Question

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.

Answer 1

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.