Let x be the first positive integer in the sequence, so the sum of the 2112 consecutive positive integers would be x + (x + 1) + (x + 2) + ... (x + 2111).
This is equal to 2112x + (2111)(2112)/2, or simplified,
1056(2x + 2111).
We know that 1056 is not a perfect square, so 2x + 2111 would have to be equal to 1056 for the whole expression to be a perfect square. If you set 2x + 2111 = 1056, you get
x = 1055.5
Since this is not a positive integer, we can conclude that the sum of any 2112 consecutive positive integers can not be a perfect square.

Posted by Marc
on 20070405 10:09:15 