Primality-testing Mersenne numbers

Report of a systematic review of Mersenne numbers 2^p-1 for p < 62982.

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.

Primality-testing Mersenne Numbers (II)

Reports the factor-filtering and primality-testing of Mersenne Numbers Mp for p < 100000, the latter using the ICL 'DAP' Distributed Array Processor.

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.

The relationships of time and temperature to body weight and numbers of endospores in Pasteuria penetrans-infected Meloidogyne javanica females

Tomato plants inoculated with Meloidogyne javanica juveniles infected with Pasteuria penetrans were grown in a glasshouse (20-32degreesC) for 36, 53, 71 and 88 days and in a growth room (26-29degreesC) for 36, 53, 71 and 80 days. Over these periods the numbers of P penetrans endospores in infected M. javanica females and the weights of individual infected females increased. In the growth room, most spores (2.03 x 10(6)) were found after 71 days. However, in the glasshouse the rate of increase was slower and spore numbers were still increasing at the final sampling at 88 days (2.04 x 10(6)), as was the weight of the nematodes (72 mug). Weights of uninfected females reached a maximum of 36.2 and 43.1 mug after 71 days in the growth room and glasshouse, respectively.

Estimating numbers of infectious units from serial dilution assays

The paper concerns the design and analysis of serial dilution assays to estimate the infectivity of a sample of tissue when it is assumed that the sample contains a finite number of indivisible infectious units such that a subsample will be infectious if it contains one or more of these units. The aim of the study is to estimate the number of infectious units in the original sample. The standard approach to the analysis of data from such a study is based on the assumption of independence of aliquots both at the same dilution level and at different dilution levels, so that the numbers of infectious units in the aliquots follow independent Poisson distributions. An alternative approach is based on calculation of the expected value of the total number of samples tested that are not infectious. We derive the likelihood for the data on the basis of the discrete number of infectious units, enabling calculation of the maximum likelihood estimate and likelihood-based confidence intervals. We use the exact probabilities that are obtained to compare the maximum likelihood estimate with those given by the other methods in terms of bias and standard error and to compare the coverage of the confidence intervals. We show that the methods have very similar properties and conclude that for practical use the method that is based on the Poisson assumption is to be recommended, since it can be implemented by using standard statistical software. Finally we consider the design of serial dilution assays, concluding that it is important that neither the dilution factor nor the number of samples that remain untested should be too large.

Numbers of antral follicles during follicular waves in cattle: Evidence for high variation among animals, very high repeatability in individuals, and an inverse association with serum follicle-stimulating hormone concentrations

The extent, causes, and physiological significance of the variation in number of follicles growing during ovarian follicular waves in human beings and cattle are unknown. Therefore, the present study examined the variability and repeatability in numbers of follicles 3 mm or greater in diameter during the follicular waves in bovine estrous cycles, and we determined if the variation in number of follicles during waves was associated with alterations in secretion of FSH, estradiol, inhibin, and insulin-like growth factor I (IGF-I). Dairy cattle were subjected to twice-daily ultrasound analysis to count total number of antral follicles 3 mm or greater in diameter throughout 138 different follicular waves. In another study, blood samples were taken at frequent intervals from cows that consistently had low or very high numbers of follicles during waves and were subjected to immunoassays. Results indicate the following: First, despite an approximately sevenfold variation in number of follicles during waves among animals and marked differences in age, stage of lactation, and season of the year, a very highly repeatable (0.95) number of follicles 3 mm or greater in diameter is maintained during the ovulatory and nonovulatory follicular waves of individuals. Second, variation in number of follicles 3 mm or greater in diameter during waves and the inverse association of number of follicles during waves with FSH are not directly explained by alterations in the patterns of secretion of estradiol, inhibin, or IGF-I. Third, ovarian ultrasound analysis can be used reliably by investigators to identify cattle that consistently have low or high numbers of follicles during waves, thus providing a novel experimental model to determine the causes and physiological significance of the high variation in antral follicle number during follicular waves among single-ovulating species, such as cattle or humans.

PDBSprotEC: A web-accessible database linking PDB chains to EC numbers via SwissProt

A mapping between chains in the Protein Databank and Enzyme Classification numbers is invaluable for research into structure-function relationships. Mapping at the chain level is a non-trivial problem and we present an automatically updated Web-server, which provides this link in a queryable form and as a downloadable XML or flat file.

Hepatozoon griseisciuri in grey squirrels (Sciurus carolinensis): changes of blood leucocyte numbers resulting from infection

Numbers of leucocytes in squirrels with gametocytes of Hepatozoon in their blood (infected) were compared with animals without gametocytes (uninfected). Typical values for leucocytes/mm(3) blood in uninfected squirrels were: leucocytes 5-7 x 10(3), granulocytes 3-4 x 10(3), lymphocytes 2-0 x 10(3) and monocytes 0-3 x 10(3) cells. Infection caused an increase in monocytes, lymphocytes and granulocytes, and there was a significant positive association between parasitaemia level and numbers of both total leucocytes and monocytes. Infected animals had more uninfected monocytes/mm(3) blood than did uninfected animals. The proportions of monocytes were more variable over time in infected animals, but no shift between infected and uninfected status was detected. Transfer of serum from infected squirrels to mice resulted in elevated counts of total blood leucocytes, monocytes and granulocytes, but not of lymphocytes, as compared with controls. Serum from squirrels with high parasitaemias had a more marked effect than serum from squirrels with low parasitaemias. Results indicate an infection - related monocytosis, possibly controlled by cytokines, that increases the number of cells available for invasion by gametocytes, thus enhancing the chances of parasite transmission.

Synthesis and fabrication of a thin film containing silica-encapsulated face-centered tetragonal FePt nanoparticles

Self-assembly of monodisperse, silica-encapsulated, face-centered tetragonal FePt nanoparticles forms closely packed 2D arrays (see figure). Placing monodisperse FePt nanoparticles in silica nanocapsules allows the transition from a disordered face-centered cubic phase to a ferromagnetic crystalline face-centered tetragonal structure at elevated temperature without severe sintering. These materials are potential candidates for the generation of ultrahigh-density magnetic recording media.