Prime Numbers

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We know what a prime number is: any natural number greater than 1 which has no positive divisors other than 1 and itself is a prime number. The first few prime numbers are: 2, 3, 5, 7, 11, 13,...

For example, 11 is a prime number because other than 1 and 11, no other natural number is a divisor of 11. On the other hand, we have composite numbers: those natural numbers which are not prime. For example, 30 is a composite number.

Prime numbers play an extremely central role in Mathematics, as we will soon see. They satisfy many interesting properties, yet many results related to primes are still conjectures which have not been proven. Prime numbers have a certain mysterious aspect for mathematicians who study them, as the following statement by a mathematician shows:

Prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, but they also exhibit stunning regularity and there are laws governing their behavior, and it is amazing that they obey these laws with almost military precision.

You will learn more about these laws and rules in higher Mathematics. For now, we are concerned with only the basic properties of prime numbers.

A very important result that must be mentioned here is the following: the number of primes is infinite. This was first proven by Euclid more than 2300 years ago. And his proof is extremely simple and instructive, so we outline it here in steps:

1. Start by assuming any finite list of prime numbers. Let us denote the finite list of primes as follows (in ascending order, so that the last number is the largest): ${p_1},\,\,{p_2},\,\,{p_3}\,,...,\,{p_n}$.

2. Consider the number N given by: $N = {p_1} \times {p_2} \times {p_3} \times ... \times {p_n} + 1$

3. First, we note the obvious fact that N is larger than P. Also, we note that none of the primes on our list will divide N, since there will always be a remainder of 1 left over after division with any of these primes.

4. Now, if N is a prime number, then we have found a prime number which is larger than the largest prime on our list.

5. If N is not a prime number, then some prime number p smaller than N must divide N (can you see why?). But this means that this prime number p is not on our list of primes, since none of the primes on that list divides N.

6. In both cases (N being prime or non-prime), we have shown that there will definitely exist a prime number which is not on our original list. This reasoning applies to any finite list of primes. For any finite list of primes, there will exist a prime number not on that list. Thus, we conclude that the number of primes is infinite. Neat, isn’t it?

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