Again I find that I need to clarify my previous post while still leaving something for the student to do and learn from.

First, what depends on the decimal system of numeration is the structure of one **possible** analysis, which happens to be the one I outlined, rather than the fact that the remainders (8^n/20) are cyclic.

Second, I have realized that the formal proof of the cyclicality indicated by SK is not itself inductive in any obvious way but can be demonstrated by first proving four lemmas of which only the first is inductive. I shall leave the last three lemmas and the theorem itself for the student. Here is the first lemma and its proof.

\(\displaystyle n \in \mathbb Z,\ n \ge 1 \implies 8^{(4n-3)} = 20j_n + 8, \text { such that } n_j \in \mathbb Z \text { and } j_n \ge 0.\)

Proof

\(\displaystyle x \in \mathbb X \implies x \in \mathbb Z,\ x \ge 1, \text { and } \exists \text { non-negative integer } j_x \text { such that } 8^{(4x-3)} = 20j_x + 8.\)

Furthermore set X is not empty because

\(\displaystyle 1 \in \mathbb Z,\ 1 \ge 1,\ j_1 = 0,\ \text { and } 8^{(4*1-3)} = 8^1 = 20j_1 + 8 \implies 1 \in \mathbb X.\)

\(\displaystyle \text {Let } k \text { be an arbitrary member of } \mathbb X.\)

\(\displaystyle k + 1 \in \mathbb Z \text { and } k + 1 \ge 0 \because \ k \in \mathbb Z \text { and } k \ge 0 \text { by hypothesis.}\)

\(\displaystyle \text {Let } j_{k+1} = 8^4j_k + 1638.\)

\(\displaystyle j_{k + 1} \in \mathbb Z \text { and } j_{k+1} \ge 0\ \because \ j_k \in \mathbb Z \text { and } j_j \ge 0 \text { by hypothesis.}\)

\(\displaystyle 8^{\{4(k+1)-3\}} = 8^{(4k + 4 - 3)} = 8^4(8^{(4k-3)}) \implies\)

\(\displaystyle 8^{\{4(k-1)-3\}} = 8^4(20j_n + 8) \ \because \ 8^{(4k-3)} = 20j_k + 8 \text { by hypothesis}.\)

\(\displaystyle \therefore 8^{\{4(k+1)-3\}} = 20 * 8^4j_k + 32768 = 20 * 8^4j_k + 32760 + 8 =\)

\(\displaystyle 20(8^4j_k) + 20(1638) + 8 = 20(8^4j_k + 1638) + 8 = 20j_{k+1} + 8.\)

\(\displaystyle \therefore \mathbb X = \mathbb Z^+.\)

I suspect this theorem is simply one case of a much more general theorem about cyclicality of low-order digits of the positive powers of all positive integers.