Well, let's say I have a function $f:\mathbb{R}\to\mathbb{R}$. This function is a polynomial of degree three under a square root sign, which means that is in the form of:
$$f(x):=\sqrt{ax^3+bx^2+cx+d}\tag1$$
Where $a,b,c$ and $d$ are integer coefficients (so they can be positive, negative or equal to zero) and $x\ge2$ and $x\in\mathbb{N}$.
Is there a (fast) way to determine for what $x$ we get: $f(x)\in\mathbb{N}$?
My work
I ran a Mathematica search, using specific values for $a,b,c$ and $d$. To be more specific I set $a=300,b=90,c=-210$ and $d=144$.
The code I used is as follows:
ParallelTable[If[IntegerQ@Sqrt[6*(24+5*x*(1+x)*(10*x-7))],x,Nothing],{x,2,10^9}]
But it will take way to long in order to finish this computation.
Is there a clever mathematical 'trick' that can be used to eliminate the number of values to check?
An equation of the form $y^2 = ax^3 + bx^2 + cx + d$ is known as an elliptic curve. The integer solutions to $y^2 = ax^3 + bx^2 + cx + d$ for $x, y \in \mathbb{N}$ are known as integral points of $y^2 = ax^3 + bx^2 + cx + d$, or simply solutions of the diophantine equation $y^2 = ax^3 + bx^2 + cx + d$. These are usually solved using Skolem's $p$-adic method (here and here), or one can use 3 other methods described in the article by Don Zagier here, two using Pell equations and one using group theory.