What use are prime numbers anyway?

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In a book he wrote after his professional career was over, the English mathematician Godfrey H. Hardy (1877 - 1947) said of his work in number theory

"No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. - Judged by all practical standards, the value of my mathematical life is nil."

In another quote from Hardy speaking in the 1940s he said,

"Here is one science (number theory) at any rate whose very remoteness from ordinary human activities should keep it gentle and clean."

By which he meant that unlike say, a physicist or a chemist, his work could not be used to make weapons, pollute the atmosphere or even to invent new technology that however useful it might seem to some, might cause harm to others.

So it would seem to be a fair question "What use are prime numbers?"

One of the fundamental theorems of maths, in fact it is so fundamental that it is called "The Fundamental Theorem of Arithmetic", states that every positive integer (except the number 1) can be expressed in exactly one way as the product of one or more primes. So that if a and b are prime, when a x b = c then a and b are the only two primes that will multiply together to make c. During the 1970s three American scientists, Ron Rivest, Adi Shamir and Leonard Adleman, used this simple theorem to create a system of cryptography that, quite possibly without even realising it, you use every day.

Most people seem to think, if they think about it at all, that cryptography is something used by spies or secret agents to send coded messages back to base. Well, it is. But it is also used by the cash register at a supermarket checkout, by cash machines, televisions and the web browser you are using to read this article. Mobile phones, credit cards, the stock exchange and banks all use cryptography to keep electronic transactions of a hundred different kinds safe and secure. Without cryptography e-commerce would not exist as we know it, and all of this security depends on prime numbers.

The system invented by Rivest, Shamir and Adleman has, for fairly obvious reasons, come to be known as RSA cryptography. This article does not attempt a complete explanation of how RSA works, but basically, if I take two large prime numbers, say 300 digits each, and multiply them together I have in effect a code number that can only be broken by finding the two primes that were used to create this composite number. Since factoring big composite numbers into primes is not a trivial matter, anyone interested in trying to break these codes may find that the code has become meaningless by the time they have cracked it.

External links

• Cryptography
• Fundamental theorem of arithmetic