Thanks for the puzzle, juki.

Oh, and kyle, I just added the "∈" up top, so you can use that if you want

(BTW kylekatarn is confirming: "is x and y a Natural Number {1,2,3,4,...} and is n an integer {... -3, -2, -1, 0, 1, 2, 3, ...} ?")

Let's plug in some numbers just to get this started:

x=1: 3x^2 + x = 4y^2 +y becomes 5 = 4y^2 +y

4y^2 +y - 5 = 0 has the solutions 1 and -1.25

Now, -1.25 is not included because y should be a Natural Number, so we are left with

x=1, y=1, and x-y=0, so I suppose 0 is the square of 0, so that is a good start.