Prime Number

What is a prime number?

Prime numbers are ones that consist of only two factors: the first is 1 and the other number is itself. Prime numbers are always whole numbers. In other terms, prime numbers are ones that cannot be split into equal groups(except 2).

Let us consider an example, number 2, Its factors are 1 and 2; therefore a prime number. Whereas take number 6, its factors are 1, 2, 3, therefore not a prime number. 

Let’s look at another example: number 1, for which we know there is just one element. As a result, it is not a prime number, as a prime number should contain exactly two elements. This means that 1 is a one-of-a-kind number because it is neither a composite number nor a prime number. 

Definition:

A prime number is any whole number larger than 1 that is divided exclusively by 1 and itself only. 

History Of Prime Number:

Humans have been fascinated with prime numbers since the ancients. Euclid presented the prime number theorem, which states that there exists an endless amount of prime numbers.

Eratosthenes, a famous scientist who lived after Euclid, devised a clever method for determining all the prime numbers up to a certain integer. The Sieve of Eratosthenes is the name given to this procedure.

Formula’s Of Prime Number

The prime numbers formula can be used to determine whether or not a given number is prime. It aids in the generation of a prime number at random. 

There are three known formulas:

1. For positive numbers/ integer (n); formula- (n+1) is prime, n! = n(mod n+1)
2. For numbers >3; 6n(+/-)1,  n=(+/-) (mod 6)
3. For numbers >40; n2 + n + 41

Rules To Find Prime Numbers:

• Even numbers in the place of any integer in the unit cannot be a prime number
• If the sum of all the digits of a huge integer is divisible by three, it is not a prime number
• All other numbers, with the exception of 2 and 3, maybe written using the prime numbers formula 6n(+/-)1, where n is a natural integer
• Subtract the final digit from the remainder of the integer. The original number will be equal to 7 if the response is divisible by 7

The Algorithm For Determining Whether Or Not X Is A Prime Number

1. Calculate x’s square root. This number should be rounded down to the whole number. This is known as truncating a number.
2. Examine all prime numbers that are less than or equal to x’s truncated square root.
3. x is prime if neither of those prime numbers split evenly into it.

How do prime numbers work?

I’m going to explain this with two examples.

1. Example 1: consider number 13, this number has only two factors. 1 and 13 are its factors, so it is a prime number.
2. Example 2: consider number 4, this number has three factors. 1, 2, and 4 are prime numbers.

Prime Numbers From 1 To 100 Are

There are 25 numbers between 1 to 100. The list of prime numbers from 1 to 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Co-Prime Number

Consider a pair of numbers: if there is no positive integer other than 1 that can divide them, the pair is co-prime number. This means if two positive numbers have no common component other than 1, they are said to be co-prime/relatively prime.

For example, take the numbers 21 and 22. 21 has 1, 3, 7, and 21 as its factors. 22 has 1, 2, 11, and 22 as its factors. Since these two numbers have only 1 in common, they are called co-prime numbers. 

Difference Between Prime Numbers And Co-Prime Numbers:

Prime numbers include 2,3,5,7,11,13,17,19, and so on. This reveals that each number in the first set contains precisely two factors: 1 and the number itself. They are prime integers by definition.

Co-primes are number pairings such as (3, 4,), (15, 61), (21, 22), and so on. This set indicates that while each group may contain prime or composite numbers, they have no common factors save 1. These are referred to as co-primes.

Conclusion:

A prime number is a natural number that can only be divided by itself and one. The rest of the numbers are composites. Every composite number has a distinct prime-factorization, which is a one-of-a-kind combination of the prime numbers or their powers that make up the number.

In mathematics and computer science, prime numbers are extremely essential. Long numbers are difficult to solve. Because finding their prime factors is challenging, the majority of the time, numbers that are most likely prime are utilized for secret codes.