UK MATHEMATICAL OLYMPIAD

[MATHPRITOM]

Prove that , the equation $ x^2+y^2=z^5+z $ has infinitely many relatively prime integral solutions .

[Individ]

In the equation: $$X^2+Y^2=Z^5+Z$$

I think this formula should be written in a more general form:

$$Z=a^2+b^2$$

$$X=a(a^2+b^2)^2+b$$

$$Y=b(a^2+b^2)^2-a$$

And yet another formula:

$$Z=\frac{a^2+b^2}{2}$$

$$X=\frac{(a-b)(a^2+b^2)^2-4(a+b)}{8}$$

$$Y=\frac{(a+b)(a^2+b^2)^2+4(a-b)}{8}$$

$a,b$ - arbitrary integers.

[Masum]

Prove that , the equation $ x^2+y^2=z^5+z $ has infinitely many relatively prime integral solutions .

For a start, use the lemma: every divisor of a sum of square is a sum of square itself. So, $z$ is a bi-square too.