Consider the diophantine equation where all variables are positive and distinct :
$$ X^6 + Y^6 + Z^6 = A^6 + B^6 + C^6 = I^6 + J^6 + K^6 = T $$
And $T$ is not of the form $V W^6$ for $W>1$.
What is the smallest solution ?
Are there any solutions ?
Does the existance of a solution imply that there are infinitely many solutions ?