X, Y, Dead?
Call an ordered pair of positive integers dead if . Find the sum of all over all dead pairs of positive integers.
If you think there are no pairs of dead integers, enter 0. If you think there are infinitely many pairs of dead integers, enter 1000.
Clarifications : If you think there are 2 pairs of dead integers, (1, 2) and (3, 4), your answer would be .
The answer is 11.
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The equation is equivalant to x 3 − y 3 − x y = 6 1 . Now we use the well known factorisation x 3 + y 3 + z 3 = ( x + y + z ) ( x 2 + y 2 + z 2 − x y − x z − y z . However, to use this factorisation, we must first multiply both sides by 2 7 to obtain ( 3 x ) 3 + ( − 3 y ) 3 + ( − 1 ) 3 − 3 ( 3 x ) ( − 3 y ) ( − 1 ) = 1 6 4 6 . Using the factorisation now, we have ( 3 x − 3 y − 1 ) ( 9 x 2 + 9 y 2 + 9 x y + 3 x − 3 y + 1 ) = 1 6 4 6 = 2 ⋅ 8 2 3 . Notice that the second term must be equivalant to 1 ( m o d 3 ) , and the first equivalant to 2 ( m o d 3 ) , so the first term is equal to 2 and the second equal to 8 2 3 . Therefore, we have x − y = 1 and 9 x 2 + 9 y 2 + 9 x y + 3 x − 3 y + 1 = 8 2 3 . We can solve this system of equations to obtain that our only solution is (5, 6). Ans 1 1