Use this definition: A prime number is a positive whole...

Use this definition: A prime number is a positive whole number with no factors other than itself and $1 .$ For example, $2,13,$ and 37 are primes, but 24 and 39 are not. $B y$ convention 1 is not considered prime, so the list of the first few primes is as follows: $2,3,5,7,11,13,17,19,23,29, \ldots$ (a) If $P(x)=x^{2}-x+17,$ find $P(1), P(2), P(3),$ and $P(4)$ Can you find a natural number $x$ for which $P(x)$ is not prime? (b) If $Q(x)=x^{2}-x+41,$ find $Q(1), Q(2), Q(3),$ and $Q(4)$ Can you find a natural number $x$ for which $Q(x)$ is not prime?

Use this definition: A prime number is a positive whole number with no factors other than itself and $1 .$ For example, $2,13,$ and 37 are primes, but 24 and 39 are not. $B y$ convention 1 is not considered prime, so the list of the first few primes is as follows:
$2,3,5,7,11,13,17,19,23,29, \ldots$
(a) If $P(x)=x^{2}-x+17,$ find $P(1), P(2), P(3),$ and $P(4)$ Can you find a natural number $x$ for which $P(x)$ is not prime?
(b) If $Q(x)=x^{2}-x+41,$ find $Q(1), Q(2), Q(3),$ and $Q(4)$ Can you find a natural number $x$ for which $Q(x)$ is not prime?

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Key Concepts

Prime Numbers

Prime numbers are positive integers greater than 1 that have exactly two distinct positive divisors: 1 and themselves. They form the basic building blocks of number theory and are essential in topics such as divisibility, factorization, and cryptography.

Polynomial Evaluation

Evaluating a polynomial involves substituting specific numerical values for its variable and then calculating the resulting expression. This technique is crucial for examining the behavior of functions, testing conjectures, and finding particular attributes of the function outputs.

Counterexample

A counterexample is an instance that disproves a general statement or rule. In mathematical investigations, finding a counterexample is a powerful method to show that a property or conjecture does not hold universally.

Quadratic Polynomials and Prime Generation

Certain quadratic polynomials have been noted for their prime-generating properties over a range of integer inputs. However, despite producing primes for many input values, these polynomials eventually yield composite numbers, emphasizing the limitations of using them as universal generators of primes.

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