To find all integral solutions of $3x^2 - 4y^2 = 11$

I have to find all integral solutions of $3x^2 - 4y^2 = 11$.

I looked at the equation $\text{mod}\ 3$, and $\text{mod}\ 4$ to see that $x^2 \equiv 1\ \text{mod}\ 4$ and $y^2 \equiv 1\ \text{mod}\ 3$

How do I proceed from here? Or is there a better way to approach this?

• Did you mean "integer solutions"? – ArsenBerk Dec 25 '17 at 0:44
• @AsBk3397 "integral" is the adjective form of "integer". It is grammatically and mathematically correct. – Kenny Lau Dec 25 '17 at 0:44
• Oh, sorry for that then. I have never seen that before so I thought it may be a typo but thank you for the information as well. – ArsenBerk Dec 25 '17 at 0:46
• program – Kenny Lau Dec 25 '17 at 0:49
• I was hoping for a more 'mathematical answer'. But thanks for this – Naweed G. Seldon Dec 25 '17 at 0:51

Two good books. I have given more detailed answers for many of this type of problem.

http://www.maths.ed.ac.uk/~aar/papers/conwaysens.pdf

https://www.math.cornell.edu/~hatcher/TN/TNbook.pdf

As $11$ is prime, there are two orbits; one sequence of $x$ values is $$3, \; \; 37, \; \; 515, \; \; 7173, \ldots$$ with $$x_{n+2} = 14 x_{n+1} - x_n,$$ with matching $y$ values $$2, \; \; 32, \; \; 446, \; \; 6212, \ldots$$ $$y_{n+2} = 14 y_{n+1} - y_n.$$

Here is the second orbit: sequence of $x$ values is $$5, \; \; 67, \; \; 933, \; \; 12995, \ldots$$ with $$x_{n+2} = 14 x_{n+1} - x_n,$$ with matching $y$ values $$4, \; \; 58, \; \; 808, \; \; 11254, \ldots$$ $$y_{n+2} = 14 y_{n+1} - y_n.$$

The linear recurrences above come from Cayley-Hamilton for the matrix $$\left( \begin{array}{cc} 7 & 8 \\ 6 & 7 \\ \end{array} \right)$$ • Can you give more context? Orbit is from group theory? Is this some generalization of Pell's equation? – mvw Dec 25 '17 at 1:05
• I'm interested. How do you know that there are two orbits? – Kenny Lau Dec 25 '17 at 1:05
• @mvw see references I put in the answer now. Worth the effort to draw the topograph for $3x^2 - 4 y^2,$ out enough layers to find all occurrences of $11.$ – Will Jagy Dec 25 '17 at 1:17
• @KennyLau see references I put in the answer now. Worth the effort to draw the topograph for $3x^2 - 4 y^2,$ out enough layers to find all occurrences of $11.$ – Will Jagy Dec 25 '17 at 1:17
• Can it be one orbit if we propagate the recursion in both directions? – Oscar Lanzi Dec 25 '17 at 1:27