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How many positive integer solutions does x 2 + y 2 = z 3 has?
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3 solutions
The very most obvious is for the case of ( x , y , z ) = ( 7 5 , 1 0 0 , 2 5 ) , ( 2 6 , 1 8 , 1 0 ) and some others like ( 9 6 0 0 , 2 8 0 , 1 0 0 )
Take x = a ( a 2 + b 2 ) , y = b ( a 2 + b 2 ) , z = ( a 2 + b 2 ) for any a , b ∈ N .
Then, LHS = x 2 + y 2 = a 2 ( a 2 + b 2 ) 2 + b 2 ( a 2 + b 2 ) 2 = ( a 2 + b 2 ) 3 = z 3 .
Hence, there are infinitely many solutions.
For any nonnegative integer k , ( x , y , z ) = ( 2 3 k + 1 , 2 3 k + 1 , 2 2 k + 1 ) is a solution. Thus, there are an infinite number of solutions.