# Richard P. Brent

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Richard Peirce Brent
Born20 April 1946 (age 74)
NationalityAustralian
Alma materStanford University
AwardsHannan Medal (2005)
Scientific career
FieldsMathematics, computer science
InstitutionsAustralian National University
Doctoral advisorsGene H. Golub
George Forsythe

Richard Peirce Brent is an Australian mathematician and computer scientist. He is an emeritus professor at the Australian National University and a conjoint professor at the University of Newcastle (Australia). From March 2005 to March 2010 he was a Federation Fellow[1] at the Australian National University. His research interests include number theory (in particular factorisation), random number generators, computer architecture, and analysis of algorithms.

In 1973, he published a root-finding algorithm (an algorithm for solving equations numerically) which is now known as Brent's method.[2]

In 1975 he and Eugene Salamin independently conceived the Salamin–Brent algorithm, used in high-precision calculation of ${\displaystyle \pi }$. At the same time, he showed that all the elementary 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 the arithmetic-geometric mean of Carl Friedrich Gauss.[3]

In 1979 he showed that the first 75 million complex zeros of the Riemann zeta function lie on the critical line, providing some experimental evidence for the Riemann hypothesis.[4]

In 1980 he and Nobel laureate Edwin McMillan found a new algorithm for high-precision computation of the Euler–Mascheroni constant ${\displaystyle \gamma }$ using Bessel functions, and showed that ${\displaystyle \gamma }$ can not have a simple rational form p/q (where p and q are integers) unless q is extremely large (greater than 1015000).[5]

In 1980 he and John Pollard factored the eighth Fermat number using a variant of the Pollard rho algorithm.[6] He later factored the tenth[7] 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):

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

The degree 6972593 is the exponent of a Mersenne 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 degree 43112609 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 published Modern Computer Arithmetic (Cambridge University Press), a book about algorithms for performing arithmetic, and their implementation on modern computers.

Brent is a Fellow of the Association for Computing Machinery, the IEEE, SIAM and the Australian Academy of Science. In 2005, he was awarded the Hannan Medal by the Australian Academy of Science. In 2014, he was awarded the Moyal Medal by Macquarie University.

## References

1. ^ Federation Fellowships Funding Outcomes 2004 Archived 2012-07-07 at the Wayback Machine. Australian Research Council
2. ^ Richard Peirce Brent (1973). Algorithms for Minimization without Derivatives. Prentice-Hall, Englewood Cliffs, NJ. Reprinted by Dover Publications, Mineola, New York, 2002 and 2013. ISBN 0-486-41998-3. Original edition is available on his own professional web page at ANU.
3. ^ Brent, Richard Peirce (1975). Traub, J. F. (ed.). "Multiple-Precision Zero-Finding Methods and the Complexity of Elementary Function Evaluation". Analytic Computational Complexity. New York: Academic Press. CiteSeerX 10.1.1.119.3317.
4. ^ Brent, Richard Peirce (1979). "On the Zeros of the Riemann Zeta Function in the Critical Strip". Mathematics of Computation. 33 (148): 1361–1372. doi:10.2307/2006473. JSTOR 2006473.
5. ^ Brent, Richard Peirce and McMillan, E. M. (1980). "Some New Algorithms for High-Precision Computation of Euler's Constant". Mathematics of Computation 34 (149) 305-312.
6. ^ Brent, Richard Peirce; Pollard, J. M. (1981). "Factorization of the Eighth Fermat Number". Mathematics of Computation. 36 (154): 627–630. doi:10.2307/2007666. JSTOR 2007666.
7. ^ Brent, Richard Peirce (1999). "Factorization of the Tenth Fermat Number". Mathematics of Computation. 68 (225): 429–451. Bibcode:1999MaCom..68..429B. doi:10.1090/s0025-5718-99-00992-8. JSTOR 2585124.
8. ^ Brent, Richard Peirce and Larvala, S. and Zimmermann, Paul (2005). "A primitive trinomial of degree 6972593". Mathematics of Computation 74 (250) 1001-1002.
9. ^ Brent, Richard Peirce and Zimmermann, Paul (2011). "The great trinomial hunt". Notices of the American Mathematical Society 58 233-239.
10. ^ Richard P. Brent, Paul Zimmermann, "Twelve new primitive binary trinomials", arXiv:1605.09213, 24 May 2016.