Bill, Gabe and I are doing a Fibonacci numbers project for his school's math expo and we discovered some interesting things that we wanted to share/ask you about. 1: up the 300th fibonacci number, all the prime fibonacci numbers have prime index numbers (for example, the 131st and 137th number in the fibonacci sequence. the only one that doesn't work is the 4th fibonacci number. Because 4 is obviously not prime but the fibonacci number is: 3.) 2. and more interestingly...up to the 300th fibonacci number, all fibonacci numbers greater than 1 with prime index numbers, are mutually prime with all previous fibonacci numbers that are greater than 1. WHY? Does it continue? Seems as awful coicidence, doesnt it? We are excited and perplexed! Best, Marnie (and Gabe) On Tue, Jun 6, 2017 at 4:15 AM, Bill Gosper <billgosper@gmail.com> wrote:
Date: 2017-06-05 17:48 From: Henry Baker <hbaker1@pipeline.com> To: math-fun@mailman.xmission.com Reply-To: math-fun <math-fun@mailman.xmission.com>
What is known about which orbit sizes are possible?
By size, you must mean period, since x₀ and y₀ can be anything. I'm pretty sure we found many a small set of multipliers (δ, ε) whose periods grew periodically with (x₀, 0), say, covering the integers. If not, I'll bet Julian can construct such a covering fairly easily. Another approach: Removing the Floor functions yields an exactly solvable recurrence of any real "period", independent of (x₀, y₀). When these are large, restoring the Floor functions has only a small effect. I'm guessing you'd only need to try a few. Which isn't much help if you need exactly a googol and one.
What is known about which orbit sizes are impossible?
The above, if correct, says no period is impossible.
How many bits of state are required for an orbit of size N ? (non-trivial max ?, non-trivial min ?)
For fixed multipliers (δ, ε), the state is entirely contained in the current point (xn,yn). Usually, a very small shift in x₀ and y₀ will simply translate the whole orbit by that shift. Often, a small shift in (δ, ε) will not change the orbit at all. E.g., the large rectangles in the (mysteriously cropped) "Monster" plot on p12 of gosper.org/g12a.pdf (un-animated part of the talk). But see the discussion of accumulation points in the book. (Speaking of the animations, the one Rohan misprepared, "The worst circle I ever drew", should have drawn gosper.org/circle.png . Try to picture a Minsky making that!)
The video <https://youtu.be/lXsVWwPa7bc> talks about parallelizing; is there any kind of "speedup" -- e.g., repeated squaring of matrices or some such?
No. NeilB figured out how to heuristically subdivide the space of possible (x,y) starting points with about a 12% chance of connecting up with the desired orbit, so breakeven needed about eight processors. Although TomR never managed to close the Minsky Stock Index, he did manage to close disjoint loops of record length, in the quadrillions.
Suppose that a Minsky orbit of size N is known; how hard is it to map the integers 1..N to the elements of the orbit, You just count. We don't have a way of jumping ahead a prescribed distance.
and how hard is it to map back? Or is a table lookup the fastest method in both directions? More generally, can I use Minsky orbits for addition mod N ?
I don't know. Which is the answer to most questions about Minskyspace.
Are there any Minsky orbits which "decompose" into direct products of Minsky orbits? (I'm not entirely certain how such a direct product would work, but perhaps the peculiarities of particular numbers might allow the separation of a "high order" part and a "low order" part that happen not to interfere with one another.)
Julian? Help!
Does Minsky/Trinsky work on any other algebraic objects other than standard integers -- e.g., polynomials over Q in a single variable ? Probably. Important: The algorithm described in the talk and the book is over the reals, not the integers.
Perhaps I didn't understand the video well enough, but are there elegant cellular automata that perform the Minsky/Trinsky computation? I'd be surprised. --Bill
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