By Melissa Rhoads Warden

A fascinating subject of number theory that continues to excite and amaze mathematicians and non-mathematicians alike is the study of the prime numbers. Although it would be impossible to discuss even a small fraction of the wealth of current knowledge about the primes within the space of this column, I hope to provide some tantilizing leads that will encourage you to do some reading on your own!

An early definition of what it means to be prime was given by Euclid in his Elements. A number is considered prime if it has no whole-number divisors other than 1 and the number itself. Using this rule, the first few elements of the sequence of primes can be generated: 2, 3, 5, 7, 11, 13, 17, 19... If you were looking at this sequence for the first time, you might wonder whether or not it continues forever. It turns out that it does. The first step in Euclid's proof of this statement is to suppose there is a number P, such that:

where P1, ... Pn are all the known primes. In order to show that there is a prime greater than Pn, you look at P twice, each time making a different assumption. The first time, you assume that P is prime. This leads to the conclusion that the list of primes can be continued, because you have found a prime greater than Pn - needless to say, this is a rather trivial case. But now you look at P and assume that it is not prime - in other words, that it must be evenly divisible by a prime. However, P is not divisible by any of the primes in the list on the right side of the equation - there is always a remainder of 1. Therefore, there must be a prime greater than Pn that divides P. Both ways of looking at P lead to the same result - that there is a prime greater than Pn! So the sequence of primes can always be continued.

Now that we know that there are infinitely many primes, we might like to know, for example, how they are distributed along the number line. How does the density of primes change as the numbers get larger? Does it approach a limit? If you look at the first few hundred primes, it seems that the density does decrease as the number gets larger - that there are more and more non-prime numbers between the primes. It is possible to prove that this pattern holds as you walk up the number line. But the density decreases in a chaotic fashion - it is not linear, but oscillates back and forth around a smoothly decreasing line - and does approach a limit of 1/log(N) as N approaches infinity.

There are many wonderful and amazing things about the primes, a set of numbers that is defined simply by the indivisibility of its members.

Who would ever have thought, for example, that between any number N and its double 2N you can always find at least one prime? There are many interesting patterns to be found within this sequence, and many questions that remain unanswered. Here are a couple:

1. The Goldbach conjecture. Is every even number the sum of two primes? This result has been shown to be true for numbers up to a billion, but it has not been generally proven.

2. The twin prime conjecture. Are there infinitely many numbers of twin primes - primes that only separated by one number? Again, there are many examples, but no conclusive proof has been found.

Perhaps you might answer one of these questions some day!

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