CONJECTURE

"the sequences of the last digit in the sequences of the powers of the integers have recurring patterns that repeat every 5 successive sequences and occur in repeating 10 digit intervals"

"the sequences of the last digit in the sequences of the powers of the integers are according to the power p, thus"

p = 1     0-1-2-3-4-5-6-7-8-9-0.........

p = 2     0-1-4-9-6-5-6-9-4-1-0.........

p = 3     0-1-8-7-4-5-6-3-2-9-0.........

p = 4     0-1-6-1-6-5-6-1-6-1-0.........

p = 5     0-1-2-3-4-5-6-7-8-9-0.........

p = 6     0-1-4-9-6-5-6-9-4-1-0.........

p = 7     0-1-8-7-4-5-6-3-2-9-0.........

p = 8     0-1-6-1-6-5-6-1-6-1-0.........

and so on.......