Properties of Higher-Order Prime Number Sequences

In green wireless communication, the wireless network is preferred to design a system for low power consumption to minimize the environmental impacts of devices. An antenna is an essential tool in the transceiver system. In a multi-user network environment, large antenna arrays are required that consume high power. This paper proposes a solution using a thinned array antenna (TAA) to minimize energy use by the antenna module. For the array thinning, linear mathematical sequences like the prime number sequence and the Fibonacci sequence are used instead of conventional optimization methods. The antennas are arranged according to these sequences, and the performances are compared with the corresponding fully populated arrays. For a thinned array of 51 elements, using the prime number sequence 36.58% power saving is achieved. For a thinned array of 34 elements, using the Fibonacci sequence 26.5% power saving is achieved. The mathematical sequences (the array sequences) are directly inserted in the program of array thinning. This method doesn’t need any optimization algorithm for array thinning and hardware to implement it. The power consumption in hardware can be avoided. Therefore, the proposed thinned array antenna design using mathematical sequences provides a low-power antenna system for green wireless communication.

During the analysis of the fractal-primorial periodicity of the natural series of numbers, presented in the form of an alternation (sequence) of prime numbers (1 smallest prime factor > 1 of any integer), the regularity of prime numbers distribution was revealed. That is, the theorem is proved that for any integer = N on the segment of the natural series of numbers from 1 to N + 2N: (1) prime numbers are arranged in groups, by exactly three consecutive prime numbers of the form: (Р1-Р2-Р3). In this case, the distance from the first to the third prime number of any group is less than 2N integers, that is, Р3–Р1 < 2N integers. (2) These same prime numbers are redistributed in a line in groups, by exactly two consecutive prime numbers, on all segments of the natural series of numbers shorter than 2Nintegers.

Sequences are fundamental mathematical objects with a long history in mathematics. Sequences are also tools for the development of other concepts (e. g. the limit concept), as well as tools for the mathematization of real-life situations (e. g. growth processes). But, sequences are also interesting objects in themselves, with lots of surprising properties (e. g. Fibonacci sequence, sequence of prime numbers, sequences of polygonal numbers). Nowadays, new technologies provide the possibility to generate sequences, to create symbolic, numerical and graphical representations, to change between these different representations. Examples of some empirical investigation are given, how students worked with sequences in a computer-supported environment.

Euclid's proof that there are infinitely many primes [3, p. 25] can be modified to yield a proof of a simple inequality: If P1, P2'... is the sequence of prime numbers and n ? 2 then PI P2. Pn > P,l+?I In 1907, Bonse [I] gave an elementary proof of a stronger inequality, now called Bonse's Inequality [4]: If n ? 4 then PI P2 . IN > Pn + I Bonse then used his inequality to prove that 30 is the largest integer mi with the following property:

A sequence of prime numbers {p, 2p+1, 4p+3,...,2n-1(p+1)-1} is called a Cunningham chain. These are finite sequences of prime numbers, for which each element is a Sophie Germain prime. It is conjectured that there are arbitrarily large such Cunningham chains, and these chains form an essential part of the study of Sophie Germain primes. In this paper, we aim to significantly improve existing bounds for the length of Cunningham chains by considering their behaviour in the framework of what we will define as rogueness.

1. The Sequence of Prime Numbers

1. The Sequence of Prime Numbers

Prime Number Sequence Within Even Numbers

<p>For any positive integer n ¸ 2, the Smarandache Inferior Prime Part fpp(n)g<br /> is de¯ned as the largest prime less than or equal to n;</p>

In this work, we propose a method for the derivation of prolate spheroidal wave functions (PSWFs) and Slepian functions on continuous and disjoint intervals on the real number line. The proposed method uses Fourier series to obtain a closed-form approximation for Slepian functions on the real line. With this closed-form expression, Slepian functions can be evaluated at arbitrary points in the region of interest with high accuracy. The conventional method uses properties of the Slepian concentration problem to evaluate PSWFs on finite number of points in an interval. The conventional method is computationally expensive and does not allow for easy storage. By approximating an interval containing regions of interest as periodic, we express the Slepian concentration problem as a finite dimensional problem using the Fourier series domain. Solutions to the Slepian concentration problem in this form are Fourier series coefficients corresponding to the Slepian functions. Reconstruction in Fourier series basis, scaling and subsequent truncation provides the closed-form expression for the Slepian problem. Upon comparison with PSWFs obtained by the conventional method, we find negligible difference.

In this work, new types of finite subgroups are given, they are called prime gaps additive subgroups (S, +p) of Zp and prime gaps multiplication subgroups (S, p) of Zp*. Also, new practical algorithm to track prime factors for any composite numbers is introduced. This new method depends on the general rule for prime numbers that has one variable (n). The new rule is very important because it exactly gives the number of primes for any interval of integer numbers. Also, the large gaps between n-th primes are given in this paper. Next, this formula is used to make application by Excel in order to exhibit the results for any integer number if it is prime or not at polynomial time. By this new method, we can find straightforwardly more than (97.5%) of the composite numbers while the rest of the percentage needs polynomial-time. In addition, we can use this rule to exhibit all prime numbers between the smallest prim number and any number x in sequence list. Finally, by this work the following points are considered; The distribution for any sequence of prime numbers is given, Our results better than Riemann Hypothesis (see Tables 3 and 4), For any large integer we can find its factorization in polynomial time by a classical computer.

Let ψ(n)be Euler function,σ(n)denote the sum of divisors of n and d(n)be the divisor function.In this paper we prove the following four theorems.Theorem 1.For any given sequence of κ non-negative numbers α_1,…,α_κand e0,there exists a prime number p such that(?) (1)There exist positive constants с-0(α,e)and Х_0(α,e)such that the numberof primes р satisfying(1)in the interval 1р≤x is greater than(?)Theorem 2 is obtained from Theorem 1 by replacing the letters ψ by σ,с_0 by с-1 and Х_0 by Х_1.Theorem 3.For any given natural number к,there exists a constant γdepending on κ only such that for any given sequence of κ+1 positive integersα_0,α-1,…,α_κ,there exists a prime number р,such that (?) (2)There exist positive constant с_2(α)and Х_2(α)such that the number ofprimes р satisfying(2)in the interval 1≤р≤x is greater than(?)Theorem 4.For any given sequence of κ numbers α_1,…,α_κ,where α_i=0or +∞(1≤i≤κ),there exists an infinite sequence of prime numbers{р_j}(j=1,2,…)such that(?)These theorems improve some results of the author,Schinzel,andShao.The proof of these theorems depends on the followingFundamental Lemma.Let(?)be given natural numbers,where q_(μυ)(0≤μ≤κ,1≤υ≤t_μ)are prime num-bers greater than κ+1 and relatively prime in pairs.If xZ(m_0m_1…m_κ)~2,and N_z(x)denote the number of positive solu-tions(р,x_0,x_1…,x_κ)of the system of equations(?)satisfying the conditions(?)where р and р′denote primes,than there exist positive constants с_3,Х_3 depen-dent only on m_i,and α,dependent only on κ such that(?)If xZ(m_0…m_κ)~2,λ is a given positive number in the interval 1≤λ≤≤(m_0…m_κ)~2 satisfying(λ,m_0…m_κ)=1 and р_1р_2…р_r≤Z are all primenumbers that do not divide m_0…m_κ and do not exceed Z,and if α_(ij)(1≤i≤≤r,1≤j≤κ+1)are given positive numbers satisfying the conditions 1≤α_(ij)р_i and j_≠j_2 impries α_(ij_1)≠α_(ij_2),then we can define M_z(x)as the numberof primes р satisfying the conditions1р≤x,р≡λ(mod(m_0…m_κ)~2),р(?)α_(ij)(mod р_i)(1≤i≤r,1≤j≤κ+1).Fundamental Lemma obviously follows from the following two lemmas.Lemma 1.There exist α_(ij) and λ such that M_z(x)≤N_z(x).Lemma 2.There exist positive constants с_4,Х_4,dependent only on m_i,and positive constant β,dependent only on κ,such that(?)for any given λ and α_(ij).The proof of Lemma 2 depends essentially on the methods of Линник andRe(?)yi.

Among the many techniques of cloning new genes, one approach involves degenerate primers (, , , , , , ). The approach usually requires the following three steps: 1. Using degenerate primers to amplify part of the gene of interest by PCR: The degenerate primers’ sequences may be designed from known protein sequences or conserved regions of a gene family (e.g., ref. ,). Because deoxyinosine can base pair with all of the four deoxyribonucleotides, it has been substituted for specific nucleic acids in degenerate primers to reduce the number of different primer sequences that would otherwise be needed in the reaction (,,). 2. A determination of which amplified PCR product(s) is from the gene of interest: If the target gene and the primers are only partially homologous, a moderate annealing stringency in PCR is usually necessary to obtain amplification. Moderate stringency may result in multiple PCR products. Although from the size of the PCR products it may be possible to predict which is from the gene of interest, sequencing analysis of the PCR products may be required. 3. The screening of a cDNA library using the correct PCR product as a probe and cloning the gene of interest.

This paper presents an exploratory analytic framework for examining the distribution of prime numbers through complex exponential embeddings and their associated Residue Recurrence Classes (RRCs). Each prime p is mapped to a unit-magnitude complex oscillator ω( p )= e ip , allowing the study of angular dynamics modulo 2π within a multiplicative and rotational setting. This representation generalises classical modular analyses and reveals local angular recurrences and clustering within the prime sequence. Residue Recurrence Classes are defined as collections of primes whose angular residues approximate those generated by the integer or fractional roots of smaller primes. Empirical computations suggest that such classes contain numerous primes and exhibit coherent alignments on the unit circle, indicating possible small-scale regularities within the global modular uniformity of primes. We further introduce a quasi-harmonic prime function, formulated as a root-weighted cosine sum over prime angular residues, and analyse the cumulative complex prime centre summation. The latter exhibits an approximate power-law relation between its real and imaginary components, consistent with self-similar or fractal-like scaling. These observations are empirical in nature and do not contradict the asymptotic equidistribution of primes modulo 2π . Rather, they suggest that within finite intervals, the prime sequence may display local harmonic organisation and interference effects that warrant further formal investigation within analytic number theory.

In a general approach to anything measurable one usually takes a real line as a model. Each point on this line belongs to one real number. This point is mathematical in nature, being infinitesimally small, so that no matter how short the interval of the line we select there is always an infinite number of points within the boundaries of the chosen interval. Fig. 1.1 shows such a line with number 4 5 marked. There is no uncertainty regarding this number; it is an ordinary number and as such an element of a familiar infinite set of real numbers r ∈ R +. The real line may in general represent negative and positive numbers as well as number zero — it represents an interval from —∞ to + ∞. The set of all real numbers is 1. = R + ∪ {0} ∪ R - and we are interested in a particular element γ.KeywordsMembership FunctionFuzzy NumberFuzzy SystemDesign ConsiderationFuzzy VariableThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.