Understanding Prime Numbers

 One of the most intriguing things in Mathematics is Prime Numbers. To put it in simple words, the numbers are all out in the "number world" all by themselves. They have no relatives i.e. factors other than themselves and the "universal factor" called 1. So, what's really unique about prime numbers?  

 

Over the history of Mathematics and especially Number Theory, many theorems based on prime numbers have been proposed. The best thing about prime numbers is that they are the building block of the whole number system. If you add two prime numbers, it always gives an even number greater than 2. Likewise, if you were to pull off a number line so infinitely long then most of the numbers except for the prime number themselves, would just be the multiple of prime numbers or in other words products of prime numbers. To put into Euclid's own words, "Every natural number can be represented as a product of prime numbers." 

 

Here's a fun activity for you.  

Think of the 1st two prime numbers, find their product and add 1. A prime number right? 

Let's try one more, take 1st five prime numbers, find their complete products, and add 1 to the final product. 

 

But is this all that makes prime numbers interesting? Definitely, not. There is much uniqueness portrayed by prime numbers but one of the basic ones being that, if you elongate the positive side of the number line, the longer you extend the line, the rarer you get to see prime numbers. Similarly, in the same elongated line, two prime numbers occasionally appear just after a single gap. In the first ten natural numbers, there are two sets of primes, {3,5} and {5,7} separated by a single number. We also have other similar sets such as {11, 13}, {17,19} and so on. We refer to such occurrences as "twin primes".  

 

Now here's the interesting part. For the first twenty numbers, you might think that in every 1st ten numbers, there is always 2 twin primes i.e. {3,5} and {5,7} between 1-10 and {11,13} and {17,19} between 11-20. In order to verify whether this proposed statement is true or not, let's see few graphs.  

 

Here's a graph showing the occurrence of prime numbers. 

 

 

Did you notice, every time the line hit a prime number, the graph rose by 1, making somewhat a picture of the 2D staircase. If we look at the figure from 3 to 7 along the X-axis, the "staircase" seems uniform but after we hit 7, the line goes blank until it meets the next prime number. 

 

If we were to zoom out a bit, this is how our graph (the below graph) should look like. Notice how the "uniform staircases" get rarer. In addition to that, we get to see a lot of blanks before the next jump.  

 

Finally, the second picture of the same image shows how the graph of prime numbers would look like if we zoom out to see their occurrence pattern in the first 1000 numbers, in a small plane. 

 

 

For those of you wondering what 

π (x)  means, it is known as a prime counting function, introduced by Gauss. Also the π has n−

 

othing to do with the fucntion. To put into simple words, since we don't have any known 

 
function for counting primes, this prime counting function is known to us only as a plot that increases by 1 whenever x hits a prime number.  

 

Here are two different conclusions from the above picture: the farther we go, 

 the rarer we see the  primes and, 

 the rarest gets the twin primes. 

 

But there's one more thing. The way they occur is different, rather confusing and that is why they are so interesting. If we were able to really understand the pattern of prime numbers, say in a number line, not only would it solve the mystery for twin primes but also give a greater understanding of numbers. 

 

Prime numbers have been studied by great mathematicians over the course of history. It started when Euclid proved that there are infinite prime numbers and all the natural numbers are a product of prime numbers. Then for over many years, prime numbers remain untouched until Gauss and Legendre's formulated the prime number theorem which was then proved by another group of mathematicians: Hadamard and da le Vallee Poussion. However, the 8-page paper published in 1859 by Bernhard Reimann made the biggest breakthrough in prime number theory, as the paper made new and previously unknown discoveries regarding the distribution of prime numbers. The same paper became the core behind the proof of what is famously known as the Prime Number Theorem (1859).  

 

If so, then we must have understood prime numbers and their behavior, right? For the fun part, the answer is a mysterious NO, just like the Reimann Hypothesis.  

 

Reimann, during his study about the behavior of prime numbers, observed that the frequency of prime numbers is very similar to the behavior of harmonic series of a general type of function i.e. 

 

ς(s)=1+12s+13s+14s+15s+……

 

 

called the Reimann Zeta function. The Reimann Hypothesis states that for all the interesting solutions of the equation 

ς(s)=0

 lie on a certain vertical straight line called the critical line. 

 

 

If there is even a single prime number that disproves what Reimann suggested, it would be groundbreaking. But at the same time, proof that is true for every interesting solution would shed light on our understanding of these strange behaving prime numbers, especially their distribution. So far, the hypothesis has been checked for the first 10 trillion solutions.   

 

For now, Reimann's hypothesis about the roots of zeta function remains a mystery and so do the prime numbers with their intriguing behavior, by which I mean distribution. 

 

Fun fact: If you solve this mysterious beautiful problem, not only would you receive immortality in Mathematics, but also a million dollars.