Problem
ALG-B1-M02-P021 When a Fourth Power Is Almost a Square
#21
★★★★☆ Level 4 of 5
Prove that for positive integer \(n\), the number \(n^4+4\) is not a square.
Assume \(n^4+4=m^2\) and factor \(m^2-n^4\).
Assume \(n^4+4=m^2\). Then \(m^2-n^4=4\), so \((m-n^2)(m+n^2)=4\).
Both factors are positive. The possible pairs are \((1,4)\), \((2,2)\), \((4,1)\). The pair \((1,4)\) gives \(2m=5\), impossible. The pair \((2,2)\) gives \(n=0\), contradicting positivity. The pair \((4,1)\) is impossible by order.
Therefore \(n^4+4\) is not a square.
A good problem on turning a square question into a product.