Problem
ALG-B1-M02-P010 Compositeness of a Quadratic Form
#10
★★☆☆☆ Level 2 of 5
Prove that for every positive integer \(n\), the number \(n^4+4n^2+3\) is composite.
Treat the expression as a quadratic trinomial in \(n^2\).
We have \(n^4+4n^2+3=(n^2+1)(n^2+3)\).
For positive \(n\), both factors are greater than \(1\), so the number is composite.
The non-triviality of the factors must be checked.