Problem
ALG-B1-M02-P020 All Prime Values of a Shifted Expression
#20
★★★★☆ Level 4 of 5
Find all integers \(n\) for which \(n^4+4n^3+6n^2+4n+5\) is prime.
First recognise \((n+1)^4+4\).
The expression is \((n+1)^4+4\). Put \(m=n+1\). Then \(m^4+4=(m^2-2m+2)(m^2+2m+2)\).
If \(|m|>1\), this is a non-trivial factorisation. If \(m=0\), the number is \(4\), not prime. If \(m=1\) or \(m=-1\), the number is \(5\), which is prime.
Hence \(n=0\) or \(n=-2\).
The shift makes the standard trick less visible.