Changing S_1 from 1 to any integer won't affect post #4 as this will just multiply all the S_n's by your new S_1.
Quote by ramsey2879
The converse is not true. One example is n = 19*59 also works.

Maybe I mucked something up, but 2 isn't a residue mod 19*59 but I get S_((19*59+1)/2) =S_1 mod 19*59. However S_((19*591)/2) is 0 mod 19*59? In any case, the converse does appear false, n=29*41=1189, 2 is not a quadratic residue and S_1190= 0 mod 1189.