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"Numbers, Prime"
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Prime numbers, friends who give problems : a trialogue with Papa Paulo
Prime numbers, friends who give problems, is written as a trialogue, with two persons who are interested in prime numbers asking the author, Papa Paulo, intelligent questions. Starting at a very elementary level, the book advances steadily, covering all important topics of the theory of prime numbers, up to the most famous problems. The humorous conversations and the inclusion of a back-story add to the uniqueness of the book. Concepts and results are also explained with great care, making the book accessible to a wide audience. -- Provided by publisher.
Book
Prime Number Sieving—A Systematic Review with Performance Analysis
The systematic generation of prime numbers has been almost ignored since the 1990s, when most of the IT research resources related to prime numbers migrated to studies on the use of very large primes for cryptography, and little effort was made to further the knowledge regarding techniques like sieving. At present, sieving techniques are mostly used for didactic purposes, and no real advances seem to be made in this domain. This systematic review analyzes the theoretical advances in sieving that have occurred up to the present. The research followed the PRISMA 2020 guidelines and was conducted using three established databases: Web of Science, IEEE Xplore and Scopus. Our methodical review aims to provide an extensive overview of the progress in prime sieving—unfortunately, no significant advancements in this field were identified in the last 20 years.
Journal Article
Prime number theorem for regular Toeplitz subshifts
We prove that neither a prime nor an l-almost prime number theorem holds in the class of regular Toeplitz subshifts. But when a quantitative strengthening of the regularity with respect to the periodic structure involving Euler’s totient function is assumed, then the two theorems hold.
Journal Article
Prime suspects : the anatomy of integers and permutations
Integers and permutations--two of the most basic mathematical objects--are born of different fields and analyzed with separate techniques. Yet when the Mathematical Sciences Investigation team of crack forensic mathematicians, led by Professor Gauss, begins its autopsies of the victims of two seemingly unrelated homicides, Arnie Integer and Daisy Permutation, they discover the most extraordinary similarities between the structures of each body.
Book
Distribution of monomial-prime numbers and Mertens sum evaluations
In this paper, we mainly study the monomial-prime numbers, which are of the form
p
n
k
for primes
p
and integers
k
≥
2
. First, we give an asymptotic estimate on the number of numbers of a general form
pf
(
n
) for arithmetic functions
f
satisfying certain growth conditions, which generalizes Bhat’s recent result on the Square-Prime Numbers. Then, we prove three Mertens-type theorems related to numbers of a more general form, partially extending the recent work of Qi-Hu, Popa and Tenenbaum on the Mertens sum evaluations. At the end, we evaluate the average and variance of the number of distinct monomial-prime factors of positive integers by applying our Mertens-type theorems.
Journal Article
A dynamical approach to the asymptotic behavior of the sequence
We study the asymptotic behavior of the sequence$ \\{\\Omega (n) \\}_{ n \\in \\mathbb {N} } $from a dynamical point of view, where$ \\Omega (n) $denotes the number of prime factors of$ n $counted with multiplicity. First, we show that for any non-atomic ergodic system$(X, \\mathcal {B}, \\mu , T)$, the operators$T^{\\Omega (n)}: \\mathcal {B} \\to L^1(\\mu )$have the strong sweeping-out property. In particular, this implies that the pointwise ergodic theorem does not hold along$\\Omega (n)$. Second, we show that the behaviors of$\\Omega (n)$captured by the prime number theorem and Erdős–Kac theorem are disjoint, in the sense that their dynamical correlations tend to zero.
Journal Article
The nth Prime Exponentially
Consider both the Logarithmic integral, Li(x)=limϵ→0∫01−ϵdulnu+∫1+ϵxdulnu, and the prime counting function π(x)=∑p≤x1. From several recently developed known effective bounds on the prime counting function of the general form |π(x)−Li(x)|Li−1n1−a(ln[nlnn])b+1exp−cln[nlnn] for n≥n∗. Herein, the range of validity is explicitly bounded by some calculable constant n∗ satisfying n∗≤maxπ(x0),π(17),π((1+e−1) exp2(b+1)c2). These bounds provide very clean and up-to-date and explicit information on the location of the nth prime number. Many other fully explicit bounds along these lines can easily be developed. Overall this article presents a general algorithmic approach to converting bounds on |π(x)−Li(x)| into somewhat clearer information regarding the primes.
Journal Article
Entropy, Periodicity and the Probability of Primality
The distribution of prime numbers has long been viewed as a balance between order and randomness. In this work, we investigate the relationship between entropy, periodicity, and primality through the computational framework of the binary derivative. We prove that periodic numbers are composite in all bases except for a single trivial case and establish a set of twelve theorems governing the behavior of primes and composites in terms of binary periodicity. Building upon these results, we introduce a novel scale-invariant entropic measure of primality, denoted p(s′), which provides an exact and unconditional entropic probability of primality derived solely from the periodic structure of a binary number and its binary derivatives. We show that p(s′) is quadratic, statistically well-defined, and strongly correlated with our earlier BiEntropy measure of binary disorder. Empirical analyses across several numerical ranges demonstrate that the variance in prime density relative to quadratic expectation is small, binormal, and constrained by the central limit theorem. These findings reveal a deep connection between entropy and the randomness of the primes, offering new insights into the entropic structure of number theory, with implications for the Riemann Hypothesis, special classes of primes, and computational applications in cryptography.
Journal Article
A Diophantine Inequality Involving Mixed Powers of Primes with a Specific Type
Let λ1,λ2,λ3 be nonzero real numbers, not all of the same sign; let λ1/λ2 be irrational; and let η be any real number. We investigate the solvability of the inequality |λ1p1+λ2p2+λ3p32+η|<(maxpj)−1/12+θ, θ>0 in the prime variables p1, p2, and p3. We require that p1+2 and p2+2 have no more than 20 prime factors, while p3+2 has no more than 42 prime factors.
Journal Article