Question

Show that 32008+42009 can be written as product of two positive integers each of which is larger than 2009182 .

Answer 1

Explanation:

We use the standard factorisation:

x4 + 4y4 = (x2 + 2xy + 2y2)( x2 - 2xy + 2y2).

We observe that for any integers x, y,

x2 + 2xy + 2y2 = (x + y)2 + y2 ≥ y2,

x2 - 2xy + 2y2 = (x - y)2 + y2 ≥ y2,

We write