Abstract: A new interpretation of the knots in the M87 jet is presented. It is proposed that they are the sites of dense blobs of gas lying within a directed supersonic jet. Jet kinetic energy is dissipated in each knot behind a strong bow shock, where particle acceleration and magnetic field amplification can occur. The momentum flux incident on these obstacles accelerates them in the direction of the jet, and they should cease to radiate efficiently when their velocity approaches that of the jet. The eight known optical knots may be interpreted as either supernova remnants or interstellar clouds. The five outermost knots could be fragments of one such object which became dynamically unstable.

Abstract: A chain of nearly doubled primes is an ordered set {a 1 , a 2 , ..., aλ} of prime numbers, interlinked by a k =2a k−1 ±1. Chains of length up to 13 have been found. Shorter chains have been counted in some restricted ranges. Some of these counts are compared with the frequencies predicted by a quantitative version of the prime k-tuples conjecture

Abstract: Let G be a finite group, p be a prime number that divides its order, P be a Sylow psubgroup of G, and x be an element of order p of P. The element x (the subgroup ) is said to be isolated in P with respect to G if x ((x)) is not conjugate in G to any element of P {x} (to any subgroup in P {r} ). Glauberman [i] has proved the Z*-theorem that for p = 2 an isolated involution of the group G, each n0ntrivial normal subgroup of which has even order, always belongs to the center Z(G). He posed the following problem: Is the analogue of the Z*-theorem for odd prime numbers p valid (see [I, 2; Problem 4.21])7

Abstract: A method is described for producing tilings with various quasicrystallographic space groups, paying particular attention to the two-dimensional space groups pnm1 and pn1m that can exist as distinct possibilities when the order of rotational symmetry n is a power of an odd prime number.

Abstract: The so-called “principal term” of Goldbach problem, i.e., the principal order of the number of representations of an even number as a sum of two primes, is evaluated here in a way quite different from the classical circle method. It is amazing that the author did this with the knowledge just at the PNT (Prime Number Theorem) level, even without using high-level theorems in analytic number theory as usual, e.g., Siegel-Walfisz Theorem.

Abstract: Based on the sieve of Eratosthenes, a faster and more compact algorithm is presented for finding all primes between 2 and N.

Abstract: In this paper we define n+1-valued matrix logic Kn+1 whose class of tautologies is non-empty iff n is a prime number. This result amounts to a new definition of a prime number. We prove that if n is prime, then the functional properties of Kn+1 are the same as those of Łukasiewicz's n +1-valued matrix logic Łn+1. In an indirect way, the proof we provide reflects the complexity of the distribution of prime numbers in the natural series. Further, we introduce a generalization Kn+1*of Kn+1 such that the set of tautologies of Kn+1 is not empty iff n is of the form pβ, where p is prime and β is natural. Also in this case we prove the equivalence of functional properties of the introduced logic and those of Łn+1. In the concluding part, we discuss briefly a partition of the natural series into equivalence classes such that each class contains exactly one prime number. We conjecture that for each prime number the corresponding equivalence class is finite.

Abstract: Let Q denote the field of rational numbers, and let p be an odd prime number. Let K be a cyclic extension of Q of degree p, and let a be a generator of Gal (KQ). Let CK denote the p-class group of K (i.e., the Sylow p-subgroup of the ideal class group of K), and let for i = 1, 2, 3, . It is well known that is an elementary abelian p-group of rank tt1, where t is the number of ramified primes in KQ. So we focus our attention on . We let

Abstract: We are concerned with the following: If k is a quadratic field and N a cyclic unramified extension of degree qn over k, q a prime number, determine N explicitely via a primitive element β, i.e., N=k(β), in the spirit of Helmut Hasse [3]. We propose a method which determines these extensions, once we are able to specify the arithmetic of a certain field\(k_{\bar \chi } \). To explicit our method, we construct the Hilbert fields of ℚ(√226) and ℚ(√646).

Abstract: Let $G$ be a finite $p$-group of order $p^{m}=p^{2n+e}$, with $n$ a non-negative integer, $p$ a prime number and $e=0$ or $1$, and let $r(G)$ be the number of conjugacy classes of elements of $G$. Then the following equality, due to P. Hall, holds ([4], p. 549) : $$r(G)=(p^{2}-1)n+p^{e}+k(p^{2}-1)(p-1) ,$$ For some non-negative integer $k$. In this paper, we obtain new properties relative to $r(G)$ by the analysis of the number $r_{G}(gN)$ of conjugacy classes of elements of $G$ that intersect the coset $gN$, where $N$ is a normal subgroup of $G$ and $g$ any element of $G$. It contains a number of equations and congruences relating $r(G)$ to other invariants of $G$. In particular, our results improve the above equality of P. Hall, when $G$ has maximal nilpotent class or $n\leq p+1$. Examples are given, which make our improvements evident.

Abstract: The present method generates machine-independent uniform random sequences of real numbers in the interval (0.,1.) excluding 1. It uses a set of up to 1024 independent multiplicative congruential generators working with:• modulii which are chosen prime numbers whose values have been fixed according to the positive 31-bit positive integer arithmetic available and in the form of 2.P'+1, where P's are also primes.• multipliers which are selected from one of their corresponding primitive elements as multipliers to achieve each full cycle independently.The "astronomical" maximum periodicity can be considered as infinite: O (106021); it can be adjusted if required by the user in the sequential version RAN01 or statistically reaching the maximum in the improved "stagger" version DAN01.An "acceptable" composite period is estimated to be O (10189) for a set of only 32 of such independent generators: this fact could find a nice application in the realization of efficient hash-functions in smart cards.An implementation in structured FORTRAN 77 shows very good results in terms of statistical proprieties, velocity and periodicity.

Abstract: Letf be an additjve arithmetical function having a distrl- Introduction. Let f be an additive function. The well known theorem of Erdiis and Wintner gives the necessary and sufficient conditions for the existence of the distribution function off; that is, there exists a distribution function F such that the density of integer m satisfyingf(m) < x exists and equals F(x). These conditions are that the two series converge, where f'(p) = f(p) if If{ p)l =< 1 and f'(p) = I otherwise. Here and in what follows p stands for prime numbers.

Abstract: As I have already stressed, the various proofs of existence of infinitely many primes are not constructive and do not give an indication of how to determine the nth prime number. Equivalently, the proofs do not indicate how many primes are less than any given number N.

Abstract: The utility model provides a '1+1' calculating ruler which not only can be used as an ordinary measuring ruler, but also can display special numbers, prime numbers and composite numbers in integers and can carry out operation. The calculating ruler is composed of a fixed ruler and a slide ruler which have opposite directions and scales, wherein, the fixed ruler and the slide ruler have the same length, the same marking method and symmetrical positions, and the calculation is carried out by making use of different scales on equant rulers and the prime numbers, the composite numbers and the special numbers which are represented by representing methods. The calculating ruler not only can develop the mathematical intelligence of students of primary and middle schools, but also can help people research the Goldbach conjecture of the difficult problem of the world mathematics.

Abstract: where a~0 is an arbitrary fixed integer, u, v are integers satisfying the condition u 2 + v 2 < n, n is a sufficiently big natural number, and m is a natural number, such that ~ (m)~<6, and if p \ m then p~n I/sS~ [~(m) is the number of prime divisors of m, accounting also the multiplicity, p is a prime number]. Equation (I) is an analog of the well-known HardyLittlewood equation [5] on the representation of a number in the form of the sum (or the difference) of a prime number and squares two numbers. A great contribution to the solution of the Hardy-Littlewood problem is that of Hooly [6], Linnik [i] using the dispersion method has unconditionally solved this problem, i.e., he has derived an asymptotic formula for the number of solutions of the equation p + u 2 + : = n .

Abstract: With this chapter we begin the process of finding the primes and factoring the composite integers. The first question that arises is whether or not the list of primes is finite. If it were then we could, at least in theory, publish a book containing all the prime numbers and anyone wanting to determine whether an integer were prime would only have to look it up. Unfortunately, there is no limit to the number of primes, a fact which was known to Euclid.

Abstract: The complexity of residue addition processors that are implemented by 3-D optical interconnections has been investigated. The complexity is measured as the number of switching elements required to perform the operation. The complexity of these processors has been determined for moduli 2 through 17 that are prime or powers of prime numbers. The results are compared with the complexity of the 2-D architecture. >

Abstract: This paper is a follow up of [B1]. It is shown that the sequence of squares {n 2|n=1, 2, ...} contains Λ(p)-subsets of “maximal density”, for any givenp>4. The proof is based on the probabilistic method developed in [B1] and a precise estimate of the Λ(p)-constant for the sequence of squares itself. Analogues of this phenomenon are obtained for other arithmetic sets, such as the sequence ofkth powers {n k |n=1, 2, ...} or the sequence of prime numbers. Sections 2 and 3 of the paper are of independent interest to orthogonal system theory.