On p.4 you argue that for integers a, b, c and n:

  (a + b − c)^n = (c − a)(c − b)g_1(n)

  => a + b − c = [(c − a)(c − b)g_1(n)]^(1/n)

  => g_1(n) | a + b - c

This doesn't follow as it stands. For example, if a=b=3 and c=n=2, then g_1(n)=16 whereas a + b - c = 4.