I was calcultaing quadratic residues. I noticed nice pattern: quadratic residues modulo $p^n$ where $p$ is prime are well-behaved; they always repeat after $\frac{p^n - 1}{2}$-th place (starting with $1^2 \equiv 1 \pmod {p^n})$.
I know this is proven for $n = 1.$ Also, I noticed that many composite numbers have random-looking quadratic residues. I found only few exception: $10$ (1, 4, 9, 6, 5, 6, 9, 4, 1), $6$ (1, 4, 3, 4, 1).
Three examples for $n = 2$: (1) $p = {2^2} = 4$: 1, 0, 1, 0, etc. (2) $p = {5^2} = 25$: 1, 4, 9, ..., 19, 19, ... 9, 4, 1. (3) $p = 3^2 = 9$: 1, 4, 0, 7, 7, 0, 4, 1.
Two examples for $n = 3$: (1') $p = 2^3 = 8$: 0, 1, 4, 1, 0, 1, 4, 1. (2') $p = 3^3 = 27$: 1, 4, 9, 16, 25, 9, ..., 7, 7 ... 9, 25, 16, 9, 4, 1.
So, I see that residues can repeat (see example (2')) but they always have this property that sequance repeat after half terms. Is there proof of this?