Prime number patterns

This dissertation focuses on studying the patterns in the prime number distribution. It visualizes the distribution of prime numbers in a graphic manner through the usage of the Microsoft Excel software. This study is restricted to the first 20,000 primes. A list of the prime numbers is first obtained from Project Gutenberg online. By performing several methods of transformations, which are gaps between the first and second numbers, logarithm of the numbers, progressive ratio, progressive mean and progressive standard deviation, the transformed data are graphed in forms of scatter charts and radar charts. These charts are analyzed and compared with control data, which are number series that are increasing because prime numbers is an increasing number series. Prime numbers have been known to appear randomly, however, through the study of the graphs, it shows some regularity. This regularity is shown when a radar plot of primes, its progressive means and progressive standard deviation display a shell-like formation. All in all, although prime numbers seem to be scattered and occur in a somewhat random form, this dissertation shows that there are more underlying patterns with regularity that have not been totally discovered

Linear patterns of prime elements in number fields

We prove that the von Mangoldt function $\Lambda _K$ of a number field $K$ is well approximated by its Cramer/Siegel models in the Gowers norm sense. Via the inverse theory of Gowers norms, this is achieved by showing that the difference of $\Lambda _K$ and its model is asymptotically orthogonal to nilsequences. To prove the asymptotic orthogonality, we use Mitsui's Prime Number Theorem as the base case and proceed by upgrading Green--Tao's type I/II sum computation to the general number field. As an application, we prove a number field analog of the Green--Tao--Ziegler theorem on simultaneous prime values of affine-linear forms of finite complexity.Comment: 94 page

Birational Mappings and Matrix Sub-algebra from the Chiral Potts Model

We study birational transformations of the projective space originating from lattice statistical mechanics, specifically from various chiral Potts models. Associating these models to \emph{stable patterns} and \emph{signed-patterns}, we give general results which allow us to find \emph{all} chiral $q$-state spin-edge Potts models when the number of states $q$ is a prime or the square of a prime, as well as several $q$-dependent family of models. We also prove the absence of monocolor stable signed-pattern with more than four states. This demonstrates a conjecture about cyclic Hadamard matrices in a particular case. The birational transformations associated to these lattice spin-edge models show complexity reduction. In particular we recover a one-parameter family of integrable transformations, for which we give a matrix representationComment: 22 pages 0 figure The paper has been reorganized, splitting the results into two sections : results pertaining to Physics and results pertaining to Mathematic

Entanglement Patterns in Mutually Unbiased Basis Sets for N Prime-state Particles

A few simply-stated rules govern the entanglement patterns that can occur in mutually unbiased basis sets (MUBs), and constrain the combinations of such patterns that can coexist (ie, the stoichiometry) in full complements of p^N+1 MUBs. We consider Hilbert spaces of prime power dimension (as realized by systems of N prime-state particles, or qupits), where full complements are known to exist, and we assume only that MUBs are eigenbases of generalized Pauli operators, without using a particular construction. The general rules include the following: 1) In any MUB, a particular qupit appears either in a pure state, or totally entangled, and 2) in any full MUB complement, each qupit is pure in p+1 bases (not necessarily the same ones), and totally entangled in the remaining p^N-p. It follows that the maximum number of product bases is p+1, and when this number is realized, all remaining p^N-p bases in the complement are characterized by the total entanglement of every qupit. This "standard distribution" is inescapable for two qupits (of any p), where only product and generalized Bell bases are admissible MUB types. This and the following results generalize previous results for qubits and qutrits. With three qupits there are three MUB types, and a number of combinations (p+2) are possible in full complements. With N=4, there are 6 MUB types for p=2, but new MUB types become possible with larger p, and these are essential to the realization of full complements. With this example, we argue that new MUB types, showing new entanglement characteristics, should enter with every step in N, and when N is a prime plus 1, also at critical p values, p=N-1. Such MUBs should play critical roles in filling complements.Comment: 27 pages, one figure, to be submitted to Physical Revie