Both are cubes
Let be a non-negative integer. For how many possible values of are and both cubes of integers?
The answer is 1.
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1 solution
If n + 1 is a cube, then 8 ( n + 1 ) = 8 n + 8 is also a cube. ( 8 n + 8 ) − ( 8 n + 1 ) = 7 , and this is only possible if 8 n + 1 = 1 and 8 n + 8 = 8 (since the distance between adjacent cubes is increasing). Solving this we get n = 0 as the only solution. Thus there is only 1 value of n that fulfills these criteria.