“Finite or Infinite is not the question; it’s not which side of the room you’re on, it’s what’s in it that matters.”
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Here is a curiously fun simple irrational number search I was thinking about.
Just fun simple search I was thinking about.
I call it PAIR: Palindromic Aligned Irrational Requirement.
PAIR is a method for analyzing the decimal expansion of an irrational number. It begins at the first digit after the decimal point (the anchor) and searches for the earliest non-trivial palindrome that reflects the initial digit sequence. This palindrome is called the PAIR Measure, or PM, and the number of digits scanned before it appears is it’s IRIS, (initial reflective identity span. It’s literally PAIR’S eye-dentity.
Formal Description
Let an irrational number be written as:
x = a₀.a₁a₂a₃a₄…
The digit a₁ is the anchor.
For each n ≥ 1, define the candidate sequence Sₙ = a₁a₂…aₙ.
Reflect Sₙ to form a palindrome:
Pₙ = Sₙ + reverse(Sₙ₋₁)
If Pₙ matches the digits of x starting at the anchor, and is not a trivial repeat (has to contain a central reflective digit, so the shortest possible palindrome is 3 digits), then:
PAIR(x) = Pₙ
IRIS = n
If no such palindrome is found within the scanned digits, there may be no closure.
For example, some irrationals “close early” like √2 → 141, while others may be a bit more stubborn , and hang on into the tens or hundreds. Sometimes though……….like within the known digits. π, it has yielded nothing, even after trillions of digits so far.
One morning as a small mental math excersize I was curious about both, non interger factorial values and finding the factorial for a relatively large number, since it’s more probable that this “factorial root” would be a non interger for most numbers, especially the larger they are, I set out to make one. Despite ankle pains, (hey math is a great passion of mine so a great relaxation as well), I decided to make a basic formula. Now based just on first principals I came up with a simple linear scaler, but decided a log linear one would be more accurate.
Here is that formula:
r = n+(ln N – ln n!) / ln(n+1)
n!<N<(n+1)!
where r=the factorial root, N=the number we are looking for the factorial of, and n is the largest expanded factorial<N
That got me curious, because this formula was so easy to create, what else already existed, and I learned of the inverse gamma function, which was designed to give exact results, but is frequently looked as reasonably math heavy. My little expression gives quite accurate results for unimportant day to day activities, especially considering its comparitive lack of great complexity.
Let’s compare it to inv gamma:
| N | Log Scaler Root | Inverse Gamma Root | Δr |
|---|