Problem
ALG-B1-M01-P023 When a Product Can Be Prime
#23
★★★★★ Level 5 of 5
Find all positive integers \(n\) for which \(n^4+4n^2+4\) is prime.
First recognise a perfect square.
We have \(n^4+4n^2+4=(n^2+2)^2\). A square of an integer greater than \(1\) is never prime.
Here \(n^2+2\ge3\), so \((n^2+2)^2\) has a divisor \(n^2+2\), different from \(1\) and from the number itself. Hence there are no solutions.
The strong move is not to complicate the problem, but to see the square.