Prime Research Articles

Distribution of the prime numbers

Distribution of the prime numbers

A class of constacyclic codes of length 2pˢ over Fp^m[u,v] / ⟨u², v², uv − vu⟩

Let p be an odd prime number. This paper investigates λ-constacyclic codes of length 2ps over the ring Rᵤ,ᵥ = Fpm[u,v] / ⟨u², v², uv − vu⟩, where λ = ψ + ρu + ηv + τuv, with ψ, ρ, η, τ ∈ Fpm, ψ ∈ F*pm, and ρ, η not both zero. When λ = μ² for some μ ∈ Rᵤ,ᵥ, the structures of all λ-constacyclic codes of length 2ps over Rᵤ,ᵥ are determined and can be expressed as the sum of a (−μ)-constacyclic code and a μ-constacyclic code of length ps. When λ is not a square, we classify these codes into four distinct types, provide the number of codewords for each type, and give the details of their dual codes.

Electronic Fourier–Galois Spectrum Analyzer for the Field GF(31)

A scheme for the Fourier–Galois spectrum analyzer for the field GF(31) is proposed. It is shown that this analyzer allows for solving a wide enough range of problems related to image processing, in particular those arising in the course of experimental studies in the field of physical chemistry. Such images allow digital processing when divided into a relatively small number of pixels, which creates an opportunity to use Galois fields of relatively small size. The choice of field GF(31) is due to the fact that the number 31 is a Mersenne prime number, which considerably simplifies the algorithm of calculating the Fourier–Galois transform in this field. The proposed scheme of the spectrum analyzer is focused on the use of threshold sensors, at the output of which signals corresponding to binary logic are formed. Due to this fact, further simplification of the proposed analyzer scheme is achieved. The constructiveness of the proposed approach is proven using digital modeling of electronic circuits. It is concluded that when solving applied problems in which an image can be divided into a relatively small number of pixels, it is important to take into account the specificity of particular Galois fields used for their digital processing.

Prime number factorization and degree of coherence of speckled light beams.

We discover a connection between a Gauss sum of number theory and the degree of coherence (DOC) of the field in a transverse plane of structured speckled light beams. We theoretically demonstrate and experimentally validate that prime number factorization can be achieved by manipulating the source beam's DOC in Young's double-slit experiment. The determination of whether a number can be factored is based solely on the visibility of the resulting interference patterns. Our findings offer new insights into information encryption and decryption, data compression, etc.

Calculation of Nielsen periodic numbers on infra-solvmanifolds

Recently, a formula for computing the Nielsen periodic numbers NFn(f) and NPn(f) of self maps f on infra-nilmanifolds and infra-solvmanifolds of type (R) was found. In this paper, we extend this formula to the case of general infra-solvmanifolds. We show that infra-solvmanifolds are essentially reducible to the GCD and essentially toral, and determine conditions under which NFn(f)=N(fn). We show that the prime Nielsen-Jiang periodic number NPn(f) of a self map f on an infra-solvmanifold M can be calculated by Nielsen numbers of lifts of suitable iterates of f to an NR-solvmanifold that finitely covers M.

A new bound for A(A + A) for large sets

For a large prime number p and a set A⊂Fp we prove the following:(1)If A(A+A) does not cover all nonzero residues in Fp, then |A|⩽p/8+o(p).(2)If A is both sum-free and satisfies A=A⁎, then |A|⩽p/9+o(p).(3)If |A|≫log⁡log⁡plog⁡pp, then |A+A⁎|⩾(1+o(1))min⁡(2|A|p,p). Here the constants 1/8, 1/9, 2 are the best possible. Proofs make use of wrappers, subsets of a finite abelian group G, which ‘wrap’ popular values in convolutions of dense sets A,B⊆G. These objects carry certain structural features, making them capable of addressing additive-combinatorial and enumerative problems.

The Riemann's Hypothesis, the Prime Numbers Theorem (PNT), and the Error

In this simple paper, a small refinement to the Prime Number Theorem (PNT) is proposed, which allows us to limit the error with which said theorem predicts the value of the Prime-counting function π(x); and, in this way, endorse the veracity of the Riemann Hypothesis. Many people know that the Riemann Hypothesis is a difficult mathematical problem - even to understand - without a certain background in mathematics. Many techniques have been used, for more than 150 years, to try to solve it. Among them is the one that establishes that, if the Riemann hypothesis is true, then the error term that appears in the prime number theorem can be bounded in the best possible way. Specifically, Helge von Koch demonstrated in 1901 that it should be:

Prime number-based multistep measurement for separation of roundness errors

Roundness measurement is critical in manufacturing, as it ensures that products conform to precise design specifications. However, traditional multistep measurements for roundness error separation are time-consuming and limited in their ability to separate specific Fourier components. In this study, we propose a novel combined multistep measurement method with prime numbers that overcomes these limitations. We demonstrate this method through three experimental cases, achieving high levels of Fourier components in error separation with a limited number of measurements. Our method combines two ( p and q) or three ( p, q, and r) steps of prime numbers to achieve high levels of Fourier components for error separation, compared to traditional multistep measurements that require more steps. In the first experimental case, we use a 2-step and 5-step measurement to achieve traditional multistep measurement in ten steps. In the second case, we use 3-step and 5-step measurements, and in the third, we combine the 2-step, 3-step, and 5-step measurements. We achieve roundness deviations (RONt) of 12.7, 7.8, and 9.9 nm, respectively, and maximum En-values of 0.8, 0.8, and 0.7, respectively. Our proposed combined multistep measurement method using prime numbers has practical applications in manufacturing, as it reduces the time and resources required for roundness error separation while achieving a higher level of Fourier components. Our results demonstrate the effectiveness of our method and its potential to revolutionize roundness measurement in industry.

Detecting nontrivial products in the stable homotopy ring of spheres via the third Morava stabilizer algebra

Let p ≥ 7 p \geq 7 be a prime number. Let S ( 3 ) S(3) denote the third Morava stabilizer algebra. In recent years, Kato-Shimomura and Gu-Wang-Wu found several families of nontrivial products in the stable homotopy ring of spheres π ∗ ( S ) \pi _* (S) using H ∗ , ∗ ( S ( 3 ) ) H^{*,*} (S(3)) . In this paper, we determine all nontrivial products in π ∗ ( S ) \pi _* (S) of the Greek letter family elements α s , β s , γ s \alpha _s, \beta _s, \gamma _s and Cohen’s elements ζ n \zeta _n which are detectable by H ∗ , ∗ ( S ( 3 ) ) H^{*,*} (S(3)) . In particular, we show β 1 γ s ζ n ≠ 0 ∈ π ∗ ( S ) \beta _1 \gamma _s \zeta _n \neq 0 \in \pi _*(S) , if n ≡ 2 n \equiv 2 mod 3, s ≢ 0 , ± 1 s \not \equiv 0, \pm 1 mod p p .

The definable content of homological invariants I: Ext$\mathrm{Ext}$ and lim1$\mathrm{lim}^1$

AbstractThis is the first installment in a series of papers illustrating how classical invariants of homological algebra and algebraic topology may be enriched with additional descriptive set theoretic information. To effect this enrichment, we show that many of these invariants may be naturally regarded as functors to the category, introduced herein, of groups with a Polish cover. The resulting definable invariants provide far stronger means of classification. In the present work we focus on the first derived functors of and . The resulting definable for pairs of countable abelian groups and definable for towers of Polish abelian groups substantially refine their classical counterparts. We show, for example, that the definable is a fully faithful contravariant functor from the category of finite‐rank torsion‐free abelian groups with no free summands; this contrasts with the fact that there are uncountably many non‐isomorphic such groups with isomorphic classical invariants . To facilitate our analysis, we introduce a general Ulam stability framework for groups with a Polish cover; within this framework we prove several rigidity results for non‐Archimedean abelian groups with a Polish cover. A special case of our main result answers a question of Kanovei and Reeken regarding quotients of ‐adic groups. Finally, using cocycle superrigidity methods for profinite actions of property (T) groups, we obtain a hierarchy of complexity degrees for the problem of classifying all group extensions of by up to base‐free isomorphism, when for prime numbers and .

A smooth version of Landau’s explicit formula

In this paper, we present a smooth version of Landau’s explicit formula for the von Mangoldt arithmetical function. By assuming the validity of the Riemann hypothesis, we show that in order to determine whether a natural number [Formula: see text] is a prime number, it is sufficient to know the location of a number of nontrivial zeros of the Riemann zeta function of order [Formula: see text]. Next we use Heisenberg’s inequality to support the conjecture that this number of zeros cannot be essentially diminished.

Generalizations of the Gandhi formula for prime numbers

Generalizations of the Gandhi formula for prime numbers

An Approximation of the Prime Counting Function and a New Representation of the Riemann Zeta Function

Determining the exact number of primes at large magnitudes is computationally intensive, making approximation methods (e.g., the logarithmic integral, prime number theorem, Riemann zeta function, Chebyshev’s estimates, etc.) particularly valuable. These methods also offer avenues for number-theoretic exploration through analytical manipulation. In this work, we introduce a novel approximation function, ϕ(n), which adds to the existing repertoire of approximation methods and provides a fresh perspective for number-theoretic studies. Deeper analytical investigation of ϕ(n) reveals modified representations of the Chebyshev function, prime number theorem, and Riemann zeta function. Computational studies indicate that the difference between ϕ(n) and the logarithmic integral at magnitudes greater than 10100 is less than 1%.

An Efficient Algorithm for Sorting and Duplicate Elimination by Using Logarithmic Prime Numbers

Data structures such as sets, lists, and arrays are fundamental in mathematics and computer science, playing a crucial role in numerous real-life applications. These structures represent a variety of entities, including solutions, conditions, and objectives. In scenarios involving large datasets, eliminating duplicate elements is essential to reduce complexity and enhance performance. This paper introduces a novel algorithm that uses logarithmic prime numbers to efficiently sort data structures and remove duplicates. The algorithm is mathematically rigorous, ensuring correctness and providing a thorough analysis of its time complexity. To demonstrate its practicality and effectiveness, we compare our method with existing algorithms, highlighting its superior speed and accuracy. An extensive experimental analysis across one thousand random test problems shows that our approach significantly outperforms two alternative techniques from the literature. By discussing the potential applications of the proposed algorithm in various domains, including computer science, engineering, and data management, we illustrate its adaptability through two practical examples in which our algorithm solves the problem more than 3×104 and 7×104 times faster than the existing algorithms in the literature. The results of these examples demonstrate that the superiority of our algorithm becomes increasingly pronounced with larger problem sizes.

Robust Left-Right Hashing Scheme for Ubiquitous Computing

Ubiquitous computing systems possess the capability to collect and process data, which is subsequently shared with other devices. These systems encounter resource challenges such as memory constraints, processor speed limitations, power consumption considerations, and the availability of data storage. Therefore, maintaining data access and query processing speed in ubiquitous computing is challenging. Hashing is crucial to search operations and has caught the interest of many researchers. Several hashing techniques have been proposed and Cuckoo Hashing is found efficient to use in several applications. There are two variants of Cuckoo Hashing: Parallel Cuckoo Hashing and Sequential Cuckoo Hashing. Cuckoo Hashing suffers from challenges like high insertion latency, inefficient memory usage, and data migration. This paper proposes two hashing schemes: Left-Right Hashing and Robust Left-Right Hashing that successfully address and solve the major challenges of Sequential Cuckoo Hashing. The proposed schemes adopt the Combinatorial Hashing technique after modification and use this with a new collision resolution technique called Left-Right Random Probing. Left-Right Random Probing is a variant of random probing and uses prime numbers and Fibonacci numbers. In addition, this paper proposes a new performance indicator, degree of dexterity to estimate the performance of hashing techniques. Sequential Cuckoo Hashing suffers from hidden switching costs which are identified and its estimation is given by a new performance indicator called, T.R.C./Key. Performance of Sequential Cuckoo Hashing is order dependent.

Representation and Generation of Prime and Coprime Numbers by Using Structured Algebraic Sums

The algebraic structure and distribution of prime numbers remain two of the most fundamental problems in mathematics. The Fundamental Theorem of Arithmetic, proved by Euclid, and Goldbach’s conjecture, while universal in scope with respect to how numbers can be represented multiplicatively or additively, do not provide insights into the structure of primes. Similarly, the definition of a prime −as a number divisible only by 1 and itself− or a sieve algorithm, commonly used to generate primes by successively eliminating multiples, offer no insight into the structure of primes. The powerful and persistent consideration of prime numbers as universal “arithmetic quanta” has not necessitated an equally powerful need for parallel research into a deeper and possibly more insightful explanation of primeness, that is, a better understanding of “why” a number is prime. In this paper, prime and coprime numbers are represented and generated by algebraic expressions. Specifically, given the first <i>n</i> primes, <i>p<sub>1</sub></i>,<i> p<sub>2</sub></i>,…, <i>p<sub>n</sub></i>, sufficient conditions are given for expressing primes greater than <i>p<sub>n</sub></i>, and coprimes with prime factors greater than <i>p<sub>n</sub></i>, as algebraic functions of <i>p<sub>1</sub></i>,<i> p<sub>2</sub></i>,…, <i>p<sub>n</sub></i>. Thus, primality and co-primality are shown to be mathematical properties with inherently evolutionary algebraic characteristics, since larger primes and coprimes can be generated algebraically from smaller ones. The methodology described in the paper can be a useful tool in the study and analysis of the complexity, structure, interrelationships and distribution of primes and coprimes.

Investigating and improving student understanding of the basics of quantum computing

Quantum information science and engineering (QISE) is a rapidly developing field that leverages the skills of experts from many disciplines to utilize the potential of quantum systems in a variety of applications. It requires talent from a wide variety of traditional fields, including physics, engineering, chemistry, and computer science, to name a few. To prepare students for such opportunities, it is important to give them a strong foundation in the basics of QISE, in which quantum computing plays a central role. In this study, we discuss the development, validation, and evaluation of a Quantum Interactive Learning Tutorial, on the basics and applications of quantum computing. These include an overview of key quantum mechanical concepts relevant to quantum computation (including ways a quantum computer is different from a classical computer), properties of single- and multiqubit systems, and the basics of single-qubit quantum gates. The tutorial uses guided inquiry-based teaching-learning sequences. Its development and validation involved conducting cognitive task analysis from both expert and student perspectives and using common student difficulties as a guide. For example, before engaging with the tutorial, after traditional lecture-based instruction, one reasoning primitive that was common in student responses is that a major difference between an N-bit classical and N-qubit quantum computer is that various things associated with a number N for a classical computer should be replaced with the number 2N for a quantum computer (e.g., 2N qubits must be initialized and 2N bits of information are obtained as the output of the computation on the quantum computer). This type of reasoning primitive also led many students to incorrectly think that there are only N distinctly different states available when computation takes place on a classical computer. Research suggests that this type of reasoning primitive has its origins in students learning that quantum computers can provide exponential advantage for certain problems, e.g., Shor’s algorithm for factoring products of large prime numbers, and that the quantum state during the computation can be in a superposition of 2N linearly independent states. The inquiry-based learning sequences in the tutorial provide scaffolding support to help students develop a functional understanding. The final version of the validated tutorial was implemented in two distinct courses offered by the physics department with slightly different student populations and broader course goals. Students’ understanding was evaluated after traditional lecture-based instruction on the requisite concepts and again after engaging with the tutorial. We analyze and discuss their improvement in performance on concepts covered in the tutorial. Published by the American Physical Society 2024

Sophie Germain prime p and the permutation of product of first p cycles

For a natural number n, the permutation (n!) is defined as the left-to-right product of the first n cycles, namely, . In this article, we prove that for any natural number n, 2 is a primitive root of 2n + 1 if and only if 2n + 1 = pk for some odd prime number p and for some natural number k such that the permutation (n!) has exactly k orbits. We also prove that a prime number p is a Sophie Germain prime if and only if the permutation (p!) has at most two orbits.

The Wiener Index of Prime Graph $PG(\mathbb{Z}_n)$

Let G = (V, E) be a simple graph and (R, +, ·) be a ring with zero element 0R. The Wiener index of G, denoted by W(G), is defined as the sum of distances of every vertex u and v, or half of the sum of all entries of its distance matrix. The prime graph of R, denoted by P G(R), is defined as a graph with V (P G(R)) = R such that uv ∈ E(P G(R)) if and only if uRv = {0R} or vRu = {0R}. In this article, we determine the Wiener index of P G(Zn) in some cases n by constructing its distance matrix. We partition the set Zn into three types of sets, namely zero sets, nontrivial zero divisor sets, and unit sets. There are two objectives to be achieved. Firstly,we revise the Wiener index formula of P G(Zn) for n = p2 and n = p3 for prime number p and we compare this results with the results carried out by previous researchers. Secondly, we determine the Wiener index formula of P G(Zn) for n = pq, n = p2q, n =p 2q2, and n = pqr for distinct prime numbers p, q, and r.

Explicit continued fraction expansion of a rational root of 1+x−1$1+x^{-1}$ over Fp$\mathbb {F}_p$

AbstractLet be a prime number, and positive integers coprime with . We provide the explicit continued fraction expansion of the th root of in the power series field . We determine its irrationality exponent.