Area of Second Smallest Square

Number Theory Level 5

Let $(x_1,y_1)$ , $(x_2,y_2)$ , $(x_3,y_3)$ , and $(x_4,y_4)$ be four distinct integer solutions to the equation

$x^4+x^3y^3=y^4+xy$

such that when connected, they form a square.

If this square is the second smallest possible square that can be made, then find the area of this square.

Details and Assumptions

The smallest square possible is made by the four solutions $(\pm 1, \pm 1)$ . This square has area $4$ .

The answer is 136.

1 solution

Daniel Liu
Aug 28, 2014

Move everything on one side to get $x^3y^3+x^4-y^4-xy=0$ . Now we can use a SFFT-like factorization: $(x^3-y)(y^3+x)=0$

Thus, all solutions of $x^4+x^3y^3=y^4+xy$ are either on the graph $y=x^3$ or the graph $y=-\sqrt[3]{x}$ .

Imgur Imgur

It is clear to see that the next smallest square forms at the coordinates $(2,8), (8,-2), (-2,-8), (-8,2)$ and the area of this square is $6^2+10^2=\boxed{136}$ .

I did the same way!

Ayush Garg - 6 years, 9 months ago

You could have tagged it in Algebra section

Ayush Garg - 6 years, 9 months ago

Good question .I did the same way too . And yeah Daniel Liu could you please tell me which software you used to get the above graph.

Thanks for the same .

A Former Brilliant Member - 6 years, 4 months ago

Area of Second Smallest Square

The answer is 136.

1 solution

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