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. By convention 1 is not considered prime, so the list of the first few primes is as follows: (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. By convention 1 is not considered prime, so the list of the first few primes is as follows:
(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?

Key Concepts

Prime Numbers

A prime number is defined as a positive integer greater than 1 that has no positive integer divisors other than 1 and itself. This concept is foundational in number theory and is used to understand the building blocks of the integers, especially in factorization and the study of divisibility.

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. Evaluating a polynomial at a given number involves substituting that number for the variable and simplifying the expression to obtain a result.

Quadratic Polynomials

Quadratic polynomials are a specific type of polynomial function characterized by the highest exponent of the variable being 2. They often take the form ax² + bx + c and are studied both in algebra for their properties, like their parabolic graphs, and in number theory for instances where the polynomial might produce prime numbers for a sequence of integer inputs.

Prime-Generating Functions

Some quadratic polynomials have been studied for their ability to generate a sequence of prime numbers when evaluated at successive natural numbers. However, despite their initial appearance, these functions eventually produce composite numbers, and finding such a value typically involves testing the function on various inputs to identify a counterexample.

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