The number 185136 has the interesting property that it is a triangular number that is also the product of 3 consecutive integers.
I square 185136 and each of the x consecutive numbers following it, and total the sum of all these squares, S.
Then I do the same with 185136+x+1 and the y consecutive numbers following it, until I again reach the same sum, S.
Find positive integer values of x,y,S compliant with the above requirements.
(In reply to
re(2): Bit different with added consideration  short step? by brianjn)
Close enough, Brian.
In fact the range x(=y+1) is always exactly half the index of the triangular number; equivalently: T = 2x^2+x; while S factors nicely as 1/6(x)(x+1)(2x+1)(12x^2+12x+1).
Note that x here is 1 less than the full range: '185136 and each of the x consecutive numbers following it'; and likewise with y: '185136+x+1 and the y consecutive numbers following it', so there is one more square in the first (full) range than there is in the second  in this case, 305 as compared to 304.
A rather neat full proof is easily worked up once the principle is understood.
Edited on January 8, 2013, 11:45 am

Posted by broll
on 20130108 07:52:31 