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Unit 1

The Primes

1.1 Introduction

"Reason is immortal, all else mortal."


"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate."

-Leonhard Euler (1707-1783)

In 1974, astronomers sent a message into space from the Arecibo radio telescope in Puerto Rico in an attempt to broadcast the presence of our intelligent life to any other potentially sentient beings that happened to be listening. The message contained information about our planet's location, the basis of our chemistry, and rudimentary information about our biological form. In sending this message, astronomers had to confront a deep problem: how does one communicate with another intelligent species that one knows nothing about?

ARECIBO MESSAGE Item 1671 / Frank Drake, ARECIBO MESSAGE OF 1974 (1974). Courtesy of Frank Drake.

At the time, astronomers thought that the best way to ensure the comprehensibility of the message was to use a concept fundamental to our own logical understanding of the world, namely, prime numbers. No known natural process creates prime numbers, yet they are at the very center of mathematical thought. The astronomers assumed that any beings intelligent enough to be able to listen to radio broadcasts would know about prime numbers. With this in mind, the message senders encoded a pictogram consisting of 1,679 bits; although the number 1,679 is not itself prime, it is the product of two prime numbers. The general idea was that any alien listener who receives a random number of blips might think to factor that number, just to get a sense of what it is. Choosing a number whose only factors are two primes might prompt the receiver to think two-dimensional," a rectangular array, in other words, with one factor representing the number of rows and the other representing the number of columns. Using this information to arrange the pixels into a rectangle would reveal the picture. This, of course, assumes that an intelligent alien would recognize prime numbers to be special in some way.

This assumption is actually quite reasonable. Prime numbers, known to humankind for at least 20,000 years, represent a fundamental concept of mathematics that has provided rich grounds for study. Primes have been described as the atoms of arithmetic, the indivisible parts from which all other numbers can be constructed. Seemingly simple, they have provided some of the most challenging problems for those willing to explore their world. Long studied for their mysteries, many of which remain unsolved, primes were, until relatively recently, solely the concern of mathematicians. With the explosion of digital communications and our dependence on Internet transactions, however, primes now play a pivotal role in other areas, as well.

The advent and growth of the Internet and the "information age" have given unprecedented convenience to millions of technology users worldwide. Technological advances and their application in numerous fields have helped to shrink perceived distances and have done much to break down barriers that divide us. Peril often accompanies progress, however, and our modern Internet economy is not immune to risks. For example, when we make a purchase, pay bills, or check our bank balance online, how do we know that someone will not intercept the transmission and steal our personal information? One of the ideas we will see in this unit is how the properties of prime numbers, in combination with modular arithmetic, or "clock math," are used to help keep our information secure.

In this unit we will see how primes are fundamental to mathematical thought. We will explore the seemingly simple concept that underlies their existence, and we'll catch a glimpse of the mystery inherent in their distribution on the number line. Finally, we'll take a brief look at how the modern standard of data security, RSA encryption, uses fundamental properties of primes to create virtually impenetrable codes.

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Next: 1.2 Math at the Dawn of Time

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