Suggested Reading: Information Gathering in Classical Greece

May 16, 2015

Frank S. Russell — Information Gathering in Classical Greece — University of Michigan Press, 2002, 268 pp. ISBN 0-472-11064

(Buy at

(Buy at

While Amazon’s blurb speaks of cloak-and-dagger and other spy clichés, this book has little to do with a thrilling espionage novel, and is as far from an ‘easy read’ as anything can be. Well documented, Russell’s book brings us back to classical times and tells us about war, politics, oracles, and “spies” (for lack of a better term) and a lot about the Greek mind. We learn, for example, that the Greeks did not consider intelligence gathering in the same way we would today with professional spies and information-gathering network, à la NSA, but rather in a rather ad hoc way.

The narrative is really fascinating but the form itself remains difficult. First, there are quite many ancient greek words to remember (will you remember what are a proxenoi and a presbeutai in two days?), and very often we find more than half of the pages being footnotes. This excruciatingly well documented book is still a must-read for one interested in classical Greece as well as one interested in the history of espionage.

Suggested Reading: How Mathematics Happened: The First 50000 Years

May 19, 2014

Peter S. Rudman — How Mathematics Happened: The First 50000 Years — Prometheus Books, 2006, 314 pp. ISBN 978-1-59102-477-4

(Buy at

(Buy at

What first got me interested in this book is the “50000 years” part. I was preparing lectures notes for my course on discrete mathematics and I wanted my students to have an idea of what prehistoric maths might have been, say, 20000 years ago. Unfortunately, you wont learn much about this in this book

The book does hint about what mathematics might have been in hunter-gatherer times, and how it might have affected later developments. But that lasts for about a chapter or so, and the remainder is devoted to historical mathematics: Ancient Egyptian, Babylonian, and Classical Greek. All kinds of numerical algorithms are covered, presented in great detail, making the book more technical than historical. Some part are speculative as the historical record is incomplete at best, but it is speculative in the best way possible, with every assumption backed by an actual historical observation.

Amicable Numbers (part I)

August 20, 2013

I know of no practical use for Amicable numbers, but they’re rather fun. And rare. But let’s start with a definition. Let n be a natural number (a positive integer), and let

\displaystyle \sigma(n)=\sum_{d|n}d

with d \in \mathbb{N} and 1\leqslant{d}\leqslant{n}, be the sum of the divisors of n. We’re in fact interested in the sum of the proper divisors of n, that is,


Now we’re ready to define amicable numbers!

Amicable numbers: Two numbers, n, m \in \mathbb{N} are amicable, if, and only if, s(n)=m and s(m)=n.

Given n, we can find m=s(n) and test if n=s(m)=s(s(n)). But to do that efficiently, we need to compute s(n) (or \sigma(n)) very rapidly. The first expression above requires O(n), but we can do much better. Let’s see how.

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