Question

If n is positive integer, find the value of n such that n^4-16n^2+100 is a prime number

Collected in the board: Number Theory

Steven Zheng posted 13 hours ago

Answer

n^4-16n^2+100

=n^4+20n^2+100-36n^2

=(n^2+10)^2-(6n)^2

=(n^2+6n+10)(n^2-6n+10)

Since n^2+6n+10>1, to make the number prime n^2-6n+10 must be equal to 1, that is

n^2-6n+10 = 1

then

(n-3)^2 = 0

n=3

Checking

Substitute n=3 to n^2+6n+10

3^2+6\times3+10 =37

Therefore

When n= 3, n^4-16n^2+100 is a prime numer

Steven Zheng posted 13 hours ago

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