Find the least positive integer such that every prime number p,p2+n is never prime
To find the least positive integer such that every prime number p, p^2 + n is never prime, we can use the process of elimination.
First, let's assume a value of n, and then check for prime numbers p and the value of p^2 + n. If we find a counterexample where p^2 + n is prime, we can conclude that the assumed value of n is not the solution.
We will start with the smallest prime number, which is 2. Let's substitute p = 2 and check p^2 + n for various values of n, starting with the smallest positive integer:
For n = 1, p^2 + n = 2^2 + 1 = 4 + 1 = 5, which is prime.
Since 5 is prime for n = 1, we can eliminate n = 1 as the possible solution.
Next, let's try n = 2:
For n = 2, p^2 + n = 2^2 + 2 = 4 + 2 = 6, which is not prime. This means that for n = 2, p^2 + n is not prime for p = 2.
Now, let's try n = 3:
For n = 3, p^2 + n = 2^2 + 3 = 4 + 3 = 7, which is prime.
Since 7 is prime for n = 3, we can eliminate n = 3 as the possible solution.
Continuing this process, let's try n = 4:
For n = 4, p^2 + n = 2^2 + 4 = 4 + 4 = 8, which is not prime. This means that for n = 4, p^2 + n is not prime for p = 2.
By following this process, we can continue checking different values of n with increasing values of p until we find the least positive integer n such that every p, p^2 + n is never prime.