Test for a Number to be Prime

“Prime number’ is a basic concept of mathematics. “Prime numbers” are considered to be whole or natural numbers which have two factors that are one or “the number itself”. A number which is greater than one is called a composite number. For instance, 7 is a prime number because the only way to write its product as 7 × 1 and 1 × 7. However, 16 is a “composite number” because its product is 4 × 4 and both of the numbers are less than the actual number.

Prime number

“Prime number” is the general concept of mathematics. “Prime numbers” are centred in number theory because of the “fundamental theorem of mathematics”. Each natural number that is greater than 1 can itself be multiplied by a prime number that is unique in their order. The feature of being prime is called “primality”. “Prime numbers” have two factors which are 1 and the number itself. In other words, a prime is to be called a prime number if it cannot be divided into equal groups. One can divide a number into subdivisions with the same numbers only when it can be factorized into “a product of two numbers”.  “Prime numbers” are always positive numbers that are greater than one. Natural numbers such as 2,3,5,7,11,13,15, etc. are considered to be prime numbers because these number products are either one or the number itself. In addition, it can be said that all the remaining numbers except for one are categorized as either prime or composite numbers. Generally, all odd numbers are considered to be prime numbers except for 2, 2 is considered to be the “smallest prime number” and only “even prime number”. 

What is the test for a number to be prime?

The test through which a number is prime or not can be identified is known as primality. Suppose a given number n whose primality is needed to be evaluated, a simple but comparatively simple method can be used. This method is to be known as trial division. Where the number n will be multiple in the range of 2 and √2. The other way through which primality can be evaluated is the “Miller-Rabin primality test”. This is a probabilistic approach. This method is simple but while applying this method a slight chance of error might happen. Another way of determining primality is through “AKS primality test” which is a slow approach but produces the correct answer. It is a deterministic approach to assessing primality for a given number. We know that prime numbers are divided by 1 and the “number itself”. Thus, for checking whether a number is to be prime or not one needs to test whether the given number is divisible by one and itself or not. 

Algorithm to see whether n is a prime number not:

• The first one needs to find a square of n and then round it down to the nearest “whole number”. This method is known as the intersection of a number.
• Examine whether all the “prime numbers” are less than or equal to “square root of n”.
• If no prime number is actually divided into n, then n will be prime. 

Example:

Is 361 a prime number?

• The square root of 561 will be approximately 23.69. We will consider it as 24.
• Prime numbers which are less than or equal to 22 will be 2,3,5,7,11,13,17
• After applying both divisibility and prime calculation, we have come to the point that these primes equally divide 561

Hence, 561 is a prime number.

Prime number definition

“Prime numbers” are those numbers that are completely divisible by one and “the number itself”. “Prime numbers” are always positive integers which have two factors 1 and “the number itself”. Any number that does not obey this principle is termed a “composite number” which can be further factorized into other “positive integers”. Through an example, the definition of “prime numbers” can easily be understood. Let’s suppose, 41 and 29 are prime numbers if they follow the criteria as they should be divisive by 1 and themselves. 41 have two products such as 1 × 41 and 41 × 1 whereas 29 have also two products such as 1 × 29 and 29 × 1. The “smallest and only even prime number” is 2 because it fulfills the criteria of being a “prime number”. The concept of prime number was first invented by mathematician Eratosthenes. Every “composite number” can be factored into “prime number” and all of them are special in nature.

Conclusion

The concept of a “prime number” is very essential in mathematics as it builds the fundamentals for number theories. Knowing prime numbers is essential for studying number theories. Prime numbers should always be positive and “whole numbers”. While understanding “prime numbers” two types of confusion arises – first one identified as 1 is not a “prime number” and 2 is a “prime number”.