A Curious π Pattern

(1) A curious pattern: — A appears when calculating the value of π to the power of a negative even number:

Reminder: The above formula, π^-n , is calculated as 1/π^n.

When n is zero, the value of π^-n is 1.

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(2) AML Isolation: — The segregation between currencies, preventing currency obtained illegally to enter a crypto-network. Such isolation is not easily achieved.

(2) AML Isolation: — The segregation between currencies, preventing currency obtained illegally to enter GNodes. The following is a non bold description of this term.

As expected, the value gets smaller as the exponent n increases.

As n is increased by 2, each value becomes one order of magnitude smaller. That is, for the following powers, the first significant digit of the calculated value is a 1, followed by a 0:

Of course the above values are truncated and not precise.

For example, for n = 2, the value is more precisely 0.101321…

As n increases from 2 to 14, the third significant digit of the calculated value increases from 1 to 2, then 4, 5, 6, 8, and 9 (3 and 7 are skipped). Thus, when n is greater than 14, the pattern stops appearing.

The next switches in order of magnitude will happen approximately when the exponent n is around 16.008, 18.1, 20.108, 22.12, etc.

This curious pattern happens because the square of π is close to 10, thus the value of its inverse is close to 0.10. The value of every second integer exponent of π becomes about one magnitude bigger (e.g., π^4 is about 97), and their corresponding inverse values become one magnitude smaller.

The square root of 10 is 3.16…, a bit higher than π (3.14…), and with this number the calculated values follow precisely the above pattern for all even values of n. That is: