Csdrhrt

I'm trying to find integer solutions to equations of the form
$$y^2 = 4x^3 - p tag{1}$$
where $p$ is a prime and $p equiv 7 (text{mod } 8)$.

1) Is there a simple way to check if solutions do not exist for a given $p$?

2) Is there a computationally efficient way to find at least one solution? or maybe for a subset of $p$ by assuming some additional property?

Eventually I'd like to solve for some large $p$ ( > 1000 bits). I do not know if this can be done efficiently, but I'm starting with smaller $p$ to try to understand properties of the equation better.

For reasonable sized values, I can use magma to test out some values of $p$. I do this by noting that if there is an integer solution to $Y^2 = X^3 - 16p$ with $X$ a multiple of 4, then I have solved the original equation. This has helped me see that sometimes there are no solutions, but I haven't figured out if there is a simple way to determine when this occurs.

47 fails. I don't know how much sage would slow down with larger $p$

jagy@phobeusjunior:~$ sage

┌────────────────────────────────────────────────────────────────────┐

│ SageMath Version 6.9, Release Date: 2015-10-10                     │

│ Type "notebook()" for the browser-based notebook interface.        │

│ Type "help()" for help.                                            │

└────────────────────────────────────────────────────────────────────┘

sage:  E = EllipticCurve([0,0,0, 0, -112])

sage: E.integral_points()

[(8 : 20 : 1)]

sage:  E = EllipticCurve([0,0,0, 0, -368])

sage: E.integral_points()

[(8 : 12 : 1),

 (9 : 19 : 1),

 (24 : 116 : 1),

 (32 : 180 : 1),

 (48 : 332 : 1),

 (944 : 29004 : 1),

 (1313 : 47577 : 1)]

sage:  E = EllipticCurve([0,0,0, 0, -496])

sage: E.integral_points()

[(8 : 4 : 1),

 (16 : 60 : 1),

 (25 : 123 : 1),

 (40 : 252 : 1),

 (113 : 1201 : 1),

 (560 : 13252 : 1)]

sage:  E = EllipticCurve([0,0,0, 0, -752])

sage: E.integral_points()



sage:  E = EllipticCurve([0,0,0, 0, -1136])

sage: E.integral_points()

[(96 : 940 : 1)]

sage: 

Partial result:

If $x$ is odd then we have modulo 8:

$$ -1equiv 4-y^2implies y ^2 equiv 5$$ which is impossible.

So, if there is a solution then $x$ must be even.