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Showing 1 to 20 of 22 for “"prime numbers"”.
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Historic development of prime numbers
… of this thesis is to investigate the history of prime numbers and development of prime number theory. There are three major sections to this thesis, Ancient times, Dark Ages, and Modern times. The ancient time's section has topics on the `Ishango Bone', `Rhine Papyrus' with an investigation of …
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Asymptotic Distribution of Beurling's Generalized Prime Numbers and Integers
… of Beurling's generalized (henceforth g-) prime numbers and integers Acta Math. 1937 . We call P = p(,i) (,i=1)('(INFIN)), where 1 ) (INFIN), a set of g-primes. The set of all products of g-primes is called the associated set of g-integers. Define summatory functions N(x), (psi)(x), (PI)(x) …
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Self Dual Codes and the Indecomposable Building Blocks
<p>Just like prime numbers are to integers, indecomposable codes are to self dual codes. This paper gives an explicit listing of the first few families of binary self dual codes, up to length 16. Binary self dual codes that are decomposable are given as a composition of indecomposable codes. The …
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Analysing rounding data using radial basis function neural networks model
… were just systematically lumped to certain based numbers reflecting the rounding process or counting behaviour. A new method to analyse rounded data would therefore be academically valuable. The neural network model developed in this study fills the gap and serves the purpose by complementing and …
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An Exposition Of Dirichlet’s Theorem
… Euclid probably knew there were infinitely many primes, Euclid was the first to provide a proof of the fact. Since then, mathematicians have asked much more detailed and difficult questions about the location and size of the prime numbers. Arithmetic progressions are very easily described subsets …
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Topics in analytic number theory
We investigate properties of prime numbers and L-functions, and interactions between these two topics. First, we discuss the problem of primes in thin sequences, expanding on work of Maynard and Friedlander-Iwaniec. Next, motivated by work of Iwaniec and Sarnak, we study the question of average …
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Higher Derivatives of the Hurwitz Zeta Function
… that ζ(s) is closely connected to the prime numbers and Riemann gave proofs of the basic analytic properties of the zeta function. Values of the zeta function and its derivatives have been studied by several mathematicians. Apostol in particular gave a computable formula for the values …
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The Factoradic Integers
… integers are then extended to a larger set of numbers, the factoradic integers. The properties of the factoradic integers are developed in detail, with particular attention to prime factorization, exponentiation, infinite series, and continuous functions, as well as to polynomials and their …
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Variations on the Theme of Higher Dimensional Weisfeiler-Leman Algorithms
… parametrised by a natural number and a set of prime numbers. By generalising the concept of coherent algebras to an arbitrary field, we show that the invertible map tests can be viewed as a representation theoretic generalisation of the Weisfeiler-Leman algorithms. We also show that the …
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On Beurling's Theory of Generalized Numbers
… 255-291), we consider a set of generalized prime numbers $P=\{1<p\sb1\le p\sb2\le\...\}$ and the set of generalized integers $N=\{n\sb1=1\le n\sb2\le\...\}$ generated by P. We let $N(x)$ be the counting function of the set N. In this thesis we give continuous versions of generalized number …
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On Weak Number Theories
… of integers of a finite extension of rational numbers. In Chapter 3 we give a necessary condition for a set to be (FOR ALL)('n)(THERE EXISTS)-diophantine definable over R. From this necessary condition we can show that many subsets of R including N and cofinite subsets, are not (FOR …
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ΠΡΩΤΟΙ ΑΡΙΘΜΟΙ ΚΑΙ ΚΥΚΛΟΤΟΜΙΑ-ΠΡΩΤΟΙ ΑΡΙΘΜΟΙ ΤΗΣ ΜΟΡΦΗΣ Χ + (Χ+1)
… NAMELY: CHAPTER 1: CRITERIA AND FORMULAE FOR PRIME NUMBERS. CHAPTER 2: PRIMES AND CYCLOTOMY. CHAPTER3: PRIME AND COMPOSITE NUMBERS OF THE FORM X2+(X+1)2. IN CHAPTER 1 NEW PRIMALITY CRITERIA ARE OBTAINED, ALONG WITH NEW FORMULAE FOR THE SEQUENCE OF PRIMES (RECURSIVE AND EXPLICIT). SUCH FORMULAE …
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Irreducibility Criteria For Polynomials With Non-Negative Integer Coefficients, and the Prime Factorization of F(N) For F(X) In Z[X]
<p>We explore two specific connections between prime numbers and polynomials. Cohn's Criterion states that if $d_nd_{n-1}\ldots d_0$ is the base $10$ representation of a prime, then the polynomial $\sum_{j=0}^n d_jx^j$ is irreducible. Let $f(x)$ be a polynomial with non-negative integer …
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A comparison of threaded versus non-threaded computing on the sieve of Eratosthenes Algorithm
… the sequential program for all threads computing prime numbers in the ranges greater than 10,000. The results of the study also determined that there existed a positive statistically significant relationship between the number of threads employed and the overhead incurred. In addition to this, the …
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Prime Number-Based Hierarchical Data Labeling Scheme for Relational Databases
… path model uses the unique characteristics of prime numbers for its labeling purposes. However, the performance and space utilization of this prime number labeling scheme could be significantly improved. This research introduces a new scheme called reusable prime number labeling (rPNL) that …
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Some topics in analytic and probabilistic number theory
… The other two problems are about smooth numbers (i.e. numbers only having small prime factors), both in their own right and in their application to finding solutions to S-unit equations over the integers. Thus all four problems are concerned, in different ways, with _understanding the …
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Digital processing of system responses
… sequences can be expressed as multiples of prime numbers the mixed-radix fast Fourier transform is suitable and has been mechanised successfully. Errors when using the procedure are presented taking quantisation levels, sampling rates, sequence lengths and smoothing techniques into …
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