Math Problem Statement

Consider the following statement.

There is an integer n such that 

2n2 − 5n + 2

 is prime.

To prove the statement it suffices to find a value of n such that 

(n, 2n2 − 5n + 2)

 satisfies the property 

"2n2 − 5n + 2

 is prime." Show that you can do this by entering appropriate values for n and 2n^2-5n+2. Enter the solution as a ordered pair.

Solution

To show that the statement is true, we need to find an integer nn such that the expression 2n25n+22n^2 - 5n + 2 is a prime number. Once we find such an nn, we can write the solution as an ordered pair (n,2n25n+2)(n, 2n^2 - 5n + 2).

As shown previously, when n=3n = 3:

2(3)25(3)+2=1815+2=52(3)^2 - 5(3) + 2 = 18 - 15 + 2 = 5

Since 5 is a prime number, the ordered pair that satisfies the statement is:

(3,5)\boxed{(3, 5)}

This pair demonstrates that there exists an integer n=3n = 3 such that 2n25n+22n^2 - 5n + 2 is prime.

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Math Problem Analysis

Mathematical Concepts

Prime numbers
Quadratic equations

Formulas

Quadratic formula

Theorems

Prime number theorem

Suitable Grade Level

Grades 10-12