Monthly Archives: March 2001

The Prime Adventure

We all look for adventures to make our lives exciting and to challenge the limits of our abilities . When we look for adventures we look at high mountains , deep seas or fast flowing rivers . But there are adventures all around us . I talk about the world of Mathematics . ANd some of the challenges you face here would put climbing Mount Everest to shame .

In VERITAS today we explore the problems related to primes that have challenged Mathematicians for centures . SOme of these are open ( unsolved ) problems and you may want to try solving them . I assure you that once you start you would get more than a hillclimb of adventure . And if you succeed in solving any one of these open problems you can assure yourself of a place in history .

       A prime number as all of us know is one that is divisible by no other number except 1 and the number itself . 2 , 3 , 5 , 7 are all examples of prime numbers .

       In Book IX of the Elements, Euclid proves that there are infinitely  many prime numbers. So we have a big playground !

Open Problem 1  : Goldbach’s conjecture : Every even number greater than

2 can be expressed as a sum of two primes . For example take the number 16 .

16 = 11 + 5  ( sum of 2 primes ) . No one has ever found any even number that violates this conjecture . But no one has ever been able to prove it .

Would you like to try ?

       The first 100 numbers have 27 primes . From 100 to 200 we have 21 . From 200 to 300 we have 19 . So the density of primes ( the number of primes for 100 natural numbers ) keep decreasing . When you go into the billions you still get about 5 primes per hundred . So the decrease is very gentle . Lots of mathematicians have tried to find a formula which would tell you how many primes you will find within a set of numbers . But none of them has succeeded .  Mathematicians have come up with an approximate formula for this . They say :

 

The number of primes till a number x is approximately given by :

  x/(ln(x) -1)               where ln(x) is the natural logarithm of x .

 

Now this is not exact . Forexample in the first million natural numbers the actual number of primes is 78,498    . Our furmula gives us : 78,030  which is very approximate . So thet gets us to the second open question :

 

Open Problem 2 : Create a formula to calculate the density of primes .

Open Problem 3 : Is there always a prime between n^2  and (n+1)^2 ?

 

For example take n =10 and n+1 is 11 . So there is a prime between

100 and 121 . The primes are 101 , 103 , 107 … Can you prove / disprove

this .

 

And now the GREATEST ( and the most difficult ) problem . THis problem is not just the greatest problem in Number theory but the greatest in The World of Mathematics today . It is called the Reimann Zeta Hypothesis .And it is not for the faint hearted .

 

The German Mathematician Reimann noted that the frequency in which you will get primes is closely related to a function called the zeta function .

The zeta function is :

1 + 1/2^x + 1/3^x + 1/4^x …….

Reimann observed that all nontrivial solutions of the equation :

zeta function = 0

lie on a straight line . 

THis is a millinium prize problem and u become real famous if you solve it .

 

And finally the adventure of primes is summed up in the words of Euler :

 

Euler, Leonhard (1707-1783)

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.