*Double Square*
Probability
Level
2
A number is a double square if it can be be expressed as the sum of 2 squares of integers. Is the product of two double squares also a double square ?
No
Yes
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1 solution
Let n = a 2 + b 2 and m = c 2 + d 2 be two double squares . Note how a 2 + b 2 can be factored as ( a + b i ) ( a − b i ) , where i 2 = − 1 . Hence,
n m = ( a 2 + b 2 ) ( c 2 + d 2 )
= ( a + b i ) ( a − b i ) ( c + d i ) ( c − d i )
= [ ( a + b i ) ( c + d i ) ] [ ( a − b i ) ( c − d i ) ]
= [ ( a c − b d ) + ( b c + a d ) i ] [ ( a c − b d ) − ( b c + a d ) i ]
= ( a c − b d ) 2 + ( b c + a d ) 2
Since a , b , c , d are all integers, ( a c − b d ) and ( b c + a d ) are too! Therefore, n m is a double square because it can be written as a sum of two squares of integers.