MHB Is 27x^6+27x^3y^3+8y^6 Composite for Positive Integers x and y?

[anemone]

Prove that the number $27x^6+27x^3y^3+8y^6$ is composite for any positive integers $x$ and $y$.

 

[kaliprasad]

Good question

Spoiler

We see that
$27x^6 + 27x^3 y^3 + 8y^ 6$
= $27x^6 - 27 x^3 y^3 + 8y^ 6 + 54 x^3y^3$
= $(3x^2)^3 + (-3xy)^3 + (2y^2)^3 – 3(3x^2)(-3xy)(2y^2)$
Above is
$a^3+ b^3 + c^3 – 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - cb)$ where $a = 3x^2, b = - 3xy, c = 2y^2$

We are not finished yet. Because one term is –ve we need to show that neither a+b+c is 1 nor other term is 1

a+b+ c = 3x(x-y) + 2y^2 >= 2

$a^2 + b^2 + c^2 – ab –bc – ca \ge ½(a-b)^2$

or $\ge 1/2(3x^2+ 3xy)^2$ so > 2

as both factors are > 1 so this is composite

edited above to correct some typo error ( metioned by anemone in PM)
thanks anemone

 

[anemone]

Well done, kali! You always show us the very insightful way to tackle just about any problem, and again, thanks for participating in my challenge thread.(Sun)