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in Olympiad by (65.1k points)

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

1 Answer

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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

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