Is 1 a prime number?

Is 1 a prime number?

 There are lots of confusion about the primality of the number one.

 So, naturally the question arises that is 1 a prime number?

The short answer is: no.
Yes it's simply  'no'!
   

Now let's prove our answer with mathematics.

Before we discuss the reason behind this , we shall recall the definition of a prime number.
    Definition: A prime number is a positive integer (natural number), which is greater than 1 and has exactly two factors 1 and the number itself. As for example, 2,3,5,7,11 etc.
    The definition can be stated in another way that: a prime number is a natural number (>1), which can not be expressed as the product of two smaller natural number both smaller than that number.
   

     Here, from the definition, the point to be noted that , 2 is the smallest prime number. So, every prime number is greater than 1. Thus  , we can't say 1 as a prime number.
 On the other hand, one can not be expressed as the product of two smaller natural numbers both smaller than 1 (as 1 is the least natural number).
 So, in this case also 1 failed to prove himself as a prime number.

  There are some other theories which  also prove that one is far more special than a prime number.
   (1) one is the unit of the positive integers.
   (2) one is the only multiple identity.
   (3) one is the smallest natural number.
   (4) one is the smallest positive integer which merits its own existence  by peano's axiom.
    (5) one is the only positive integer  which has only one factor or divisor 1.
    (6) The fundamental theorem of arithmetic states that,
"Every positive integer greater than one can be written uniquely as a product of primes, with the prime factors in the product written in order of nondecreasing size".
   Here we find the most important use of primes. They are the unique building blocks of the multiplicative group of integers. In discussion of warfare you often hear the phrase "divide and conquer." The same principle holds in mathematics. Many of the properties of an integer can be traced back to the properties of its prime divisors, allowing us to divide the problem into smaller problems. The number one is useless in this regard because a = 1 ×a = 1 ×1 ×a =1×1×1×a= ... That is, divisibility by one fails to provide us any information about a .

     Here, the interesting fact is , 1 is also not a composite number(a number which is not a prime number).
  So 1 is neither a prime number nor a composite number; it's unity!!!

  

 If you find out any incorrect information or know anything more about this , please write it in the comment section!