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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 an a (probaby negative) integer $n$, where $3n -3n < x_1+x_2+x_3$, satisfying:

$x_1x_2x_3=-n^3-an-b,$ and

$x_1x_2+x_1x_3+x_2x_3=a+3n^2.$

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!)

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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 an integer $n$, where $3n < x_1+x_2+x_3$, satisfying:

$x_1x_2x_3=n^3+an+b,$ x_1x_2x_3=-n^3-an-b,$and$x_1x_2+x_1x_3+x_2x_3=a+3n^2.$I am not expecting a solution to this (although that would of course be the ideal outcome)! However, I don't really know where to start. How might one go about solving something like this? Are there any tried and tested methods I should know about? And finally, given the unsolvability of Hilbert's tenth problem, is it possible that there is no way to know whether or not this is true? (edit: equations corrected. Sorry for time-wasting!) 3 added 11 characters in body I have reduced a knotty research problem to the following reasonable looking form: Given any two integers$a$and$b$, show that there are natural numbers$x_1,x_2,x_3$and an integer$n$, where$3n < x_1+x_2+x_3$, satisfying:$x_1x_2x_3=n^3+an+b,$and$x_1x_2+x_1x_3+x_2x_3=a+3n^2.\$

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!)

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