Integer Solutions to an Ellipse

Question

I'm trying to find positive integer solutions to the ellipse $$x^2 - xy + y^2 - k^2 = 0$$ where $k$ is a constant. Specifically, I already have two solutions for a given $k$, and I'm trying to find a third, possibly by using the two known solutions. (I'm trying to copy the technique of producing rational solutions to a conic from one known rational point on the conic.)

I have searched a lot on the Internet, but most resources either suggest checking all values within the range of the ellipse's 'box', or give methods for a particular type of ellipse. Edit: I found some questions which are similar to mine, but I cannot apply the technique used in them to my question mainly because I do not understand the technique. They all mention work by Fricke and Klein (1897).

My questions are:

How many [positive] integer solutions does a general ellipse have?
How can we find them? (From scratch, or knowing a few solutions beforehand?)

@Ghartal That question, and many other linked questions, are similar to mine but since none of them tell me what the "Fricke and Klein (1897)" method/theorem is, I cannot apply the same technique here. — shardulc, Jun 23 '16 at 11:25
@individ I am having a little difficulty in understanding the method. What do you mean by "if the root of the whole", "if a root $\sqrt{fa}$", etc.? — shardulc, Jun 23 '16 at 12:18
In this case. $\sqrt{fa}=k$ It is possible to factorize and to consider all solutions. — individ, Jun 23 '16 at 12:22
@individ Then what are $p$ and $s$? You cannot solve the Pell equation because $b^2 - 4ac$ is negative. — shardulc, Jun 23 '16 at 12:54
@individ So how do I find those solutions? How can there be infinitely many integer solutions? — shardulc, Jun 23 '16 at 13:01
Given $k$ the number of solutions of course. You can use some brute force to find the desired number http://math.stackexchange.com/questions/816681/find-all-integers-satisfying-m2-n-12n-1n-2n-22/816685#816685 — individ, Jun 23 '16 at 13:08

score 3 · Accepted Answer · edited Apr 13 '17 at 12:20

3

The answer depends on the number of prime factors of $k$ when we consider a factorization over the ring of Eisenstein integers $\mathbb{Z}[\omega]$. Have a look at this answer, too. For instance, there are just trivial solutions if $k$ is a square-free number and a product of primes of the form $3m-1$, that do not split over $\mathbb{Z}[\omega]$. So the number of integer points on such ellipses has a very irregular arithmetic behaviour, but a quite regular behaviour on average, just like in the Gauss circle problem.

edited Apr 13 '17 at 12:20

Community

1

answered Jun 23 '16 at 15:44

Jack D'Aurizio

353,855

Thanks! By the way, the linked question had prompted me to ask this question :) – shardulc Jun 24 '16 at 04:55

score 0 · Answer 2 · edited Jun 26 '17 at 10:57

0

The equation $x^2-xy+y^2=k^2$ has a parametric solution given below,

$$x=p^2-q^2$$

$$y=2pq-q^2$$

$$k=p^2-pq+q^2$$

For $(p,q)=(3,2)$ we get $(x,y,k)=(5,8,7)$.

edited Jun 26 '17 at 10:57

Glorfindel

3,955

answered Jun 26 '17 at 09:21

Sam

11

3

Consider using Mathjax for posting here. – StubbornAtom Jun 26 '17 at 10:26
See this answer to another question. – Jan-Magnus Økland Jun 30 '17 at 13:58

Integer Solutions to an Ellipse

2 Answers2