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Given a complex number $z$ and a positive integer $n$, I would like to be able to find integer solutions $\alpha,\beta,\gamma$ to the Diophantine equation $$0 < a^2\vert z \vert^2 + \beta\textrm{ Re}(z) + \gamma^2 \leq n.$$ As an example, I tried

z := 2 + 3*i
Solve[0 < a^2*Norm[z^2] + b*Re[z] + c^2 <= 100,{a,b,c},Integers]

But when I hit return+shift I got "This system cannot be solved with the methods available to Solve". Can anyone explain why this doesn't work and offer a method which will work? Thanks.

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    $\begingroup$ You have a couple of errors there. For one thing, Mma uses I not i. $\endgroup$ – bobbym Jan 31 '17 at 1:54
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See if this is what you want:

z = 2 + 3*I;
rul = FindInstance[0 < a^2*Norm[z^2] + b*Re[z] + c^2 <= 100, {a, b, c}, Integers, 5]

Change the 5 for more or less.

Check the answers:

0 < a^2*Norm[z^2] + b*Re[z] + c^2 <= 100 /. rul

To store the answers:

a^2*Norm[z^2] + b*Re[z] + c^2 /. rul
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  • $\begingroup$ Great, thanks. Is there a way to take the solutions $(a,b,c)$ and plug them back into the equation to obtain and store the values of $a^2\vert z \vert^2 + b\textrm{ b}(z) + c^2 $ at those points? $\endgroup$ – Ethan Alwaise Jan 31 '17 at 3:54
  • $\begingroup$ I have added it to the answer. $\endgroup$ – bobbym Jan 31 '17 at 4:10
  • $\begingroup$ Is there a way to directly list out the finitely many real numbers $x \in (0,n]$ such that $$x = a^2\vert z \vert^2 + b\textrm{Re}(z) + c^2$$ for some integers $a,b,c$? It appears that there are infinitely many solutions to the Diophantine equation I posted, but only finitely many such $x$. $\endgroup$ – Ethan Alwaise Feb 1 '17 at 2:00
  • $\begingroup$ The x you have there is what is inside of FindInstance. If there are a finite number of real x (which I doubt) then it is a large number. $\endgroup$ – bobbym Feb 1 '17 at 2:31
  • $\begingroup$ There must be a finite number because the set of such points $x$ is a lattice in a bounded region. $\endgroup$ – Ethan Alwaise Feb 1 '17 at 3:24

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