The Mystery of Prime Numbers

 

The Mystery of Prime numbers.  

Solving the problem of prime numbers is an exceedingly challenging task for mathematicians.

A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Using Euclid’s theorem, there are an infinite number of prime numbers.

A prime number is any number that cannot be expressed as the product of two numbers other than itself and one. 

For example, 15 = 3 x 5, so 15 is not a prime number; and 12 = 6 x 2 = 4 x 3, so 12 is not a prime number. On the other hand, 13 = 13 x 1, and is not the product of any other pair of numbers, so 13 is a prime number.

There is no way for us, even with a university advanced training in pure mathematics can we tell, just by looking at some numbers whether they are prime or not. 

We can tell at once that certain numbers are not prime. Any number, however long, which ends in a 2, 4, 5,6,8 or 0, or whose digits do not add up to a sum divisible by 3, is not a prime. 

But if a number ends in 1,3,7, or 9, and if its digits do not add up to the sum divisible by 3, it may be a prime or may not. There is no formula for us calculate out this. We just have to and see if we can make it by the product of two smaller numbers.

One way of looking for a prime randomly is to begin by entering all the numbers beginning with 2 and going as high as we possibly can, say up to 100,000. The first number is 2, which is prime. Omit that and go up the list by removing every second number. That eliminates all the numbers divisible by two, which are therefore not prime. The smallest number left after 2 is 3. That’s the next prime and, leaving that in place, you just cross out every third number thereafter to get rid of all the numbers divisible by 3. The next untouched number is 5 so you cross off every fifth thereafter. The next is 7, every seventh; then 11, every eleventh; then 13… and so on.

We might ponder that as we keep crossing out more and more numbers, we may ultimately reach a point where all the numbers greater than the same particular number will be crossed out, so there will be no more prime numbers after some particular highest prime number.

Unfortunately, it is not that simple.  It just cannot happen. No matter how high up we go into the millions, billions, perhaps into trillions (I have not tried), there are always more prime numbers higher up that have escaped all the crossing-out.

We already know as long ago as 300 BC., the Greeks mathematician Euclid demonstrated that no matter how high up we go there must be prime numbers higher still.

For instance, if we take the first six prime numbers and multiply them together: 2 x 3 x 5 x 7 x 11 x 13 = 30,030. Now add 1 to get 30031. That number cannot be divided evenly by either 2,3, 5,7, 11, or 13, since in each case we will get a result that will leave a remainder of 1.

If 30,031 can’t be divided by any number except by itself, it is a prime number. If it can, then the numbers of which it is a product must be higher than 13.

In fact, 30,031 = 59 x 509.

We can do this for the first hundred prime numbers or the first trillion, or the first any amount. If we calculate the product and add 1, the final figure is either a prime number itself or the product of the prime number higher than those we’ve included in the list. In other words, it does not matter how far we go , there are still prime numbers higher still, so that the number of prime numbers is infinite.

Occasionally, we may come to pairs of consecutive odd numbers, both of which are prime: 5, 7; 11, 13; 17, 19; 29, 31; 41, 43. As high as mathematicians have searched, such prime pairs are found. Are there an infinite number of such prime pairs? I don’t think we know. Mathematicians think otherwise, but they have never been able to prove this.  

That’s why we are interested in prime numbers. Prime numbers overture simple sounding problems that are very tough to work out and mathematicians can’t resist this challenge.

What use are these prime numbers we may ask? None we admit, except they offer us a huge academic challenge to the greatest of our mathematical minds. 

Perhaps it is easier for us to apply tensor calculus in astrophysics and in theorectical cosmology, areas of studies I am familiar with at Oxford than trying to deal with this mystery on prime numbers! 

Any mathematician wanting to take up this mind-blowing challenge?