1. The Sequence of Prime Numbers

In this paper, we analyze properties of prime number sequences produced by the alternating sum of higher-order subsequences of the primes. We also introduce a new sieve which will generate these prime number sequences via the systematic selection and elimination of prime number indices on the real number line.

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.

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.

1. The Sequence of Prime Numbers

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.

Prime Number Sequence Within Even Numbers

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.

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.

This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together with theorem of Dirichlet, Siegel and Euler allow to prove the infinity of twin primes.