# high order integer equation

Find all tuple $$(x,y)$$ such that $$x,y$$ are integers and $$(x^2-y^2)^2=20y+1$$.

First i see that $$x^2-y^2$$ is odd and from the fact that a difference between square of two odd is multiple of $$8$$ and thus $$y$$ is a multiple of $$2$$.

Moreover, we have $$(x^2-y^2+1)(x^2-y^2-1)=20y$$.

Somebody can give some hint! Whether the first information is useful?

• Where does $z$ enter into this? Commented Jun 20, 2021 at 16:27
• ops! only $x,y$. Sorry. I will edit now! Thanks! Commented Jun 20, 2021 at 16:31
• The first thing that occurs to me is that $(x^2-y^2)^2\equiv1\pmod{20}$ gives $x^2-y^2\equiv1,9,11,\text{ or }19\pmod{20}$ That gives $4$ congruences to analyze. There are only $6$ squares mod $20$ so it should be easy. I don't know whether it helps, though. Commented Jun 20, 2021 at 16:50
• The congruence $x^2-y^2\equiv 1\pmod{20}$ is equivalent to the pair of congruences $x^2-y^2\equiv 1\pmod5$ and $x^2-y^2\equiv 1\pmod4$ by the Chinese remainder theorem; each of these is much easier to analyze than the congruence modulo $20$. Commented Jun 20, 2021 at 16:55

Since $$x^2-y^2$$ is odd then $$x\not=y$$ and $$20y+1=(x^2-y^2)^2\geq ((y-1)^2-y^2)^2=(-2y+1)^2=1-4y+4y^2.$$ Hence $$0\geq 4y^2-24y\Leftrightarrow y(y-6)\leq 0$$ which implies that $$0\leq y\leq 6$$.

Moreover $$20y+1$$ is a perfect square, and therefore $$y\in\{0,4,6\}$$:

1. if $$y=0$$ then $$(x^2-0)^2=1^2$$ and $$x=\pm 1$$;

2. if $$y=4$$ then $$(x^2-16)^2=9^2$$ and $$x=\pm 5$$;

3. if $$y=6$$ then $$(x^2-36)^2=11^2$$ and $$x=\pm 5$$.

Therefore the integer solutions are $$(\pm1,0),(\pm5,4),(\pm 5,6).$$

• can you explane why $(x^2-y^2)^2\ge ((y-1)^2-y^2)^2$? I see that it equavalent to $[x^2-(y-1)^2][x^2-y^2-2y+1]\ge 0$. I may not occur! Commented Jun 20, 2021 at 17:21
• The absolute value of the difference of two distinct perfect squares is minimal when the two perfect squares are the squares of two consecutive integers. Therefore $|x^2-y^2|\ge |(y-1)^2-y^2|$. Commented Jun 20, 2021 at 17:24
• oh! nice solution! i forgot this property! thank you! Commented Jun 20, 2021 at 17:30
• Are you watching the Italy vs Wales match? :) Commented Jun 20, 2021 at 17:31
• No, but I know the score so far... Commented Jun 20, 2021 at 17:35