A Pattern Among Powers - Part 4

This post is the last in a series discussing
a pattern I found in sequences of powers,
or exponents. I won't explain the pattern
again in this post, so if you'd like, taking
a look back at the previous posts would
be helpful.

In the previous posts in this series, I had
manually checked whether the pattern
applied for number systems 2, 3 and 10,
or binary, decimal and ternary bases.
However, in order to mathematically
explain this pattern, I have to find a valid
generalisation of the rule, and prove that
it applies to certain bases. Because the
rule was consistent only for bases 2 and 10,
I have to check whether it applies for
even-numbered bases.

But first, I'll find more exceptions to the
rule.

Base 4:

Using a little reasoning, I found the
following pattern for base 4.

Ending digit    E.D. of powers    Loop Length

        0                            0                            1
        1                            1                            1
        2                          4, 2                          2
        3                          1, 3                          2

I don't think this follows with the
previous bases' pattern, but we'll
continue to check other bases.

Base 5:

Ending digit    E.D. of powers    Loop Length

        0                            0                            1
        1                            1                            1
        2                      4, 3, 1, 2                      4
        3                      4, 2, 1, 3                      4
        4                          1, 4                          2

Although this still doesn't show the kind
of pattern we wan't, it's interesting because
the base 10 pattern is the exact same thing,
but replicated once, creating a 1_1_4_4_2
pattern, but twice.

Base 6:

Ending digit    E.D. of powers    Loop Length

        0                            0                            1
        1                            1                            1
        2                          4, 2                          2
        3                            3                            1
        4                            4                            1
        5                          1, 5                          2

Finally! Apart from base 2 and 10, base 6
is the only number system to show this
pattern. The loop length repeats in a 1_1_2
pattern.

Unfortunately, this pattern is too irregular
to show any promise, so I will leave it at
this point.