“Finite or Infinite is not the question; it’s not which side of the room you’re on, it’s what’s in it”.
Before we get started on our adventure, if anyone decides they like what they see on here and want to make a contribution, I would very much appreciate any amount that you choose. You can do so by clicking the link below.
Thank you very much
Blog
Hello visitors, i would like to welcome all of you to my math site. It’s been a long voyage to get where i am right now, and here.
I excelled in HS AP calculus (“top” student). Spending many hours on weekends and summer, I chose to study, explore, and ponder what I called simple and relaxing, yet the world may typically view as deep math, science, and philosophical ideas.
After about half way through my HS senior year, I started to develope a severe mental illness. Many Universities offered for me to go, no payment, but I could not pursue. In the following times, I avoided using the internet to “look up” more on math stuff. Instead, over and excess of a three decade span I independently produced conceptual models from my own ideas and logicical thinking.
I’m just learning recently, that some of mine are similar to those formally established and in some cases different ones have perhaps no existing equivalents.
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.
Here is a peak at something else I’ve been working on below.
Whole Theory
Now to something else:
One morning as a small mental math exercise I was curious about both, non integer factorial values and finding the factorial for a relatively large number, since it’s more probable that this “factorial root” would be a non integer 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)!
Formula Created by Russell B
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 comparative lack of great complexity.
Let’s compare it to inv gamma:
| N | Log Scaler Root | Inverse Gamma Root | Δr |
|---|
Some people may have heard of a google plex or Graham’s number, which a far larger than “ordinary” numbers. I was thinking about a possible growth rule myself, and created one that I found interesting.
I call it PIE CHEF, it’s a pretty quick growing function.
Acronym: PIE = Pi + Irrational + Euler’s number; CHEF = Cyclic Hierarchical Exponential Factorial
Abstract
PIE CHEF is a giant‑class operator law defined by a factorial‑cadence exponential coil and an irrational digit alignment test. It runs on two clocks: a dynamic loop end value (W) that drives the coil forward, and a static streak requirement (w) frozen at the first match (but reset to (W) on failure). Each cycle promotes itself until the frozen streak is satisfied, at which point the process halts finitely.
Step Structure (Coil Backbone)
The coil is built step by step, nesting each result into a descending exponential tower. The initial run uses 104 steps (a-z, a-Z, A-z, and A-Z…):
(a = 2^1)
(b = a^{2^1})
(c = b^{3^{2^1}})
(d = c^{4^{3^{2^1}}})
(e = d^{5^{4^{3^{2^1}}}})
… continuing through the alphabet …
up to the 104th step (Z).
The value at the end of the coil is the dynamic loop end value (W).
The process is
Output (W):
Evaluate the coil to produce a decimal string (W).
This is the “establishment” cycle.
Do not compare output yet to first equal number of the irrational’s digits.
New cycle, step count W from first output. Now after 2nd cycle:
Comparison:
Compare the first (|W|) digits of (W) against the first (|W|) digits of an irrational constant (π by default, but any irrational is valid).
Static Freeze Law:
On the first successful match, freeze that (W) as static (w), the required number of consecutive matches.
If a later cycle fails before the streak is satisfied, reset (w := W) (the most recent output) until the next successful match.
Promotion Law:
After each cycle, set the step count for the next coil to the current (W). Thus the coil length is always dynamic.
Halting:
Continue until exactly (w) consecutive matches occur. The final (W) is the output of PIE CHEF.
Growth Rules
Rule 1 — Dynamic promotion: Each cycle’s output (W) becomes the step count for the next coil.
Rule 2 — Static freeze: The first successful match freezes (W) as (w), the streak requirement.
Rule 3 — Reset on failure: If a cycle fails before the streak is satisfied, reset (w := W) until the next successful match.
Rule 4 — Closure: The process halts exactly when the streak requirement (w) is met.
Rule 5 — Universality: Any irrational constant may be used for comparison; the law is invariant under substitution.
Example Trace (π)
Cycle 1: (W = 3) → matches “3” → freeze (w = 3), step count = 3.
Cycle 2: (W = 314) → matches “314” → streak = 2, step count = 314.
Cycle 3: (W = 31415936) → matches “31415936” → streak = 3, step count = 31415936.
Halt after 3 consecutive matches (static (w = 3)).
Features
Irrational universality: Works with π, e, φ, ζ(3), √2, or any irrational expansion.
Dual clocks:
Dynamic (W) = loop end value, promoted each cycle as the new step count.
Static (w) = streak requirement, frozen at first match but reset to (W) on failure.
Finite closure: Always halts once the streak requirement is satisfied.
Factorial cadence: The “F” in CHEF of course refers to the integer‑stepped rhythm of the coil, not a literal factorial at each stage — the factorial grammar is structural, I thought it had a nice “feel” to it.
Explore the finite horizon with an infinitive walk.
Just a notable curiosity, it may not have a deeper meaning other that structural numerical alignment, but it is interesting nonetheless:
Pi^(sq root 2) is very close to the value of e^phi.
Another number that I created in about a second that felt right, (when I was considering near-intergers related to irratuonals) is Phi^(7(4^1/3)).
A couple close to pi:
((2^1/2)^phi)/(ln(2^1/2)(ln5))
And 306^1/5
Off topic, but here is an acronym:
Appearing Condensed Revealing Obvious Names You Memorize