A cubic Diophantine equation in two variables: $x^3+2x+1=y^2$

Question

Find all POSITIVE integer solutions to the following cubic equation:

$x^3+2x+1=y^2$.

Notice how the left side of the equation resembles $x^2+2x+1=(x+1)^2$.

The only solutions I've been able to find are:

$(x,y)=(0,1), (1,2),(8,23)$.

I'm interested in knowing if there are any more solutions (or for that matter infinitely many), or if these are the only ones. I don't know how to program, so computers aren't on my side for this one.

Thanks for your help!

Answer 1

The given elliptic curve $E:y^2=x^3+2x+1$ has rank $1$ (see Cremona database), and hence infinitely many rational points. For integer points, see for example the discussion at Integral points on an elliptic curve. The bounds of Baker give explicit bounds for the integral points. But they are huge. Zagier gives three different algorithmic methods to find out whether there are very large integral points, in his article Large integral points on Elliptic curves. The algorithms however are doubly exponential.

Answer 2

(Edit: You can sign up for a SAGE account here. It is free.)
These commands on SAGE

E = EllipticCurve([0,0,0,2,1]);
print(E);
print(E.integral_points())

Prints the results

Elliptic Curve defined by y^2 = x^3 + 2*x + 1 over Rational Field
[(0 : -1 : 1), (0 : 1 : 1), (1 : -2 : 1), (1 : 2 : 1), (8 : -23 : 1), (8 : 23 : 1)]

Which corresponds to $$(x,y)\in\{(0,\pm 1),(1,\pm 2),(8,\pm 23)\}$$ According to the documentation, integral_points() takes argument proof=True, i.e. it ensures all integral points are found. So your list is complete.