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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?

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    $\begingroup$ Mh, you didn't say that $x$ must be integer, so you are essentially asking for the real roots of $ax^3+bx^2+cx+d-n^2=0$ it seems. $\endgroup$ – Yves Daoust Jan 10 '20 at 15:49
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    $\begingroup$ $y^2=$ a cubic in $x$ is generally what's called an elliptic curve. Occasionally there are elementary ways to find all the integer solutions, but usually more advanced methods must be used. A websearch will bring up lots of helpful material. $\endgroup$ – Gerry Myerson Jan 10 '20 at 16:22
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    $\begingroup$ You may find mathoverflow.net/questions/142220/fermats-proof-for-x3-y2-2 informative, although it mostly talks about how an elementary method doesn't work. $\endgroup$ – Gerry Myerson Jan 10 '20 at 20:03
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    $\begingroup$ I think you'll enjoy kconrad.math.uconn.edu/blurbs/gradnumthy/mordelleqn1.pdf $\endgroup$ – Gerry Myerson Jan 10 '20 at 20:08
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    $\begingroup$ Well, as I wrote, finding all the integer solutions to an equation of the form $y^2=ax^3+bx^2+cx+d$ can get you into very deep mathematical waters very quickly. You can do a degree in math at a good college and never get to see the math you need to solve such equations. All I can suggest is that you keep searching, maybe you'll come across something helpful. $\endgroup$ – Gerry Myerson Jan 12 '20 at 14:28
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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.

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