What is a prime number? How to check if a number is prime or composite?

Prime Numbers

Mathematics is just like our universe. It continues to expand as we grow our understanding in the subject. Mathematics came into existence with the need of man to quantize things. It gives a set of rules to follow when we use numbers and quantities to explain science. It is the base of science. Most of science would only be crazy theories without proving them with the help of mathematics. We can understand mathematics as a language. Just as people require a language to express their thoughts, they need mathematics to express science. Let us go into the topic of interest which is “Prime Numbers”.

 

We learn about prime numbers in the early stages of gaining a good foundation on mathematics. It falls under the learning of the Arithmetic theories of mathematics. When ancient people learnt the number system they started studying the properties and characteristics of the basic number system. The classification of the numbers was first studied by the Ancient Greek mathematicians such as Euler, Euclid and other mathematicians.

What is a prime number?

Natural numbers which are greater than one and have only two positive divisors which are one and the number itself are called prime numbers

Eg: The divisors of 5 are only 1 and 5.

      The divisors of 7 are only 1 and 7.

      The divisors of 11 are only 1 and 11.

 

1 is the conspiracy number of the number system. Even though it cannot be divided further into other numbers, 1 is not considered to be a prime number since there are no other numbers smaller than itself to divide it into.

 

Although even numbers are expected to be prime numbers, 2 is the only even prime making it unique among all the primes. All the other even numbers cannot be prime numbers as they can be divided by two.

*note: even numbers are represented by (2*n) where n is any natural number.

How can we check the Primality of a Number?

There are many ways and algorithms developed by various mathematicians like Fermat and Euclid and other later scientists. But although higher mathematical methods are used to check the Primality of larger prime numbers, we can use basic mathematic operations (like division and multiplication) and analytical skills to test the Primality of smaller numbers when we require the answer as a quick response.

           

Consider a number, split it into its positive divisors 1 and the number. Check if the number can be further divided. If that case is satisfied then the number that we took initially would be a non-prime number. Let us understand this through an example by observing and odd non-prime number 15.

Example:        step 1: 15 = 1*15

                     step 2: 15 = 1*3*5

Therefore the 15 is further divided into 3 and 5 making it a non-prime.

 

One more way of analysing it analytically is by checking the numbers ability to form a matrix. A matrix is basically a representation of quantities in the form of rows and columns. A prime number can never form a multi-dimensional matrix. Conversely non-prime numbers can form two dimensional matrices.

Example: suppose we consider the numbers 7 and 10

The number 10 (non-prime) can be formed as

* * * * *

* * * * * this can be evenly distributed into 2 rows and 5 columns.

If we try to make a two dimensional shape of 7 (prime) it would look like

            * * *

            * * * *               as we notice this cannot be evenly distributed into rows and columns.

Importance of Prime Numbers

According the fundamental theory of arithmetic, every composite number (numbers greater than 1 and which are not prime numbers) can be divided into two or more non-prime numbers. In the above segment, when we divided 15 as 3*5 and 10 as 2*5, it can be noticed that those numbers were consequently broken down into prime numbers. Therefore prime numbers can be considered as the building blocks of the number system. Combinations of prime numbers can be used to create any number of our choice.

Applications of prime numbers

It is hard to find a practical application for prime numbers. Until a long time people just used prime numbers as a base of arithmetic that is only in abstract mathematics. But lately prime numbers are being used in encryption software and technology. They find use in such fields due to their unique character making it indivisible to other formats. Therefore combinations of certain prime numbers of large size form a unique set of codes which are hard to cipher. Also cycles of certain rotor engines follow prime number orders to give more efficiency.

 

As a conclusion here is a list of the prime numbers from 1 to 1000 numbers which are 168 in number

 

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997

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