Find lattice points on a planar curve

Question

I have the following curve in the plane:
$$y = \frac{c-x}{6x+1}$$ Given a constant value $c \in \Bbb N$; is there a technique(s) I can apply to find lattice points on this curve?

Answer 1

One possibility would be to rewrite as

$$6y=\frac {6c-6x}{6x+1}=-1+\frac {6c+1}{6x+1}$$

The technique here is to divide through by the denominator to leave a constant in the numerator, so that the variable denominator has to be a factor of the constant numerator. A constant numerator only has a finite number of definite factors to try.

Doing that straightforwardly here leaves a fraction $\frac 16$ which is inconvenient, so multiplying through by $6$ tidies up the expression. Obviously not every possible integer value of the fraction will do on the right-hand side - you have to look for ones which allow for division by $6$ to obtain the value of $y$.

As a further hint, there is a neater way of writing this:

$$6y+1=\frac{6c+1}{6x+1}$$ or $$(6x+1)(6y+1)=6c+1$$

And you can then see all the solutions if you look at it correctly:

So you will find a solution for each factorisation of $6c+1$ into two factors which are both congruent to $1$ modulo $6$. [I've taken the order of factors to matter - note that if $6c+1 = (6a+1)^2$ this factorisation gives only one solution even where order doesn't matter]. The trivial solutions relate to the cases when one of the factors is $1$, and of course such a factorisation always exists.

Answer 2

Hint: For a fraction to be an integer, the numerator needs to be at least as large as the denominator in absolute value.

Solve for $|c-x| > |6x+1|$ to restrict the values of $x$ that you need to look at.

Remember to include $x=c$ as a solution.