The branch of mathematics which deals with the study of natural numbers, is called number theory.
The study of natural numbers which do not have any factors (factors are the numbers which exactly divide with remainder zero) other than 1 and itself, is a part of it.
They have been the subject of intense research in pure mathematics.
They have also found application outside of pure mathematics. They formed the basis of the algorithms in public-key cryptography .
To see first 10,000 primes (the 10,000th is 104,729), look at the
List Of Primes.
Note : (i) 1 is not a prime. (ii) 2 is the only even prime. (iii) There are only 4 single digit primes.
Composite Numbers
A number which has atleast one factorother than 1 and itself is called a Composite Number.
Examples : 4, 6, 8, 9, 12 etc. are composite numbers.
Note : (i) Every composite number will have at least 3 factors. (ii) 1 is not a composite number. (iii) 9 is the only single digit odd composite number. (iv) There are only 4 single digit composite numbers.
1 is neither a prime nor a composite number
Division of Natural Numbers into three disjoint sets
Thus we can divide the set of Natural Numbers into three disjoint sets like (i) { 1 } (ii) { Prime Numbers } (iii) { Composite Numbers }.
Method of finding Prime Numbers by Sieve of Eratosthenes
Eratosthenes was a Greek Mathematician, of 3rd century B.C.
He suggested a method to find out Prime Numbers. It is called the SIEVE OF ERATOSTHENES.
This method helps to find out prime numbers upto a given natural number n.
Let us see an Example.
Solved Example 1 of Prime Numbers :
Find out the Primes upto first 100 natural numbers using the Sieve of Eratosthenes.
Solution:
STEP 1 :
Write all natural numbers from 1 to 100 in 10 rows of 10 numbers each.
1
2
3
4
5
6
7
8
9
10
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12
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100
STEP 2 :
Strike off 1 which is not a Prime.
STEP 3 :
First Prime is 2. Leave 2 and strike off all the multiples of 2 which are 4, 6, 8..........100.
STEP 4 :
The next number that is left is 3 and 3 is a prime. Leave 3 and in the remaining numbers, strike off all the multiples of 3 which are 9, 15, 21..........99.
STEP 5 :
The next number that is left is 5 and 5 is a prime.
Leave 5 and in the remaining numbers, strike off all the multiples of 5 which are 25, 35, 55..........95.
STEP 6 :
The next number that is left is 7 and 7 is a prime.
Leave 7 and in the remaining numbers, strike off all the multiples of 7 which are 49, 77, 91.
STEP 7 :
This process will end at this stage.
Observe all the remaining numbers are primes only.
Note that, for finding primes upto 100, 7 is the highest prime which does not exceed √100 (= 10).
So the process ended after striking off multiples of 7.
In general
For finding primes upto n, We have to continue this process upto the highest prime which does not exceed √n.
Method of finding Prime Numbers by Formulas
Formulating the set of primes is far from easy.
There are several attempts to arrive at formulae for Prime Numbers. Some of them use concepts of advanced mathematics.
Here is a simple one.
Euler's Formula
Euler's formula states that the quadratic polynomial n2 + n + 41 is a prime number for all non-negative integers less than 40.
The primes for n = 0, 1, 2, 3... are 41, 43, 47, 53, 61, 71...
The differences between the terms are 2, 4, 6, 8, 10...
For n = 40, it produces a square number, 1681, which is equal to 41×41, the smallest composite number for this formula.
Fermat's Formula
Fermat proposed that the numbers of the form 22n + 1 are primes.
But it is true, when n = 0, 1, 2, 3 and 4 and the corresponding Primes, called Fermat's Primes are 3, 5, 17, 257, 65537.
When we observe the set of primes, the composite numbers between any two primes is continuosly increasing.
Look at the
List of Primes. We observe that there are a mere 80 primes between 103,000 and 104,000.
It is suspected that the process of getting a prime may cease at any stage.
But Euclid proved that the set of Primes is infinite.
Scientists are finding larger and larger Primes using Super Computers.
Primes having more than nine million digits were also found.
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A pair of primes are said to be twin primes if they differ by 2.
Examples : 3 and 5 are primes and they differ by 2. So 3, 5 are twin primes.
Similarly 5 and 7 are twin primes.
17 and 19, 29 and 31, 71 and 73 are some more examples.
A question arises : Are there infinitely many twin primes ? This is still an unsolved problem in number theory.
To see more about twin primes and other types of primes go to the
List Of Primes.
Relatively Prime Numbers or Co-Prime Numbers
Two natural numbers which do not have a common factor other than 1 are relatively prime to each other and are called co-prime numbers or simply co-primes..
(1) The
G.C.F.
of two relatively primes is 1 and their
L.C.M.
is the product of the numbers.
(2) Any two primes are always relatively prime to each other.
(3) Two relatively prime numbers need not be primes. See the pair (4, 9). Neither 4 nor 9 is a prime number. But, they are relatively prime to each other.
(4) We can observe that,
If two co-primes are factors of a number, their product is alsoa factor of the number.
(5) We can also observe that,
If a and b are co-primes, c is another number such that,a is a factor of bc, then a is a factor of c.
This is known as theorem of Gauss.
(6) The integer 1 is coprime to every natural number, including itself.
Prime Factors and Prime Factorization
The factors of a number which are prime numbers are called the Prime Factors of that number.
Expressing a given number as the product of Prime Factorsis called Prime Factorization.
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