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 20130211 14:58:35 