Hello,
Greetings for the day!
In yesterday’s post, we discussed the basic properties of Prime Numbers and a couple of methods to identify them. To view the post, please click on this link:
https://bit.ly/2Yu4JMYBut, both these methods had their own limitations in being able to help us in identifying a prime number.
We had mentioned in our post yesterday that we will be discussing a foolproof approach to identifying Prime numbers. Therefore, today, let’s look at a method that can be applied on any number to ascertain whether it is prime or not.
Let’s suppose we are trying to ascertain if a number X is prime or not. Using a very simple algorithm, involving a few simple steps, you will be able to find out if X is prime. So, what’s this algorithm? We have listed it out in a step by step manner so that it becomes easy for you to perform the calculations:
First – Find out the nearest perfect square to X, lesser than X.
Second – Calculate the square root of this perfect square that you just found out in the previous step.
Third – List out all prime numbers which are lesser or equal to the square root you calculated in the second step.
Fourth – Check whether the given number X is divisible by any of the prime numbers listed down by you in the third step.
Fifth – If it is divisible by even ONE of the prime numbers, then it means X has more than 2 factors (remember the fact that any positive integer will definitely have 2 factors – 1 and itself) and hence is NOT prime.
On the other hand, if X is not divisible by any of the prime numbers listed down by you, then X is a prime.
This method can be applied to any number and does not have exceptions like the previous two methods. The only pre-requisites, for you to apply this method effectively, are that you need to know basic divisibility rules and at least the first 15 prime numbers by heart.
Having said that, this is not to say that the methods discussed in our previous post, are totally ineffective. In a lot of questions where you need to identify prime numbers, we would still recommend you to start off by using the (6k) - 1 and the (6k) + 1 concept, to eliminate at least 1 or 2 options. Post this, you can use the method that we described today to zero in on the Prime number, of the remaining options.
In the third part of this post, tomorrow, we will look at a few important application areas of Prime numbers so that you get a hang of how to apply the concepts that we have discussed till now.
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