finding integer coordinates on a circle?

Question

How can I find points with integer coordinates on a circle with the equation 10=x(squared) + y(squared)  ?

Answer 1

x²+y²=10, y²=(10-x²). Perfect squares of integers ≤10 are 0, 1, 4, 9.

Subtract each from 10: 10, 9, 6, 1. Of these only 9 and 1 are perfect squares (9=3² and (-3)²; and 1=1² and (-1)². So x=±3, ±1 making y=±1, ±3.

(x,y)=(3,1), (3,-1), (-3, 1), (-3,-1), (1,3), (1,-3), (-1,3), (-1,-3) are the only integer solutions.