$X^6 + Y^6 + Z^6 = A^6 + B^6 + C^6 = I^6 + J^6 + K^6 = T$

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 ?

• So you're looking for integers that can be written as the sum of 3 powers of 6 in 3 different ways with the conidition that the integer doesn't contain a power of six itself. – Ali Caglayan Dec 26 '17 at 12:27
• Basically you are looking for generalised taxicab numbers $\mathrm{Taxicab}(6, 3, 3)$ for which no solutions have been found here and thats within a radius of 17,871. – Ali Caglayan Dec 26 '17 at 12:43
• It seems to me that it is very unlikely that you will find a solution to this question, given that we know so little about generalised taxicab numbers. – Ali Caglayan Dec 26 '17 at 12:45
• Credit to tommy1729 , and Yes 1729 is a taxi Number too :) – mick Dec 26 '17 at 21:54