H. J. Woodall

Last updated May 18, 2022

Herbert J. Woodall was a British mathematician.

In 1925 Lt.-Col. Allan J.C. Cunningham and Woodall gathered together all that was known about the primality and factorization of such^{[ clarification needed ]} numbers and published a small book of tables.^[1]^{[ failed verification ]} "These tables collected from scattered sources the known prime factors for the bases 2 and 10 and also presented the authors' results of thirty years' work with these and the other bases."

Since 1925 many people have worked on filling in these tables. It is likely that this project is the longest, ongoing computational project in history. Derrick Henry Lehmer, a well known mathematician who died in 1991 was for many years a leader of these efforts. Professor Lehmer was a mathematician who was at the forefront of computing as modern electronic computers became a reality. He was also known as the inventor of some ingenious pre-electronic computing devices specifically designed for factoring numbers. These devices are currently in storage at the Computer Museum in Boston.

A generalized Woodall number is defined to be a number of the form

nbⁿ− 1,

where n + 2 > b; if a prime can be written in this form, it is then called a generalized Woodall prime.

Related Research Articles

In number theory, a Carmichael number is a composite number $which satisfies the modular arithmetic congruence relation:$

The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers.

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form $M n = 2 n - 1$ for some integer $n$ . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If $n$ is a composite number then so is $2 n - 1$ . Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form $M p = 2 p - 1$ for some prime $p$ .

Prime number Number whose only smaller divisor is 1

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation. The method represents one of the oldest and best-known pseudorandom number generator algorithms. The theory behind them is relatively easy to understand, and they are easily implemented and fast, especially on computer hardware which can provide modular arithmetic by storage-bit truncation.

In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers.

Mathematical tables are lists of numbers showing the results of a calculation with varying arguments. Trigonometric tables were used in ancient Greece and India for applications to astronomy and celestial navigation, and continued to be widely used until electronic calculators became cheap and plentiful, in order to simplify and drastically speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks, and specialized tables were published for numerous applications.

In number theory, a Woodall number (W_n) is any natural number of the form

A Wilson prime, named after English mathematician John Wilson, is a prime number p such that p² divides (p − 1)! + 1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1.

Derrick Henry "Dick" Lehmer, almost always cited as D.H. Lehmer, was an American mathematician significant to the development of computational number theory. Lehmer refined Édouard Lucas' work in the 1930s and devised the Lucas–Lehmer test for Mersenne primes. His peripatetic career as a number theorist, with him and his wife taking numerous types of work in the United States and abroad to support themselves during the Great Depression, fortuitously brought him into the center of research into early electronic computing.

Prime95, also distributed as the command-line utility mprime for FreeBSD and Linux, is a freeware application written by George Woltman. It is the official client of the Great Internet Mersenne Prime Search (GIMPS), a distributed computing project dedicated to searching for Mersenne primes. It is also used in overclocking to test for system stability.

John Lewis Selfridge, was an American mathematician who contributed to the fields of analytic number theory, computational number theory, and combinatorics.

Ralph Ernest Powers was an American amateur mathematician who worked on prime numbers.

TheCunningham project is a project, started in 1925, to factor numbers of the form bⁿ ± 1 for b = 2, 3, 5, 6, 7, 10, 11, 12 and large n. The project is named after Allan Joseph Champneys Cunningham, who published the first version of the table together with Herbert J. Woodall. There are three printed versions of the table, the most recent published in 2002, as well as an online version.

Samuel Standfield Wagstaff Jr. is an American mathematician and computer scientist, whose research interests are in the areas of cryptography, parallel computation, and analysis of algorithms, especially number theoretic algorithms. He is currently a professor of computer science and mathematics at Purdue University who coordinates the Cunningham project, a project to factor numbers of the form bⁿ ± 1, since 1983. He has authored/coauthored over 50 research papers and three books. He has an Erdős number of 1.

PrimeGrid is a volunteer distributed computing project which searches for very large prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing (BOINC) platform. PrimeGrid offers a number of subprojects for prime-number sieving and discovery. Some of these are available through the BOINC client, others through the PRPNet client. Some of the work is manual, i.e. it requires manually starting work units and uploading results. Different subprojects may run on different operating systems, and may have executables for CPUs, GPUs, or both; while running the Lucas–Lehmer–Riesel test, CPUs with Advanced Vector Extensions and Fused Multiply-Add instruction sets will yield the fastest results for non-GPU accelerated workloads.

Derrick Norman Lehmer was an American mathematician and number theorist.

In mathematics, specifically in number theory, a Cunningham number is a certain kind of integer named after English mathematician A. J. C. Cunningham.

References

↑ Herbert J. Woodall (1911). "Mersenne's Numbers". Nature. 87 (2177): 78. Bibcode:1911Natur..87...78W. doi:10.1038/087078f0. S2CID 35858272.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[nejmreview-1] Herbert J. Woodall (1911). "Mersenne's Numbers". Nature. 87 (2177): 78. Bibcode:1911Natur..87...78W. doi:10.1038/087078f0. S2CID 35858272.

[1]