Prime Numbers in Discrete Mathematics
Last Updated :
27 Aug, 2021
Overview :
An integer p>1 is called a prime number, or prime if the only positive divisors of p are 1 and p. An integer q>1 that is not prime is called composite.
Example –
The integers 2,3,5,7 and 11 are prime numbers, and the integers 4,6,8, and 9 are composite.
Theorem-1:
An integer p>1 is prime if and only if for all integers a and b, p divides ab implies either p divides a or p divides b.
Example –
Consider the integer 12.Now 12 divides 120 = 30 x 4 but 12|30 and 12|4.Hence,12 is not prime.
Theorem-2 :
Every integer n>=2 has a prime factor.
Theorem-3 :
If n is a composite integer, then n has a prime factor not exceeding √n.
Example-1 :
Determine which of the following integers are prime?
a) 293 b) 9823
Solution –
- We first find all primes p such that p2< = 293.These primes are 2,3,5,7,11,13 and 17.Now, none of these primes divide 293. Hence, 293 is a prime.
- We consider primes p such that p2< = 9823.These primes are 2,3,5,7,11,13,17, etc. None of 2,3,5,7 can divide 9823. However,11 divides 9823.Hence, 9823 is not a prime.
Example-2 :
Let n be a positive integer such that n2-1 is prime. Then n =?
Solution –
We can write, n2-1 = (n-1)(n2+n+1). Because n3-1 is prime, either n-1 = 1 or n2+n+1 = 1.Now n>=1, So n2+n+1 > 1,i.e., n2+n+1 != 1.Thus, we must have n-1 = 1.This implies that n=2.
Example-3 :
Let p be a prime integer such that gcd(a, p3)=p and gcd(b,p4)=p. Find gcd(ab,p7).
Solution –
By the given condition, gcd(a,p3)=p. Therefore, p | a. Also, p2|a.(For if p2| a, then gcd (a,p3)>=p2>p, which is a contradiction.) Now a can be written as a product of prime powers. Because p|a and p2| a, it follows that p appears as a factor in the prime factorization of a, but pk, where k>=2, does not appear in that prime factorization. Similarly ,gcd(b,p4)=p implies that p|b and p2|b. As before, it follows that p appears as a factor in the prime factorization of a, but pk, where k>=2, does not appear in that prime factorization. It now follows that p2|ab and p3|ab. Hence, gcd(b,p7) = p2 .
Primality Test Algorithm :
for i: [2,N-1]
if i divides N
return "Composite"
return "Prime"
Example –
Let’s take an example and makes the algorithm more efficient, 36=
1×36 |
2×18 |
3×12 |
(a=4)x(b=9) |
6×6 |
9×4 (repeated) |
12×3 (repeated) |
18×2 (repeated) |
36×1 (repeated) |
Take the inputs of a and b until,
a<=b
a . b = N
a . N/a = N
Modified algorithm :
for i : [2,√n]
if i divides N
return "Composite"
return "Prime"
C++
#include <bits/stdc++.h>
using namespace std;
bool is_prime( int n){
for ( int i = 2; i * i <= n; i++){
if (n % i == 0){
return false ;
}
}
return true ;
}
int main() {
int n;
cin >> n;
cout << is_prime(n) ? "prime" : "composite" ;
return 0;
}
|
Java
import java.util.*;
class prime{
public Boolean is_prime( int n){
for ( int i = 2 ; i * i <= n; i++){
if (n % i == 0 ){
return false ;
}
}
return true ;
}
}
class GFG {
public static void main (String[] args) {
Scanner sc = new Scanner(System.in);
int n = sc.nextInt();
prime ob = new prime();
System.out.println(ob.is_prime(n) ? "Prime" : "Composite" );
}
}
|
Python3
import math
def is_prime(n):
for i in range ( 2 , int (math.sqrt(n))):
if (n % i = = 0 ):
return False
return True
if __name__ = = "__main__" :
n = int ( input ())
if is_prime(n):
print ( "prime" )
else :
print ( "composite" )
|
Javascript
<script>
function is_prime(n)
{
for ( var i = 2; i * i <= n; i++)
{
if (n % i == 0){
return false ;
}
}
return true ;
}
var n;
document.write(is_prime(n) ? "prime" : "composite" );
</script>
|
Algorithm to find prime numbers :
In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit.
Algorithm Sieve of Eratosthenes is input: an integer n > 1.
output : all prime numbers from 2 through n.
let A be an array of Boolean values, indexed by integers 2 to n,
initially all set to true.
for i = 2, 3, 4, ..., not exceeding √n do
if A[i] is true
for j = i2, i2+i, i2+2i, i2+3i, ..., not exceeding n do
A[j] := false
return all i such that A[i] is true.
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