Mathematical Reflections 2012, Issue 6 - Problem S247

Problem:
Prove that for any positive integers $m$ and $n$, the number $8m^6 + 27m^3n^3 + 27n^6$ is
composite.

Proposed by Titu Andreescu.

Solution:
For the homogeneity of the expression, we set $t=m/n$. Then,
$$\begin{array}{rcl} 8t^6+27t^3+27 &= & 8t^6-27t^3+27+54t^3\\ &=& (2t^2)^3+(-3t)^3+3^3-(2t^2)(-3t)(3)\\&=&(2t^2-3t+3)(4t^4+6t^3+3t^2+9t+9). \end{array}$$ The second factor is clearly greater than $1$ and $2t^2-3t+3=2(t-1)^2+t+1>1$, so multiplying both sides of the equation by $n^6$ we obtain the conclusion.