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Prime Numbers

Prime numbers are a special category of natural numbers (or positive integers) which are exactly divisible by 1 and by itself. So, a prime number has exactly two divisors. Prime number is a number which is greater than 1 and which can be efficiently divided by 1 and by itself, not by any other number. It is also a whole number.

An Integer P > 1 is called a prime number when its only divisors are 1 and P. Any number m > 1 which is not a prime is called a composite.

Theorem: Every composite number could be factored into prime factors and each of these is unique in nature.Simple properties of primes:

(a) A prime 'p' is either relatively prime to a number 'n' or divide it.
(b) A product is divisible by a prime 'p' only when 'p' divides one of the factors.
(c) Every n > 1 is divisible by some prime.

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What is a Prime Number?

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Prime numbers are a special category of natural numbers or positive integers which are exactly divisible by 1 and the number itself. So, a prime number has exactly two divisors.
Largest Prime Number 
The largest known prime number is 2$^{74,207,281}$ - 1 as of January 2016. It is said to have 22,338,618 digits.

Smallest Prime Number
As we know, 2 is the smallest prime number.


Are all Prime Numbers Odd?

Yes, all prime numbers are odd, but only even prime number is 2. Since every other even number can be divide by a 2.

List of Prime Numbers

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Till now, we have explained the concept of prime numbers and given below is list of prime numbers for your better knowledge.
The list from 1 to 200 are as follows:

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


Is 1 a Prime Number?

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According to the prime number definition, we can say that 1 is not a prime number. The definition says that a prime number should have exactly two divisors. But 1 has only one divisor. So, 1 is not a prime number.

Is 2 a Prime Number?

Yes, 2 is a prime number because it is divisible by itself and 1.

How to Find Prime Numbers

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Let's identify whether the following numbers are prime or not
The number 2 is exactly divisible by 2 and 1.

The number 3 is exactly divisible by 3 and 1.

The number 4 is exactly divisible by 4, 2 and 1


The number 5 is exactly divisible by 5 and 1


The number 6 is exactly divisible by 6, 3, 2 and 1


The number 7 is exactly divisible by 7 and 1


The number 8 is exactly divisible by 8, 4, 2 and 1


The number 9 is exactly divisible by 9,3 and 1


The number 10 is exactly divisible by 10, 5, 2 and 1


The numbers 4, 6, 8, 9, 10 are not prime numbers since it has more than 2 factors.


The numbers 2, 3, 5, 7 are prime numbers since it has exactly two factors, 1 and the number itself. You can learn more about prime numbers online with the help of our tutors.

Twin Prime Numbers

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A twin prime numbers are defined as a pair of prime numbers having a difference of two. In other words, A pair of prime numbers which have a gap of 2 are called twin prime numbers. It means that twin primes are consecutive odd numbers which are both prime.

For example: (3, 5), (5, 7), (11,13), (17, 19), (41, 43) etc. Whether this list ends or not, i
t is unknown.

Also, (2, 3) are not categorized as twin prime, since they do not have a difference of two.

Prime Number Theorem

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Prime Number Theorem precisely gives us the value to which the density of prime numbers less than a given number approaches as the numbers grow larger and larger i.e. tend to infinity. There are many proofs easily accessible on the internet or otherwise. Hence, only its statement is being given here.

Statement of Prime Number Theorem:
Let P(x) be the number of prime numbers less than x, where x > 1 and x is a real number. Then the prime number theorem states

$\lim_{x \to \infty }$ $\frac{P(x)}{\frac{(x)}{(log x)}}$ =1.

This asymptotic formula precisely gives the value to which prime number density approaches as the numbers grow larger and larger. It has been termed as one of the most beautiful and important theorems in the history of mathematics and number theory.
More topics in Prime Numbers
Prime Numbers up to 100 Properties of Prime Numbers
Relatively Prime Numbers
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