2009 AMC 12B Problem 19

Problem 19 of 25HarderAlgebra Number Theory

For each positive integer

n,

let

f(n) = n^4 - 360n^2 + 400.

What is the sum of all values of

f(n)

that are prime numbers?

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Solution

Write

\begin{aligned} f(n) &= n^4 + 40n^2 + 400 - 400n^2 \\ &= (n^2 + 20)^2 - (20n)^2 \\ &= (n^2 + 20n + 20) \\ &\quad {}\cdot (n^2 - 20n + 20). \end{aligned}

For

f(n)

to be prime the smaller factor must be

1

: solving

n^2 - 20n + 20 = 1

gives

(n - 1)(n - 19) = 0,

n = 1

n = 19.

Then

f(1) = 41

and

f(19) = 761

are both prime, summing to

802.

Thus, the correct answer is E.