Its in between a square & a cube
What is the smallest possible positive integer which lies exactly between a square and a cube number such that the three numbers are consecutive?
The answer is 26.
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1 solution
Couldn't 0 be a solution as it lies between -1 and 1?
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Oh yes! I am sorry to make the question unclear , I have edited it . I hope it's fine now!
This is in fact the only such positive integer, which I think is the most special thing about the number 2 6 . :)
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Is it? If yes then how can we say that?
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The general equation y 2 − x 3 = k , k ∈ Z − { 0 } , is known as Mordell's equation , and in this case we are looking for positive solutions ( x , y ) for k = 2 and k = − 2 . As you can see from the link this isn't that easy to solve. In the case of k = − 2 the only positive solution is the one you have found, namely ( x , y ) = ( 3 , 5 ) , and in the case k = 2 there are no positive solutions, (and only two integral solutions, namely ( − 1 , 1 ) , (which is the one I mentioned in my report), and ( − 1 , − 1 ) ).
A list of integral solutions for many values of k is given here .
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@Brian Charlesworth – Oh I see , that's really interesting! Thanks .
3^3 = 27 and 5^2= 25
26 lies between 25 and 27 which are square and cube numbers respectively. So the answer is 26.