Problem
ALG-B1-M01-P006 Sophie Germain
#6
★★☆☆☆ Level 2 of 5
Factor \(x^4+4y^4\) and prove that for positive integers \(x,y\), \(x>1\), the expression is composite.
Add and subtract \(4x^2y^2\).
Transform: \[ x^4+4y^4=x^4+4x^2y^2+4y^4-4x^2y^2=(x^2+2y^2)^2-(2xy)^2. \] Hence \[ x^4+4y^4=(x^2-2xy+2y^2)(x^2+2xy+2y^2). \]
For \(x>1\), \(y\ge1\), both factors are greater than \(1\). Thus the number is composite.
This is stronger than a standard identity: one must create the difference of squares.