## How to find integer solutions for $x^3 - (ay^2 +by+c)= 0$? [closed]

How can I find integer solutions for $x^3 - (ay^2 +by+c)= 0$?

An example for that is : "How many trigonometric numbers are also cube ?"
This produces the following formula $x^3 - (0.5y^2 + 0.5y)= 0$?
So how to find solutions for it and for that general formula.

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copy of math.stackexchange.com/questions/198894/… – Will Jagy Sep 20 at 4:28

## closed as not a real question by Will Jagy, Steven Landsburg, Franz Lemmermeyer, Lee Mosher, Dan PetersenSep 20 at 17:58

As Lubin pointed out on math.stackexchange, this is an elliptic curve for most choices of a,b,c. Let me rewrite the equation in such a way that you can use the functionality in Sage which attempts to compute integral points on elliptic curves.

(There is an unconditional, and SLOW, algorithm which computes integral points, but the one in Sage goes by way of computing a Mordell-Weil basis, and Sage doesn't always know how to do this. There are conjectural algorithms, but I don't think any of them are fully implemented.)

Here is your curve rewritten: $$\left(\dfrac{y}{a}\right)^2 + \dfrac{b}{a^2}\left(\dfrac{y}{a}\right ) = \left(\dfrac{x}{a}\right)^3 - \dfrac{c}{a^3}$$

Letting $X = x/a$ and $Y = y/a$ and letting $S$ be the set of prime divisors of $a$, then the integers $(x,y)$ solving your problem will give rise to $S$-integral points on the elliptic curve $$Y^2 + \dfrac{b}{a^2} Y = X^3 - \dfrac{c}{a^3}$$

So here's how you find them, say $(a,b,c) = (1,2,3)$...

sage: a = 1
sage: b = 2
sage: c = 3
sage: S = a.support()
sage: E = EllipticCurve([0,0,b/a^2,0,-c/a^3])
sage: points = E.S_integral_points(S)
sage: for P in points:
....:     try:
....:         print "x = " + str(ZZ(P[0]*a)) + ", y = " + str(ZZ(P[1]*a))
....:     except TypeError:
....:         print "S-Integral Point " + str(P) + " does not give a soln."....:


This gives the output:

x = 3, y = 4


As I've written it, the code will only work for $a = 1$, because Sage needs an integral model for the curve. This is an easy fix, I'll try to do it later.

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Put in more effort if you like, but first see math.stackexchange.com/questions/198772/… and the profile on MSE for this OP. – Will Jagy Sep 20 at 18:41
@Will Jagy: Thanks. If I'm ever on MSE I'll try to post an improved answer. – Jamie Weigandt Sep 20 at 20:28
Dear Jamie, please see math.stackexchange.com/questions/203484/… which asks for exactly that. I also emailed you because, well, I do that. – Will Jagy Sep 27 at 16:17