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Short and sweet (Posted on 2011-02-21) Difficulty: 3 of 5
Let n be any natural number whose square is also the difference between two consecutive cubes.

Prove that 2n-1 is also a square.

See The Solution Submitted by broll    
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spoiler | Comment 1 of 4

3a^2 + 3a + 1 = n^2

If you multiply this equation by 4 and subtract 1, you can recast it as:

3(2a+1)^2 = (2n+1)(2n-1)

3 can factor only 1 of the factors of the RHS, which gives us 2 cases:

1) 2n+1 = p^2

2n-1 = 3q^2

2) 2n+1 = 3p^2

2n-1 = q^2


1) implies 3q^2 + 2 = p^2, which is impossible as no square = 2mod3.

So 2n-1 = q^2, which answers the problem.

Going further and setting q=2r+1 gives n = r^2 + (r+1)^2.

So you have the neat conclusion that when the difference of 2 consecutive cubes is a square, the root of that square is the sum of 2 consecutive squares.

  Posted by xdog on 2011-02-23 18:48:46
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