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Consecutive Contemplation II (Posted on 2013-02-10) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
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An example Comment 2 of 2 |
I started with the long list
(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
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