The Numbers , Part 1

 Numbers and especially the whole numbers are of special significance because as a child we are first taught about them only. The most common facet of mathematics is counting and it makes sense to revisit such a fundamental concept again.

So lets start with something familiar, A positive number, lets say 25. It is a perfect square. It has divisors in 1 , 5 and in 25 itself. So it is breakable number , meaning it can be broken down as a product of lower integers , i.e 25= 5*5. 

Now what about the number 23, you can see that it is not divisble by any integer except 1 and 23 itself and hence it is unbreakable. Such type of numbers are called the PRIME Numbers. Prime numbers are the building blocks of the number system because every whole number can be uniquely expressed as a product of prime numbers. (This statement is known as fundamental theorem of Airthmetic)

See for example 100 = 2*2*5*5

44 = 2*2*11, 99= 3*3*11 etc

(Here you see the integer 2 as a prime number, indeed it is and it remains the only even integer which happens to be a prime number. It is because that it is only divisible by 1 and itself)

Whole numbers other than the prime numbers are called the composite numbers. For example 26, 49, 65 etc.

The first milestone in the discussion of the prime numbers is the theorem that there are infinitely many prime numbers. The proof of this theorem is due to euclid and is of pristine nature. It is based on negation, i.e the proof by contradiction.

We assume that are finite prime numbers say p1, p2, p3....pn. Now construct a number N such that

N= (p1*p2*p3....*pn) + 1

N can be either a prime number or a composite number. If it prime then our proof is complete.

So lets assume that N is a composite number, in this case fundamental theorem of airthemetic will ensure an unique prime factorization of N. Since none of the given prime numbers from p1 to pn divides N, it implies that there must be prime numbers other than the assumed finite ones, which concludes the proof.

Another commonly stated fact is that any prime number (except 2 and 3) can be expressed in the form '6k+1' or '6k-1', where  k is some whole number. Why?

It can be seen in this way

Let p is a prime number, now consider the following 3 numbers

p-1, p ,p+1

Clearly p is odd, that will make both p-1 and p+1 the even number. 

Also these 3 numbers are consecutive integers and hence either p-1 or p+1 will be divisible by 3. ( here remember that p being a prime number cant be divisible by 3)

Combining these above two observations will give that either p-1 or p+1 will be of the form '6k' and that concludes the proof.

Another way to get this is to see that any six consecutive integers will look like as 

6m, 6m+1,6m+2,6m+3,6m+4,6m+5 (Where m is some integer and the this order doesnt hold any water)

Out of these, integers 6m, 6m+2, 6m+3 and 6m+4 are all composite numbers so a prime can only assume the form of 6m+1 or 6m+5.

Well on the same lines any 4 consecutive integers can be written as 4m, 4m+1, 4m+2 and 4m+3. Out of these integers 4m and 4m+2 are both composite. So any prime number will be of the form of 4m+1 or 4m+3 (or 4k-1). Indeed it can be proved that there are infinitely many prime numbers of the form 4k+1 and also of form 4k-1.

( to be contd..thanks for paying attention)