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Prime Numbers

 Prime Numbers 

What is the first thing that comes to your mind when someone says prime number? A number that is divisible by nothing except 1 and itself. Well, that is the definition of prime numbers but is there something in primes beyond the indivisibility. It turns out the answer is yes. Any natural number from the number line can be represented as the product of the primes. It seems so obvious but it implies that all the other numbers are just the result of primes or prime are the original numbers and all others are just the copy of prime. Due to this reason I like to think of prime as the building block of the number line, they are like the atoms of our body. One might ask if primes are so important what has been done to understand the primes and do we actually know all properties of primes? No, we don’t but there are some formulas to approximate the number of primes upto a certain number N. One of such formula, which is also the first such formula, to approximate the primes is ฯ€(๐‘) =N/logN. Let X be the actual number of prime upto N. Then, l๐‘–๐‘š ๐‘ → ∞ . ๐‘‹/(๐‘/๐‘™๐‘œ๐‘”๐‘) = 1 

Let’s use this formula to calculate the number of prime upto N. 

Put N=100, then 100/๐‘™๐‘œ๐‘”100 = 21. 7

 And the real value of prime (X) is 25.

Let’s try for when N=10000 

10000/๐‘™๐‘œ๐‘”10000 = 1085. 7 whereas the X in this case is 1229.  The difference seems to be increasing but the ratio of the difference and the N is decreasing. The ratio of percentage decreases from 3.28% to 1.43% as N goes from 100 to 10000. Above shown formula gives a good approximation but there is another formula which gives a far better approximation and is most used by mathematicians to approximate the number of primes.




As it requires calculus to understand this, we will not be going through with this. The logarithmic formula also shows the distribution of prime as N gets bigger. From the formula, when N is large enough, the probability of a random integer less than N being prime is approximately 1/logN. Using this, we can figure out that an integer with y digit is about twice more likely to be a prime than an integer with 2y digits. For example:

When the integer is 500 digit long, the probability of it being a prime is . 1/ 500๐‘™๐‘œ๐‘”10.

 And, when the integer is 1000 digit long, the probability of being a prime is 1/1000๐‘™๐‘œ๐‘”10. 

As we can see the probability is dependent on the digit of the number so we can draw a conclusion that prime gets more and more scarce when N approaches infinity.


-Navaraj Pandey

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