Remember when you were back in grade school and your teacher was going over basic arithmetic and numbers? You learned basic facts about division, as for example, how to know whether a given number is divisible by 2 or 3. You also learned about composite numbers and prime numbers. You sat there scratching your head wondering how in God’s name such a thing as a prime number would ever have any use other than to give young children homework headaches. But then again God has a funny way of letting each one of his creations participate in the grand scheme of things—yes even creations like numbers.
If you don’t remember from back in your elementary school days, a prime number is any number greater than 1 which has as its only divisors 1 and itself. Thus 5 is prime because its only divisors are 1 and 5. Similarly 7, 11, and 13 are prime as well; 2 is the only even prime. From the time of Euclid and his Elements (circa 300 B.C.), these numerical creatures have been intensely studied and classified, and all kinds of mathematical theories and conjectures regarding them have been formulated. Euclid was the first to show that the prime numbers formed an infinite set, and if you’ve read my articles on infinity (see my series Dabbling in Infinity), then this means that there are as many primes as all the counting numbers, hard as that might be to believe!
The study of prime numbers falls under the branch of mathematics known as Number Theory. This area of pure mathematics treats the study of numbers and their inherent properties as well as the interrelationships among broad classes of numbers and even number systems. Given all the elegant branches of mathematics such as topology, abstract algebra, and complex analysis, one would think that the study of just plain old numbers might be too simple or even boring. Yet two of the most famous unsolved problems in mathematics deal directly or indirectly with prime numbers: the Riemann Hypothesis and the Goldbach Conjecture.
People continue to this day in the search of ever larger prime numbers. The largest prime discovered to date was found in 2006 by two professors, and this prime number has almost ten million digits! But outside pure mathematical interest, what is the point of all this fascination with finding larger and larger primes? Well that’s where cryptography comes in and the idea of internet commerce.
Prime numbers found little practical value until about the 1970's when public key cryptography was discovered. This procedure allows information to be coded so that only the person who knows the secret key can decipher the message. This protocol allows private information to be sent over the internet securely without the fear of such being intercepted and read by somebody for whom the information is not intended. This methodology allows us to conduct secure banking and financial transactions over the internet today.
Well, lo and behold, this type of cryptography depends on prime numbers—very large prime numbers! Essentially, the theory behind this field hinges on the inherent difficulty of factoring the product of two very large prime numbers without knowing one of the primes. Thus by using the theory of prime numbers, cryptographers can encode sensitive data and thereby permit secure transactions over a public network like the internet. And the bigger the prime numbers, the harder it is to crack the code. Thus the search for ever bigger primes.
Remember this the next time you think a number is just a number is just a number. You never know what discoveries lie within the realm of mathematics and particularly within its domain of numbers. Think about this the next time you use the internet and give out your credit card number to buy that new item you want.
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