Crazy Math Stuff

I love math. So, here are a few random things I have found or am working on. The newest things are at the end (other than new additions to things already present).

Given that the half iterate of the exponential function (that is f such that f(f(x))=e^x) is given by f(x)=hexp(hlog(x)+1/2), it would be nice to have the exact coefficients of the hyperexponential function. Bell matrices can also be used to compute an arbitrary iterate, but that requires computing matrix powers. Iterating e^x-1 can be done exactly using infinite sums, but it is not the real exponential function. It also appears that the coefficients of, say, its half iterate increase so quickly that it might have a radius of convergence of 0, necessitating a more advanced summation method. The coefficient of x⁶⁴ is already about 2.617*10¹⁵ (when computed exactly using rationals), & the coefficients seem to grow faster than any exponential but slower than n!.

However, there is a way to get exact coefficients for f when f ⋅ f = exp (or really, any function which has at least one fixed point, & meets a certain other condition - the reciprocal function is one that does not work). One inputs the coefficients for the target function. The equation then returns the coefficients for the half-iterate of the original function. I discovered this equation while pondering series reversion (I do not know if anyone else has found it before, but it does follow from the Bell matrix equations). See this PDF file for details & the general formula.
The function e^x has infinitely many fixed points in the complex plane. Choosing the least one with positive imaginary part (here called λ) yields this:

It is also possible to get exact coefficients for the function f(x)=e^x-1. This function has the added benefit of only having real coefficients. It turns out that the coefficients seem to have a nice pattern to them. I have the expressions for the coefficients up to x¹⁵, but I only put the first few here due to size. If you want the rest, e-mail me.

I was pondering the equation x^xx=c, where c is a constant. If we rearrange the equation to this: x=log(log c/log x)/log x, & apply the function repeatedly, we get successive approximations of the true value of x. When c=10¹⁰¹⁰⁰ (a googolplex), x is about 56.849708988558 (according to my TI-86).

A while back I figured out the antifactorial of 1 googol, about 69.9575744573539. I found this by trial & error, but that will not help me find the antifactorial of 1 googolplex. This problem seems slightly intractable for my TI-86, though through the inverse of Stirling's Formula I get something around 1.03*10⁹⁸.

31,415,926,535,897,932,384,626,433,832,795,028,841 is prime, or so my TI-92+ says (after about 25 seconds). I saw this number in the book The Joy of π.

<CRACKPOT>I think I may have found a way to compress random data...current results look promising (I know, that is what they all say).</CRACKPOT> Later: Oops... it does not work on random data, only the ZIP files on which I happened to test it. I guess I still cannot get the Bible on my calculator. Much later: now I have a TI-89 Titanium, which has enough flash to fit a ZIP file with the Bible in it.

I think I have found the Ultimate Equation of Small Numbers:

This combines Euler's famous equation (e^πi+1=0) with the numbers Phi & phi (where Phi is the golden ratio & phi=1/Phi), 42 (the answer to Life, the Universe, & Everything), 3 (to represent the Trinity), & 7 (which is said to be God's number). I say small numbers because Big Omega is of course the most important number, & it will not fit into any mere equation, but it is a large number (or number-like thing, anyway).

In HAKMEM item 174 it says that "21963283741. = 243507216435 is a fixed point of the float function on the PDP-6/10, i.e., it is the only positive number whose floating point representation equals its fixed."
So, I decided to find the fixed points for IEEE floating point. -834214802 (hex CE46E46E), 0, & 1318926965 (hex 4E9D3A75) are the fixed points of the IEEE 32-bit convert-to-float function, & for 64-bit, they are -4337501956902952561 (hex C3CE18F3863CE18F), 0, & 4886674138783273204 (hex 43D0F43D0F43D0F4).

I was attempting to demonstrate the ability of MuPAD to factor large numbers when, upon entering a bunch of random digits, it returned the same number. I did an isprime() check, & it returned TRUE. So, here is my random prime: 1943294820398320983204923049283409284019231

Generating functions are used to represent series. Basically, one sums over the products of the series terms with powers of the independent variable. Here are a few cool identities using them. The document requires a PostScript viewer.

I am working on derivatives in the λ-calculus. There is a paper somewhere about doing them using nondeterministic choice for the sum, but I chose to use the Church numeral addition function instead (to get pure λ-terms & to avoid nondeterminism). Some of these identities can be derived from the others. Here is what I have so far: