**Example of prime number:**

** ****1 .** The number 7 can only be divided by 1 and 7, so 7 is a prime number.

- 1 is not prime number, since it has only one divisor, namely 1.
- However, 2 and 3 are prime number, since they have exactly two divisors, namely 1 and 2, and 1 and 3, respectively.

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Below are the properties of prime numbers:

- Numbers of prime numbers are uncountable. i.e., the number of primes is infinite.

- Every Prime number have exactly two factors or divisors

- Number of even primes is ONE

- G.C.D of co-prime numbers is always ONE.

- The largest known prime is 2
^{43,112,609}- 1

- The General formula for generate the primes

** Formula:** N = (2^{p} - 1) with p equal to the primes 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 etc

- If p is a prime, is 2
^{p}- 1 always square free.

- The Twin Primes Conjecture that there are infinitely many pairs of primes only 2 apart.

- The term odd prime refers to any prime number greater than 2.

** Example: ** 3, 5, 7, ..............

- All prime numbers above
*q*are of form q =*n*+*m*,

Where 0 < *m* < *q*,

And *m* has no prime factor ≤ *q*.

- If
*p*is a prime number and a, b are the integers .

* If p* divides *ab* ,

Then *p* divides *a * or *p* divides *b*.

**Twin prime numbers**

Example: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43)

**Cousin prime numbers**

Examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41)

**Balanced prime numbers**

**Palindromic prime numbers**

**Reversible prime numbers**

Example: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157

**Pythagorean prime numbers**

**Permutable prime numbers**

**Mersenne prime numbers**

** **Example:** **3, 7 , 31, 127, 511,** **2047** **

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