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 scientist who specializes in the academic study of computer science. Computer scientists typically work on the theoretical side of computation. Although computer scientists can also focus their work and research on ...

. He is an emeritus professor at the

Australian National University The Australian National University (ANU) is a public university, public research university and member of the Group of Eight (Australian universities), Group of Eight, located in Canberra, the capital of Australia. Its main campus in Acton, A ...

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

. His research interests include

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

(in particular

factorisation In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a p ...

random number generators Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...

, computer architecture, 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 numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function is a number such that . As, generally, the zeros of a function cannot be computed exactly nor ...

(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-reliab ...

. 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 (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...

. 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 The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...

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 pure ...

. 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 to produce a transuranium element, neptunium. For this, he shared the 1951 Nobel Prize in Chemistry with Glenn Seaborg. ...

found a new algorithm for high-precision computation of the

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

\gamma

using

Bessel function Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...

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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s) unless ''q'' is extremely large (greater than 10¹⁵⁰⁰⁰). In 1980 he and John Pollard factored the eighth

Fermat number In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a natural number, positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers ...

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 th ...

algorithm. He later factored the tenth and eleventh Fermat numbers using Lenstra's

elliptic curve factorisation The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the thi ...

algorithm. In 2002, Brent, Samuli Larvala and Paul Zimmermann discovered a very large primitive trinomial over GF(2): :

x^ + x^ + 1.

The degree 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 1 ...

. 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 was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

), 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 membe ...

, the

IEEE The Institute of Electrical and Electronics Engineers (IEEE) is an American 501(c)(3) organization, 501(c)(3) public charity professional organization for electrical engineering, electronics engineering, and other related disciplines. The IEEE ...

SIAM Thailand, officially the Kingdom of Thailand and historically known as Siam (the official name until 1939), is a country in Southeast Asia on the Mainland Southeast Asia, Indochinese Peninsula. With a population of almost 66 million, it spa ...

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 Soci ...

. In 2005, he was awarded the

Hannan Medal The Hannan Medal in the Mathematical Sciences is awarded every two years by the Australian Academy of Science to recognize achievements by Australians in the fields of pure mathematics, applied and computational mathematics, and statistical scienc ...

by the

. In 2014, he was awarded the Moyal Medal by

Macquarie University Macquarie University ( ) is a Public university, public research university in Sydney, New South Wales, Australia. Founded in 1964 by the New South Wales Government, it was the third university to be established in the Sydney metropolitan area. ...

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 1995 fellows of the Association for Computing Machinery Living people Fellows of the Australian Academy of Science Fellows of the Society for Industrial and Applied Mathematics

See also

References

External links