@Carl Love  Here is the proved theorem:

Given an odd integer N = pq, where p and q are odd integers such that (p, q) = 1 and λp < q < (λ + 1)p with λ ≥ 1 being an integer; let IN = [1, N − 1] be an integer interval. Then in IN,  each of G^0, G^1, ..., and G^(p−2) occurs symmetrically twice, while G^(p−1) symmetrically occurs q − p − 1 times in p distinct subintervals. Furthermore, among the p subintervals, there are r − 1 ones in each of which G^(p−1) occurs λ times, whereas it occurs λ−1 times in each of the rest p − r+ 1 ones.