Prove that if integer number q is neither prime nor composite, then q² + 2 is prime?

Answer 1

The only integers that are neither prime nor composite are 1, 0, and -1 (the latter depending on context). In these three cases, q^2 + 2 is 3, 2, and 3, respectively, all of which are prime.

Answer 2

The quality of being prime or composite only applies to positive integers greater than 1. There are numerous negative integers 'q' that are neither prime nor composite where q² + 2 will *not* be prime:

For example:
-5 is an integer that is neither prime, nor composite. (-5)² + 2 = 27 which is not prime.

Did you mean to say q was a *non-negative* integer? If so then zero and one would neither be prime nor composite. 0² + 2 = 2 and 1² + 2 = 3 which are both prime results.

Answer 3

The only integer that is neither prime nor composite is 1, so 1²+2 = 3, which is prime.

Answer 4

People who are saying "a number has to be prime or composite" are wrong. The integer 1 is neither prime nor composite. For that matter, all non-integers are neither prime nor composite, since it only applies to integers. Try to find factor for any of these numbers other than 1 or itself. If you've found one, then the number is composite. Otherwise, it's prime.

Answer 5

The only positive integer q that is neither prime nor
composite is 1 and 1²+2 = 3, which is prime.

Answer 6

Most of the prior answers are correct. The incorrect one was the one with the claim that negative numbers can't be prime.

In fact, if something is prime, and you multiply it by a divisor of 1 (called a "unit"), it's still prime. To see why that's so, check either of the prevailing definitions of "prime"

(The true one) -- If p divides ab, then p divides a or p divides b.

(Really the definition of "irreducible") -- If p = cd, then one of c or d (but not both) is a unit.

In the integers the only divisors of 1 are 1 and -1, but in other systems the concept of "unit" is more interesting.

So with that error dismissed -- to be neither prime nor composite, you need to be -1, 0, or 1. So q^2 + 2 is either 2 or 3, both of which are prime.