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Square and Consecutive Cubes (Posted on 2009-12-17) Difficulty: 3 of 5
N is a positive integer such that N2 is expressible as the difference of two consecutive perfect cubes, and 2N + 79 is a perfect square.

Determine the maximum value of N.

No Solution Yet Submitted by K Sengupta    
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Solution Solution Comment 4 of 4 |

If n^2 is expressible as the difference of two consecutive perfect cubes, then 2n-1 is a square, as was proved elsewhere, see Short and sweet

Now we have 2n-1=a^2, 2n+79=b^2; a^2=b^2-80, with exactly two solutions: {a,b,n} {1,9,1} and {19,21,181}. So the latter is the maximal value of N.

QED

Edited on May 9, 2011, 6:41 am
  Posted by broll on 2011-05-09 06:35:48

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