I won't reply further to this question about g, i do think i've been clear. and at this point you can be on your merry way still thinking it's wrong. but you simply misunderstood.
it's a proof by contradiction. g would divide a+b-c IF a+b-c are integers.
for n=2, g(2)=(c-a)(c-b)g_1(2) and g_1(2)=2.
So only when n=2 is it true that g divides a+b-c.
Otherwise we get a contradiction that it divides. since then, g_1(n) for n>2 is not a factor of a+b-c, we can safely assume at least one of them was not an integer.