/2}$ are equidistributed on the interval $[-1,1]$ as $n\to \infty$. This short note has two goals: 1. Proposes a correction to this result on the distribution of the traces of Frobenius. 2. Provides a simple elementary proof of this result. More precisely, it is shown that the sequence $\{α_n:n\geq1\}$ is dense but not equidistributed on the interval $[-1,1]$.
This article determines a lower bound for the number Germain primes $p$ and $2p+1$ up to a large number $x$.
Let $α=0.a_1a_2a_3\ldots$ be an irrational number in base $b>1$, where $0\leq a_i<b$. The number $α\in (0,1)$ is a \textit{normal number} if every block $(a_{n+1}a_{n+2}\ldots a_{n+k})$ of $k$ digits occurs with probability $1/b^k$. A proof of the normality of the real number $\sqrt{2}$ in base $10$ is presented in this note. Three different proofs based on different methods are given: a conditional proof, and two unconditional proofs.
This note shows that the product $e π$ of the natural base $e$ and the circle number $π$ is an irrational number.
The equidistribution of the double sign patterns of the Liouville function $λ$ is proved unconditionally. As application, it is shown that the computable real number $$\sum_{n \geq 1} \frac{1+λ(n)}{2^{n}}$$ is a simply normal number in base 4.
The article focuses on the problems of prime gaps and zero spacings. Possible solutions of several related problems such as the greatest lower bound, the least upper bound of the zero spacings, and the least upper bound of the prime gaps are considered.
The existence of infinitely many consecutive prime triples $p_n$, $ p_{n+1}$, and $p_{n+2}$ as $n \to \infty$, is sufficient to prove that the Catalan constant $β(2)=0.9159655941\ldots $ is an irrational number. This note provides the detailed analysis. Moreover, the numerical data suggests that the irrationality measure is $μ(β(2))=2$, the same as almost every irrational real numbers.
This article provides new asymptotic results for the summatory Mobius function $\sum_{p \leq x} μ(p+a) =O \left (x(\log x)^{-c} \right )$ and the summatory Liouville function $\sum_{p \leq x} λ(p+a) =O \left (x(\log x)^{-c} \right )$ over the shifted primes, where $a\ne0$ is a fixed parameter, and $c>1$ is an arbitrary constant. These results improve the current estimates $\sum_{p \leq x} μ(p+a)=(1-δ)π(x)$, and $\sum_{p \leq x} λ(p+a)=(1-δ)π(x)$ for $δ>0$, respectively. Furthermore, a conditional proof for the autocorrelation function $\sum_{p \leq x} μ(p+a)μ(p+b) =O \left (x(\log x)^{-c} \right )$, and an unconditional proof for the autocorrelation function $\sum_{p \leq x} λ(p+a)λ(p+b) =O \left (x(\log x)^{-c} \right )$ over the shifted primes, where $a\ne b$, are also included.
The purpose of this note is to improve the current theoretical results for the correlation functions of the Mobius sequence $\{μ(n): n\geq 1 \}$ and the Liouville sequence $\{λ(n): n\geq 1 \}$.
The first estimate of the upper bound $μ(π)\leq42$ of the irrationality measure of the number $π$ was computed by Mahler in 1953, and more recently it was reduced to $μ(π)\leq7.6063$ by Salikhov in 2008. Here, it is shown that $π$ has the same irrationality measure $μ(π)=μ(α)=2$ as almost every irrational number $α>0$.
A new integer deterministic factorization algorithm, rated at arithmetic operations to $O(N^{1/6+\varepsilon})$ arithmetic operations, is presented in this note. Equivalently, given the least $(\log N)/6$ bits of a factor of the balanced integer $N = pq$, where $p$ and $q$ are primes, the algorithm factors the integer in polynomial time $O(\log(N)^c)$, with $c \geq 0$ constant, and $\varepsilon > 0$ an arbitrarily small number. It improves the current deterministic factorization algorithm, rated at arithmetic operations to $O(N^{1/5+\varepsilon})$ arithmetic operations.
Let $p \geq 2$ be a large prime, and let $k \ll \log p $ be a small integer. This note proves the existence of various configurations of $(k+1)$-tuples of consecutive and quasi consecutive primitive roots $n+a_0, n+a_1, n+a_2, \ldots, n+a_k$ in the finite field $\mathbb{F}_p$, where $a_0,a_1, \ldots, a_k$ is a fixed $(k+1)$-tuples of distinct integers.
Let $z\ne \pm1,w^2$ be a fixed integer, and let $f(t)\ne g(t)^2$ be a fixed polynomial over the integers. It is shown that the subset of primes $p\geq 2$ such that $z$ and $f(z)$ is a pair of simultaneous primitive roots modulo $p$ has nonzero density in the set of primes. The same analysis generalizes to \textit{admissible} $k$-tuple of polynomials $z$, $f_1(z)$, $f_2(z), \ldots$, $f_k(z)$, such that $f_i(z)\ne g_i(z)^2$, and $k\ll \log p$ is a small integer.