Each of n positive integers x+100, x+200, ..., x+100*n, which are n consecutive terms of an arithmetic sequence with common difference of 100, is expressible as the sum of squares of two distinct positive integers.

Determine the maximum value of n and prove that no higher value of n is possible.

I started with the long list http://oeis.org/A001481/b001481.txt
(Which allows for the number to be the sum of two equal squares but that's ok.  I'll just check after.)


We need the starting number to be 2 less than a multiple of 9 and either 1 or 2 more than a multiple of 7.  (Trying to avoid 11, 19 etc. didn't seem worth the effort.)
A list then of possible starting numbers is 16, 43, 79, 106, 142, 169, 205, 232, 268, 295, 331, 358, 394...
I had a hit at 205.  All of {205, 305, 405, 505, 605} are in the list (but not 105 or 705.)

In retrospect I got lucky since I forgot that adding 100 won't increase the residual mod 7 by 1.  So the above process needs to be refined.

Posted by Jer on 2013-02-11 14:58:35