Richard Peirce Brent is an Australian

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

and

computer scientist A computer scientist is a person who is trained in the academic study of computer science. Computer scientists typically work on the theoretical side of computation, as opposed to the hardware side on which computer engineers mainly focus ( ...

. He is an emeritus professor at the

Australian National University The Australian National University (ANU) is a public research university located in Canberra, the capital of Australia. Its main campus in Acton encompasses seven teaching and research colleges, in addition to several national academies and ...

. From March 2005 to March 2010 he was a Federation Fellow at the

. His research interests include

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

(in particular

factorisation In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kind ...

), random number generators,

computer architecture In computer engineering, computer architecture is a description of the structure of a computer system made from component parts. It can sometimes be a high-level description that ignores details of the implementation. At a more detailed level, the ...

, and

analysis of algorithms In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that r ...

. In 1973, he published a

root-finding algorithm In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbe ...

(an algorithm for solving equations numerically) which is now known as

Brent's method In numerical analysis, Brent's method is a hybrid root-finding algorithm combining the bisection method, the secant method and inverse quadratic interpolation. It has the reliability of bisection but it can be as quick as some of the less-reliabl ...

. In 1975 he and Eugene Salamin independently conceived the Salamin–Brent algorithm, used in high-precision calculation of

\pi

. At the same time, he showed that all the

elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, a ...

s (such as log(''x''), sin(''x'') etc.) can be evaluated to high precision in the same time as

\pi

(apart from a small constant factor) using the arithmetic-geometric mean of

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...

. In 1979 he showed that the first 75 million

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

zeros of the Riemann zeta function lie on the critical line, providing some experimental evidence for the

Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pu ...

. In 1980 he and Nobel laureate

Edwin McMillan Edwin Mattison McMillan (September 18, 1907 – September 7, 1991) was an American physicist credited with being the first-ever to produce a transuranium element, neptunium. For this, he shared the 1951 Nobel Prize in Chemistry with Glenn Seabo ...

found a new algorithm for high-precision computation of the

Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural ...

\gamma

using

Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

s, and showed that

\gamma

can not have a simple rational form ''p''/''q'' (where ''p'' and ''q'' are

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s) unless ''q'' is extremely large (greater than 10¹⁵⁰⁰⁰). In 1980 he and John Pollard factored the eighth Fermat number using a variant of the

Pollard rho Pollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. It uses only a small amount of space, and its expected running time is proportional to the square root of the smallest prime factor of the ...

algorithm. He later factored the tenth and eleventh Fermat numbers using Lenstra's elliptic curve factorisation algorithm. In 2002, Brent, Samuli Larvala and Paul Zimmermann discovered a very large primitive trinomial over GF(2): :

x^ + x^ + 1.

The

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...

6972593 is the exponent of a

Mersenne prime 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 for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17 ...

. In 2009 and 2016, Brent and Paul Zimmermann discovered some even larger primitive trinomials, for example: :

x^ + x^ + 1.

The degree 43112609 is again the exponent of a Mersenne prime. The highest degree trinomials found were three trinomials of degree 74,207,281, also a Mersenne prime exponent.Richard P. Brent, Paul Zimmermann
"Twelve new primitive binary trinomials"
arXiv:1605.09213, 24 May 2016. In 2011, Brent and Paul Zimmermann published ''Modern Computer Arithmetic'' (

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...

), a book about algorithms for performing arithmetic, and their implementation on modern computers. Brent is a Fellow of the

Association for Computing Machinery The Association for Computing Machinery (ACM) is a US-based international learned society for computing. It was founded in 1947 and is the world's largest scientific and educational computing society. The ACM is a non-profit professional member ...

, the

IEEE The Institute of Electrical and Electronics Engineers (IEEE) is a 501(c)(3) professional association for electronic engineering and electrical engineering (and associated disciplines) with its corporate office in New York City and its operati ...

SIAM Thailand ( ), historically known as Siam () and officially the Kingdom of Thailand, is a country in Southeast Asia, located at the centre of the Indochinese Peninsula, spanning , with a population of almost 70 million. The country is bo ...

and the

Australian Academy of Science The Australian Academy of Science was founded in 1954 by a group of distinguished Australians, including Australian Fellows of the Royal Society of London. The first president was Sir Mark Oliphant. The academy is modelled after the Royal So ...

. In 2005, he was awarded the Hannan Medal by the

. In 2014, he was awarded the

Moyal Medal Moyal may refer to: People *Ann Moyal (1926–2019), Australian historian *Damien Moyal (born 1976), American vocalist, musician and designer * Diana López Moyal, Cuban flutist *Eliyahu Moyal (1920–1991), Israeli politician *Esther Moyal (1874� ...

Macquarie University Macquarie University ( ) is a Public university, public research university based in Sydney, Australia, in the suburb of Macquarie Park, New South Wales, Macquarie Park. Founded in 1964 by the New South Wales Government, it was the third univer ...

References

External links

Richard Brent's home page
* {{DEFAULTSORT:Brent, Richard 1946 births Australian computer scientists Australian mathematicians Academic staff of the Australian National University Complex systems scientists Fellows of the Association for Computing Machinery Living people People from the Australian Capital Territory Fellows of the Australian Academy of Science Fellows of the Society for Industrial and Applied Mathematics

See also

References

External links