Polynomial system of equations over integers

Question

I want to solve the system of equations:

$$\begin{cases} x^4+4y^3+6x^2+4y = -137 \\ y^4+4x^3+6y^2+4x = 472 \end{cases} $$

$x, y \in \Bbb{Z}$.

It most definitely amounts to messing around with algebra and completing the $4$-th powers. Like adding the two equations gives:

$(x+1)^4 + (y+1)^4 = k \in \Bbb{Z}$.

I don't see an obvious way of obtaining another such equation.

But besides simply solving this system I'm also particularly interested in a general way to approach problems like this, preferably avoiding the messy algebra or, if that's not possible, having a general algorithm instead of trying and failing many times to arrive at the appropriate algebraic expression.

Answer 1

If we subtract the equations we get to

$$(y−1)^4−(x−1)^4=(y-x)(x+y-2)(x^2+y^2-2x-2y+2)=609=3\times 7\times 29$$

Since $3,7,29$ are primes then we have either:

• $y=x+3$ which solves to $(3,6)$ but not solution of original system
• $y=x-3$ which solves to $(-1,-4)$ but not solution of original system
• $y=x+7$ which solves to $(-1,6)$ but not solution of original system
• $y=x-7$ which solves to $(3,-4)$ also solution of original system
• $y=x\pm 29$ which has no integer solutions

Therefore the only solution is $(3,-4)$

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Edit:

I though of this refinement today.

Calling $X=x-1$ and $Y=y-1$ then $$Y^4-X^4=(Y^2-X^2)(X^2+Y^2)=(Y-X)(Y+X)(X^2+Y^2)=3\times 7\times 29$$

Since $|X^2-Y^2|<|X^2+Y^2|$ strictly (unless one is $0$ but not possible since $609$ is not a perfect power of $4$) then considering that $3\times 7<29$ the only possibility is $X^2+Y^2=29$.

Whose decomposition is immediate as $29=25+4$ and $X,Y\in\{\pm 2\ ;\ \pm 5\}$

Among these only $X=\pm 2,Y=\pm 5$ gives $+3,+7$ or $-3,-7$ as sum and difference.

After testing in original equation in $x,y$ only $(3,-4)$ susbsists.

Answer 2

Taking the resultant over the complex numbers we obtain that either $(x,y)=(3,-4)$, or that $x$ is a root of an irreducible polynomial of degree $15$, having no rational root. So $x=3,y=-4$ is the only integral solution.

Added: The resultant equation is the following: $$ (x-3)(x^{15} + 3x^{14} + 33x^{13} + 99x^{12} + 1061x^{11} + 2287x^{10} + 17589x^9 + 57503x^8 + 453731x^7 + 1121833x^6 + 291875x^5 - 826007x^4 + 33699383x^3 + 776909093x^2 + 2564361287x + 8370351445)=0. $$