I have reduced a knotty research problem to the following reasonable looking form:
Given any two integers $a$ and $b$, show that there are natural numbers $x_1,x_2,x_3$ and a (probaby negative) integer $n$, where $-3n < x_1+x_2+x_3$, satisfying:
I am not expecting a solution to this (although that would of course be the ideal outcome)! However, I don't really know where to start. How might one go about solving something like this? Are there any tried and tested methods I should know about?
And finally, given the unsolvability of Hilbert's tenth problem, is it possible that there is no way to know whether or not this is true?
(edit: equations corrected. Sorry for time-wasting!)