SOLUTION: Find the difference {{{ y - x }}}, where x and y are positive integers and {{{ 0 = 7x^3 - x^2y^2 + 14399 }}}.

Your starting equation is 

    .


Factor 14399 = .  Re-write equation in this equivalent form

     = .


Right side is divisible by 7, and one term in the left side is a multiple of 7;
hence, the term  is divisible by 7.


So, I write y = 7z, where z is some positive integer number.

    I can not write x = 7z, since then the degree of 7 will be too great on the left side.


Then the last equation takes the form

     = ,  or

     = .


Cancel factor 7 in both sides and get

     = ,

     = .


Recall about the uniqueness of decomposition of integer numbers into the product of primes.

It implies  that  x= 11.


Then  7z^2 - x = 17;  7z^2 - 11 = 17;  7z^2 = 17 + 11 = 28;  z^2 = 28/7 = 4;  z=  = 2.


Thus x= 11;  z= 2.  Hence, y = 7z = 7*2 = 14.


The difference y-x is  14-11 = 3.


ANSWER.  y-x = 3.