Versatile Sum and Difference of Squares
How many integers cannot be expressed in the following form, where x , y , and z are integers?
x 2 − y 2 + z 2
The answer is 0.
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2 solutions
Thank youo!
Just a note: x 2 − y 2 = ( x + y ) ( x − y ) can take on any even integer divisible by 4 , not just zero. Just choose x = n + 1 and y = n − 1 to get 4 n . x 2 − y 2 = ( n + 1 ) 2 − ( n − 1 ) 2 = ( n 2 + 2 n + 1 ) − ( n 2 − 2 n + 1 ) = 4 n But it is impossible to get even integers like ± 2 , ± 6 , ± 1 0 , . . . from x 2 − y 2 .
( n + 1 ) 2 − n 2 = 2 n + 1 can cover all odd.
let z = 0 , x = n + 1 , y = n , then ( x 2 − y 2 + z 2 ) can express all odd
let z = 1 , x = n + 1 , y = n , then ( x 2 − y 2 + z 2 ) can express all even