mathematics

Richard P. Brent

Richard P. Brent is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Richard P. Brent rather than just read about it. In short: Richard Peirce Brent is an Australian mathematician and computer scientist. He is an emeritus professor at the Australian National University.

Key takeaways

Richard P. Brent belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Richard P. Brent to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Richard P. Brent from memory before moving on to harder problems.

Reference excerpt

Richard Peirce Brent is an Australian mathematician and computer scientist. He is an emeritus professor at the Australian National University. From March 2005 to March 2010 he was a Federation Fellow 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. In 1975 he and Eugene Salamin independently conceived the Salamin–Brent algorithm, used in high-precision calculation of π {\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 π {\displaystyle \pi } (apart from a small constant factor) using the arithmetic-geometric mean of Carl Friedrich Gauss. 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. In 1980 he and Nobel laureate Edwin McMillan found a new algorithm for high-precision computation of the Euler–Mascheroni constant γ {\displaystyle \gamma } using Bessel functions, and showed that γ {\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). In 1980 he and John Pollard factored the eighth Fermat number using a variant of the Pollard rho 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 6972593 + x 3037958 + 1. {\displaystyle x^{6972593}+x^{3037958}+1.}

The degree 6972593 is the exponent of a Mersenne prime. In 2009 and 2016, Brent and Paul Zimmermann discovered some even larger primitive trinomials, for example:

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

Worked examples

Example 1 — a first encounter with Richard P. Brent

Start with the simplest possible case. Write down what Richard P. Brent claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Richard P. Brent before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Richard P. Brent ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Richard P. Brent

In research: Richard P. Brent appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Richard P. Brent in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Richard P. Brent is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1946 births, Academic staff of the Australian National University, Australian computer scientists, so understanding it makes those chapters shorter.
In everyday life: Look for Richard P. Brent outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Richard P. Brent in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Richard P. Brent means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Richard P. Brent out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Richard P. Brent in simple terms?

Richard Peirce Brent is an Australian mathematician and computer scientist. He is an emeritus professor at the Australian National University.

Why does Richard P. Brent matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Richard P. Brent?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Richard P. Brent.