Modular Arithmetic: Solving Equations with 22x^2 = 11 mod 13

[rashida564]

Homework Statement

22x^2 + 13(x + 2y)^3 = 11 has no integer solutions x and y

Relevant Equations

Module asthmatic.

I am thinking of taking modular of 13 to both sides of equation. So it will become
22x^2 is 11 mod 13.
And try all the values from x equal to zero to x equal to 11.
Is their easier way to solve it

 

[Gaussian97]

I think it is a good idea, what I would do is to define $z=x^2$ to have the equation $az=c \pmod{m}$ which is very easy to solve without trying an error (which is not a problem for this small numbers, but its always nice to have a method to solve things), and then you only need to solve $z = x^2 \pmod{m}$ which again can be done without trying an error.

 

[rashida564]

How can I find that there's no solution using your method, there's no integer solution

 

[Gaussian97]

Well, as I said, the equations $ax=b \pmod{m}$ and $x^2=a \pmod{m}$ are very common and important and you can find the solutions of both without trying and error (if you have not studied this then it's OK). Of course, that means that, if they don't have a solution you can prove it without having to try all the possibilities. (As I said, since you are working $\pmod{13}$ try and error is not a big deal, but if you were working at $mod{97}$ it would be really long, and a number with 10 digits would be impossible except maybe for a computer.)

 

[HallsofIvy]

"Modular asthmatic"? Doesn't modular arithmetic have enough problems without being asthmatic as well?