Prime Year!

by Dr Temba Shonhiwa

For this issue of , my intention is to introduce you to what are known as the prime numbers. Prime numbers are defined as those natural numbers that are only divisible by 1 and themselves. For convenience we exclude the number 1 from the list of primes. Note that we restrict ourselves to the positive integers only. An integer that is not prime is called composite, eg the number 6 is composite. It is divisible by 1, 2, 3 and 6. So, here is a short list of some of the primes:

Now see if you can list all the primes below 100. These magical numbers have fascinated Mathematicians for centuries. For example, here is a natural question one could ask: " How many Primes are there? Do they go on forever?". Euclid, the Greek Mathematician who lived around 350 B.C proved that there were infinitely many Primes. To the layperson it may seem "obvious" at first that there are an infinite number of primes - but you keep listing them for a while and you will begin to have your doubts! They get fewer and fewer! For the professional mathematician a logical proof is imperative. "It is obvious," is an unacceptable argument. Euclid's proof is an example of the beauty and elegance of the mathematical method. Although centuries old, his proof is still considered by many the best as regards the question above. (See the article following this, for Euclid's proof.)

So, what motivated this article you may ask? Well, in case you haven't guessed already from the title, it is the fact that 1999 is a prime number! Well, how does one check that a given number is indeed prime? To answer this question, I will state here two theorems without proving them.

Theorem 1 If n > 1, then n has a prime divisor p, say.

You may want to experiment with some numbers here to test this theorem. Remember, the fact that it checks out is not a proof. On the other hand, if one number fails to check out, then the theorem is false. Now here is the second result we shall also need.

Theorem 2 If n is composite and p is the least/smallest prime factor of n, then p £ Ön.

Here is how we test whether a given number is prime or not. Suppose we wish to test whether 10 001 is a prime. Well, suppose it were composite. Then by theorem (1) it has some prime divisor p. Let us choose the smallest such prime (it may have several of course). By theorem (2), p £ [Ö10 001], that is p < 100. So, all we have to do is check whether or not any of the primes less than 100 divides 10 001! And, since you have this list above, check to see which prime divides 10 001. If no such prime exists, then it must follow that our number n is not composite! Now apply this to the number 1999.

Here is another very interesting fact concerning primes. Suppose that q and p are odd primes such that; p < q and q-p = 2. Then, we say that p and q are twin primes. For example, 5 and 7, 11 and 13, ., ., ., 197 and 199, ., . are twin primes. Now here is a bombshell: the years 1997 and 1999 are twin primes! Indeed we do live in very special times! When, if I may ask, will we have the next twin prime years? Now here are some truly huge twin primes; the numbers 1 000 000 009 649 and 1 000 000 009 651! You would need quite a long list of primes in order to check these two out. Already I can imagine someone asking; "How many twin pairs are there"? This is a good question, and you know what? The answer to this question is unknown!! All we can do is continue counting, and sometimes we go through huge gaps before hitting on a snuggly pair. So, it could be that the twin primes are finite and maybe they are infinite!

So, here's the challenge to you: come up with a proof to this question. You would become famous overnight!