Try this beautiful problem from Integer based on Prime number useful for ISI BStat Entrance.
The number of integers \(n>1\), such that n, n+2, n+4 are all prime numbers is ......
But try the problem first...
TOMATO, Problem 70
Challenges and Thrills in Pre College Mathematics
taking n=3, 5, 7, 11, 13, 17....prime numbers we will get
Case of n=3
Case of n=5
n+4=9 which is not prime....
Case of n=7,
n+2=9 which is not prime ...
Can you now finish the problem ..........
We observe that when n=3 then n,n+2,n+4 gives the prime numbers.....other cases all are not prime.Therefore any no can be expressed in anyone of the form 3k, 3k+1 and 3k+2.
can you finish the problem........
If n is divisible by 3 , we are done. If the remainder after the division by 3 is 1, the number n+2 is divisible by 3. If the remainder is 2, the number n+4 is divisible by 3
The three numbers must be primes! The only case n=3 and gives\((3,5,7)\)