Richard Peirce Brent is an Australianmathematician andcomputer scientist. He is an emeritus professor at theAustralian National University. From March 2005 to March 2010 he was aFederation Fellow^[1] at theAustralian National University. His research interests includenumber theory (in particularfactorisation),random number generators,computer architecture, andanalysis of algorithms.

In 1973, he published aroot-finding algorithm (an algorithm for solving equations numerically) which is now known asBrent's method.^[2]

In 1975 he andEugene Salamin independently conceived theSalamin–Brent algorithm, used in high-precision calculation of${\displaystyle \pi }$. At the same time, he showed that all theelementary functions (such as log(x), sin(x) etc.) can be evaluated to high precision in the same time as${\displaystyle \pi }$ (apart from a small constant factor) using thearithmetic-geometric mean ofCarl Friedrich Gauss.^[3]

In 1979 he showed that the first 75 millioncomplexzeros of theRiemann zeta function lie on the critical line, providing some experimental evidence for theRiemann hypothesis.^[4]

In 1980 he and Nobel laureateEdwin McMillan found a new algorithm for high-precision computation of theEuler–Mascheroni constant${\displaystyle \gamma }$ usingBessel functions, and showed that${\displaystyle \gamma }$ can not have a simple rational formp/q (wherep andq areintegers) unlessq is extremely large (greater than 10¹⁵⁰⁰⁰).^[5]

In 1980 he andJohn Pollard factored the eighthFermat number using a variant of thePollard rho algorithm.^[6] He later factored the tenth^[7] and eleventh Fermat numbers using Lenstra'selliptic curve factorisation algorithm.

In 2002, Brent, Samuli Larvala andPaul Zimmermann discovered a very largeprimitive trinomial overGF(2):

${\displaystyle x^{6972593}+x^{3037958}+1.}$

Thedegree 6972593 is the exponent of aMersenne prime.^[8]

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

${\displaystyle x^{43112609}+x^{3569337}+1.}$

The degree43112609 is again the exponent of a Mersenne prime.^[9] The highest degree trinomials found were three trinomials of degree 74,207,281, also a Mersenne prime exponent.^[10]

In 2011, Brent and Paul Zimmermann publishedModern Computer Arithmetic (Cambridge University Press), a book about algorithms for performing arithmetic, and their implementation on modern computers.

Brent is a Fellow of theAssociation for Computing Machinery, theIEEE,SIAM and theAustralian Academy of Science. In 2005, he was awarded theHannan Medal by theAustralian Academy of Science. In 2014, he was awarded theMoyal Medal byMacquarie University.

• Brent–Kung adder

• Richard Brent's home page
• Richard P. Brent at theMathematics Genealogy Project

• 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

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