Since we introduced "Tipping point", which is same as the length of the sequence, let's formalize its notation.
Define a function L in two variables L(power, last n digits) whose value is the length of the sequence for a given power (p) and
for last n digits.
Thus, reviewing the examples presented earlier, we can write,
L(2,1) = 10, (10 is the length of sequence for power=2 and for last 1 digit)
L(2,2) = 50, (50 is the length of sequence for power=2 and for last 2 digits)
L(2,3) = 500, so on.
So, how does it work for cubes of natural numbers? i.e. What are the values of L(3,1), L(3,2), etc.?
We can use Excel again, or a computer program helps.
As the figure above shows, L(3,2) = 100.
Similarly (no images for this), L(3,3) = 1000, L(3,4)=10000, and so on.
It appears that same the pattern remains true for all powers of natural numbers. To confirm a few more cases, a Java program was used to generate the following tables.
Sequence lengths for EVEN powers,
Sequence lengths for ODD powers,
These 2 separate tables depict the sequence lengths for even and odd powers upto 26.
(Meaning 26th power of a first few thousand natural numbers were computed.
Actually, the program calculated upto the 100th power, but we limit our discussion to the first quartile).
The tables reveal a few interesting properties:
- All odd powers, except odd multiples of 5, have sequence lengths that are powers of 10.
- All even powers exhibit sub patterns of length L/2.
- All sequence lengths, odd or even powers, are multiples of 10.
- Not evident from the tables, but the 5th power of a number has the same last digit as the original number. And, it appears, (subject to further inquiry) same is true for powers 9, 13, 17, 21, so on.
There are many more patterns of interest within the tables themselves. We will discuss them at some other time.
Meanwhile, all the conclusions drawn so far were verified emprtically. An important question thus remains -
Is it possible to find a closed form expression that allows us to find the correct value of L(p, n) for any power and last n digits?
That is L(p, n) = Some expression involving p and n.
I have made some attempts at this, however, most expressions gave correct answers for only a few combinations of p and n.
The formula is work in progress, as time permits. However, the reader is welcome to provide any insights that might help us get closer to the answer.