User blog:Edwin Shade/In The Pursuit Of Organization

At the moment I have over 200 papers in my room, almost all of which contain a scrap of math on them. To explain how my stack of papers grew to this point, I should mention that I like to keep everything as organized as possible, yet lately I have not been allocating enough time for this, and so naturally things fell into a slight state of disrepair.

I could not stand this for long though, because I am of the sort who cannot think as efficiently when my surroundings are cluttered as when things are well-organized, so I decided to compile all my dicoveries into this one blog post, which from now on will be the "Hub" of those mathematics which have lain for too long on the floor of my room.

There are some things in here which may prove interesting, such as the pattern I believe I discovered within the prime numbers, and a way to encode the information for almost any conceivable origami creation in just one symbol. For the most part though, this blog post will be incomplete for now, and will only reach it's full completion after transcribing everything I felt was worth sharing in those papers. Feel free to comment below and ask questions if you want to.

The Leading Digits of $$a^{b}$$
$$L(a^{b})=\left \lfloor{10^{\left \lfloor{b(log(a))}\rfloor \right - {b(log(a))}}}\rfloor \right$$

Hyper Subscription
Hyper Subscription is denoted by two adjacent underscores, $$\(_ _\)$$, and is to denote successive multiple subscriptions. $$G_{G_{G_{G_{64}}}}$$ can be rewritten as $$(G\(__\)4)\(_\)64$$. For a large row of subscrpts this is more practical, and also allows us to define ordinals such as $$epsilon_0$$ simply as $$(\epsilon\(_ _\){\omega})_0$$.

Paradoxicals
Paradoxicals are a new type of number I invented meant to be the solution to problems which entail division by 0. Naturally, as $$i$$ is the solution to $${x^2}+1=0$$, $$\hro$$ is to be the solution to $$x-{\frac{1}{0}}=0$$, which is currently undefined.

More information will be typed here as I discover more of the properties of paradoxicals.

Proof $$a\uparrow\uparrow{\infty}$$ Converges for $$a\leq {e^{\frac{1}{e}}}$$
First, let $$a^{a^{a^{\ddots}}}=x$$.

From this, it follows that $$x=a^x$$.

Now take the x-th root of both sides. $$x^{\frac{1}{x}}=a$$

Calculate the derivative of a.

a'=\frac{d}{dx}x^{\frac{1}{x}}=\frac{d}{dx}e^{\frac{1}{x}(ln(x))}={x^{\frac{1}{x}-2}}(ln(x)-1)

Now set the derivative equal to 0 so the maxima of $$a=x^{\frac{1}{x}}$$ can be found.

$$x^{\frac{1}{x}-2}(ln(x)-1)$$

$$ln(x)-1=0$$, the other part of the equation may be crossed out because if we find an x such that $$ln(x)-1=0$$ the value of $$x^{\frac{1}{x}-2}$$ will be irrelevant.

$$ln(x)=1$$

$$x=e$$

Now insert $$e$$ in the expression $$x^{\frac{1}{x}}$$.

$$a=e^{\frac{1}{e}}$$, therefore the maxima of x is $$e^{\frac{1}{e}$$.

Hence, the finite maxima of $$a\uparrow\uparrow\infty$$ is $$e$$.

Q.E.D.

A Pattern In The Prime Numbers
Arrange the prime numbers in ascending succession, beginning with 2.

2  3   5   7  11  13  17  19  23  29  31  . ..

Now take the absolute differences of successive primes, and take the absolute differences of these differences ad infinitum.

2  3   5   7  11  13  17  19  23  29  31  . . .   1   2   2   4   2   4   2   4   6   2     1   0   2   2   2   2   2   2   4       1   2   0   0   0   0   0   2         1   2   0   0   0   0   2           1   2   0   0   0   2             1   2   0   0   2               1   2   0   2                 1   2   2                   1   0                     1

Observe the diagonal of 1's. I hypothesize this to be unceasing in it's regularity, and what is more, I conjecture you will cycle through a finite series of numbers by doing this same process to the squares, cubes, and so on, of the primes.

To continue, (to be continued).

Encoding (Almost) Any Origami Creation In One Symbol
(To be continued.)

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