Problem
ALG-B1-M01-P016 Prime or Composite
#16
★★★☆☆ Level 3 of 5
Find all positive integers \(n\) for which \(n^4+4\) is prime.
Use Sophie Germain's factorization and check \(n=1\) separately.
Use \(n^4+4=(n^2-2n+2)(n^2+2n+2)\).
If \(n>1\), both factors are greater than \(1\), so the number is composite. For \(n=1\), \(1^4+4=5\), which is prime. Answer: \(n=1\).
The problem trains checking boundary cases.