Thank you for responding to my previous question. Here is another question that puzzles me:

Prove that it is not possible to have the sum of the squares of 13 consecutive positive integers be a square.

Your help is very much appreciated. Thank you.

Hi again Wallace,

Suppose that the smallest of the 13 integers is x then the sum of their squares is

x2 + (x + 1)2 + (x + 2)2 + ... + (x + 11)2 + (x + 12)2

which, when expanded, can be written

13 x2 + 2x(1 + 2 + ... + 11 + 12) +(12 + 22 + ... + 122)

I expect that you know expressions for the two sequences above. Substituting these expressions shows why the sum of the squares of 13 consecutive positive integers can't be a square.

Cheers,
Harley
Go to Math Central