Math Proofs : Need Help

Every day I walk the dog a couple of miles to get her and me some fresh air and exercise.. it is usually the same route, meeting the same people, etc etc.. so I come up with things to keep my head busy. Sometimes its remembering as many elements and their isotopes, other times working out factors of numbers. Last week I started on the series (n^2+1) to see what their factors were. The first thing I realized was that while half of the numbers would be divisible by 2, around 40% would be divisible by 5 (because any number ending in 2,3,7,8 would square to a number that adding 1 would make it divisible by 5 (eg 13^2+1 = 169+1 = 170). Then after a lot of ticking off things I also realized that none of the numbers I could do (which isn't a lot .. I do this because math and me have a long bad history) were not divisible by 3.

Coming home I whipped up a python scriplet that basically did that for the first million digits and found that none of them were divisible by 3 (they weren't divisible by 7 either but I figured I would stick with 3 first). Now we know that n^2 is sometimes divisible by 3 (9, 36, 81 all being examples) and we know that a series that is always divisible by 3 is 3m (two series never divisible by three are 3m+1 and 3m+2). So from this I infer that the series n^2+1 only falls inside of the series 3m+1 and 3m+2 and never on 3m... and n^2 falls on 3m or 3m+1 (since if it fell on 3m+2 then n^2+1 would push those values to 3m). Ok but why?

Looking over the series I came up with the following:

n 1 2 3 4 5 6 7 8 9 10
n^2 1 4 9 16 25 36 49 64 81 100
m{3m+1/3m} 0 1 3 5 8 12 16 21 27 33
diff between m 1 2 2 3 4 4 5 6 6

Ok now I just have more silly numbers and probably a wild goose chase but it is interesting that the series of 3m/3m+1 growth grows in a regular pattern. Anyway, just an odd thing non-fedora related (i hope).
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