Still 2018
Number Theory
Level
2
How many points with integer coordinates are on the circle with radius and it center at the origin ?
The answer is 12.
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1 solution
To be on the circle, the coordinates of a point P ( m , n ) have to verify m 2 + n 2 = 2 0 1 8 2 i.e. we are looking for all Pythagorean triples ( a , b , 2 0 1 8 ) .
There are 12 solutions : ( 0 , 2 0 1 8 ) , ( 1 1 1 8 , 1 6 8 0 ) , ( 1 6 8 0 , 1 1 1 8 ) , ( 2 0 1 8 , 0 ) , ( 1 6 8 0 , − 1 1 1 8 ) , ( 1 1 1 8 , − 1 6 8 0 ) , ( 0 , − 2 0 1 8 ) , ( − 1 1 1 8 , − 1 6 8 0 ) , ( − 1 6 8 0 , − 1 1 1 8 ) , ( − 2 0 1 8 , 0 ) , ( − 1 6 8 0 , 1 1 1 8 ) and ( − 1 1 1 8 , 1 6 8 0 ) .