On numbers which contain no factors of the form p(kp+u),

Cover title.

Exercise sheet 7 pdf

Exercises and solutions in PDF

Exercise sheet 7

Exercises and solutions in LaTex

Exercise sheet 2 pdf

Exercises and solutions in PDF

Ration--a rational arithmetic package

Includes bibliographical references.

Über die Eulerschen und Gaussschen Methoden der Primzahlbestimmung /

Inaug.-diss.--Kaiser-Wilhelms-Universität Strassburg.

New mathematical tables containing the factors, squares, cubes, square roots, cube roots, reciprocals, and hyperbolic logarithms, of all numbers from 1 to 10000; tables of powers and prime numbers; an extensive table of formulæ, or general synopsis of the most important particulars relating to the doctrines of equations, series, fluxions, fluents, &c. &c. &c.

Mode of access: Internet.

Mersenne numbers

These notes have been issued on a small scale in 1983 and 1987 and on request at other times. This issue follows two items of news. First, WaIter Colquitt and Luther Welsh found the 'missed' Mersenne prime M110503 and advanced the frontier of complete Mp-testing to 139,267. In so doing, they terminated Slowinski's significant string of four consecutive Mersenne primes. Secondly, a team of five established a non-Mersenne number as the largest known prime. This result terminated the 1952-89 reign of Mersenne primes. All the original Mersenne numbers with p < 258 were factorised some time ago. The Sandia Laboratories team of Davis, Holdridge & Simmons with some little assistance from a CRAY machine cracked M211 in 1983 and M251 in 1984. They contributed their results to the 'Cunningham Project', care of Sam Wagstaff. That project is now moving apace thanks to developments in technology, factorisation and primality testing. New levels of computer power and new computer architectures motivated by the open-ended promise of parallelism are now available. Once again, the suppliers may be offering free buildings with the computer. However, the Sandia '84 CRAY-l implementation of the quadratic-sieve method is now outpowered by the number-field sieve technique. This is deployed on either purpose-built hardware or large syndicates, even distributed world-wide, of collaborating standard processors. New factorisation techniques of both special and general applicability have been defined and deployed. The elliptic-curve method finds large factors with helpful properties while the number-field sieve approach is breaking down composites with over one hundred digits. The material is updated on an occasional basis to follow the latest developments in primality-testing large Mp and factorising smaller Mp; all dates derive from the published literature or referenced private communications. Minor corrections, additions and changes merely advance the issue number after the decimal point. The reader is invited to report any errors and omissions that have escaped the proof-reading, to answer the unresolved questions noted and to suggest additional material associated with this subject.

Mersenne Numbers: consolidated results

This document provides and comments on the results of the Lucas-Lehmer testing and/or partial factorisation of all Mersenne Numbers Mp = 2^p-1 where p is prime and less than 100,000. Previous computations have either been confirmed or corrected. The LLT computations on the ICL DAP is the first implementation of Fast-Fermat-Number-Transform multiplication in connection with Mersenne Number testing. This paper championed the disciplines of systematically testing the Mp, and of double-sourcing results which were not manifestly correct. Both disciplines were adopted by the later GIMPS initiative, the 'Great Internet Mersenne Prime Search, which was itself one of the first web-based distributed-community projects.

On the variations of the Betti numbers of regular levels of Morse flows

We generalize results in Cruz and de Rezende (1999) [7] by completely describing how the Beth numbers of the boundary of an orientable manifold vary after attaching a handle, when the homology coefficients are in Z, Q, R or Z/pZ with p prime. First we apply this result to the Conley index theory of Lyapunov graphs. Next we consider the Ogasa invariant associated with handle decompositions of manifolds. We make use of the above results in order to obtain upper bounds for the Ogasa invariant of product manifolds. (C) 2011 Elsevier B.V. All rights reserved.

RIEMANN ZETA ZEROS AND PRIME NUMBER SPECTRA IN QUANTUM FIELD THEORY

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Immunologic characteristics of immunogenic variants generated by {\it in vitro\/} treatment of a methylcholanthrene-induced murine fibrosarcoma MCA-F with 1 methyl-3-nitro-1-nitrosoguanidine, 5 -aza-2$\sp\prime$-deoxycytidine, or ultraviolet radiation

This study addresses the questions of whether the frequency of generation and in vivo cross-reactivity of highly immunogenic tumor clones induced in a single parental murine fibrosarcoma cell line MCA-F is more closely related to the agent used to induce the Imm$\sp{+}$ clone or whether these characteristics are independent of the agents used. These questions were addressed by treating the parental tumor cell line MCA-F with UV-B radiation (UV-B), 1-methyl-3-nitro-1-nitrosoguanidine (MNNG), or 5-aza-2$\sp\prime$-deoxycytidine (5-azaCdR). The frequency of Imm$\sp{+}$ variant generation was similarly high for the three different agents, suggesting that the frequency of Imm$\sp{+}$ generation was related more closely to the cell line than to the inducing agent used. Cross-reactivity was tested with two Imm$\sp{+}$ clones from each treatment group in a modified immunoprotection assay that selectively engendered antivariant, but not antiparental immunity. Under these conditions each clone, except one, immunized against itself. The MNNG-induced clones engendered stronger antivariant immunity but a weaker variant cross-reactive immunity could also be detected.^ This study also characterized the lymphocyte populations responsible for antivariant and antiparental immunity in vivo. Using the local adoptive transfer assay (LATA) and antibody plus complement depletion of T-cell subsets, we showed that immunity induced by the Imm$\sp{+}$ variants against the parent MCA-F was transferred by the Thy1.2$\sp{+}$, L3T4a$\sp{+}$, Lyt2.1$\sp{-}$ (CD4$\sp{+}$) population, without an apparent contribution by Thy1.2$\sp{+}$, L3T4a$\sp{-}$, Lyt2.1$\sp{+}$ (CD8$\sp{+}$) cells. A role for Lyt2.1$\sp{+}$T lymphocytes in antivariant, but not antiparent immunity was supported by the results of LATA and CTL assays. Immunization with low numbers of viable Imm$\sp{+}$ cells, or with high numbers of non viable Imm$\sp{+}$ cells engendered only antivariant immunity without parental cross-protection. The associative recognition of parental antigens and variant neoantigens resulting in strong antiparent immunity was investigated using somatic cells hybrids of Imm$\sp{+}$ variants of MCA-F and an antigenically distinct tumor MCA-D. An unexpected result of these latter experiments was the expression of a unique tumor-specific antigen by the hybrid cells. These studies demonstrate that the parental tumor-specific antigen and the variant neoantigen must be coexpressed on the cell surface to engender parental cross-protective immunity. (Abstract shortened with permission of author.) ^

On the Asymptotic Behavior of the Ratio between the Numbers of Binary Primitive and Irreducible Polynomials

This work was presented in part at the 8th International Conference on Finite Fields and Applications Fq^8 , Melbourne, Australia, 9-13 July, 2007.

An Invitation to Additive Prime Number Theory

2000 Mathematics Subject Classification: 11D75, 11D85, 11L20, 11N05, 11N35, 11N36, 11P05, 11P32, 11P55.