Cosmology

[desiresjab]

I have no familiarity with vectorizing components. Once January comes and I get a new system I will be doing some programming. All my programming will involve number functions, as I am not the least interested in programming my heater to go on and off et al. I hate learning a new language, because it is only a tool to get at what I want, but I will dive right in because I do want what I want.

Today I have to try and replace the side mirror on my car. An elk knocked it off as her head smashed into my windshield. If it had been a bull a horn would have killed me. It doesn't do much good to slow down because they will run right to your headlights anyway. You have to lay on the horn. Who can do that when they are swerving with their foot hard on the brake? These are the largest elk in the word--Roosevelt. I hope it did not become bear food. I was only going about 5mph when it hit me. I have lived in deer and elk country all my life and never hit one before now, though I have seen scores along the roads that others killed.

Well, it is not like you simply screw the mirror on from the outside. You have to take the door panel off. I could pay someone, but I refuse to do that on a job I should be able to do myself.

Back to math. I am hesitant to start all the work on 3's, because I do not believe there is much future there. I need one formula that shakes every prime out of a number. Actually, I think there is a way to do that by making some adaptations to the DeuceHound. I think that is hard. I know my brain is going to suffer.

When I look at the Ruler Sequences for 2 and 3, the opening cycle on 2's is so short it makes it hard to write the common pattern. I know it is there and I can see it. Writing it down successfully will be like a Chinese water torture. But when that is done, it should be fairly routine to write code for the other odd primes. That is what I am hoping, anyway.

Knowing myself, It will likely be another few days before I start. I have to rev myself up to attack the hard ones. A fine treasure waits at the end of this rainbow, if I can just find the end. First, I have to make myself look for it.

[desiresjab]

I forsee another difficulty ahead. The DeuceHound requires a number it can convert to decimal. The only way it knows to handle an exponential number is to convert it to decimal first. This would make it a wonderful machine for what today are considered fairly small numbers like billions, trillions or quadrillions, and it is especially designed to handle factorials. But what about numbers that are so large there is no possibility of the whole thing sitting inside a computer at once? I think this is the magnitude of number the calculational number theory guys are working on these days. I believe they have a way of breaking these numbers up, just the way tasks are broken into many modules on some of the large, shared research projects such as searching for the latest champ of primes, or even the Serpenski project you are contributing to (is that shared?), and delivering these modules to home computers across the world to work on while their owners surf the web, or something like that.

Theoretically, the DeuceHound (mod ∞) would still be a useful accomplishement, I guess, because it can handle powers factorialized, of a restricted kind (powers of primes). Otherwise, it can handle large decimal numbers factorializrd. 400,000,000,000!, the current ceiling for Brocard solutions, would be do-able, I believe, on the DeuceHound. A large power of 2 might not be nearby, but a large power of some prime might lie extremely close to 400000000000!, once the machine is (mod ∞). Finding that would be an extra problem. I expect to devise a better way eventually, i.e. to attack any number regardless of the form it is found in. That is a ways ahead.

The problem I forsee with the attempt to add bits is that every measure is not the same. If you just had to add 7 or 8 for each measure, that would be easy, but the values keep growing. Of course, we know what the values will be for even longer stretches than single measures. Fancy formulas or no, it may come down to adding up these stretches systematically until one reaches the desired last summand, anyway, as you have proposed. I simply do not know yet. I am still amazed that I got the DeuceHound (mod 2) to work correctly. I am not used to success.

[YesNo]

I do have one old computer running a Sierpinski sieve for PrimeGrid a distributive computing platform.

That you have broken this up into sums should allow it to be done piecemeal. I added another implementation of the DeuceHound. It only works for 2. Unfortunately it isn't vectorized. I also haven't done any performance tests on it, but I do have a few "old_ways" procedures coded for comparison and testing. https://drive.google.com/file/d/0B96...ew?usp=sharing

[desiresjab]

I do have one old computer running a Sierpinski sieve for PrimeGrid a distributive computing platform.

That you have broken this up into sums should allow it to be done piecemeal. I added another implementation of the DeuceHound. It only works for 2. Unfortunately it isn't vectorized. I also haven't done any performance tests on it, but I do have a few "old_ways" procedures coded for comparison and testing. https://drive.google.com/file/d/0B96...ew?usp=sharing

I am quite impressed with your DeuceHound implementation. You understand. I believe what you are calling piecemeal I called the Descent Method--where you got the triangular shape. Remainders are treated the same way as any number. That method never says, "Okay, thirty-one is small enough, I will just add up the Ruler Sequence from there." No sir, that is what a human would do. The DeuceHound finds the smallest power under a remainder and goes to work again, a perfect slave to its method, and perfectly accurate.

Sometimes figuring out an explicit formula for something is harder than devising an algorithm. I have not tried the explicit formula yet, but I will next. Right now I want to show you the rough schematic for the DeuceHound (mod ∞). It is what the formula will say and the algorithm do. It is not exactly a flowchart either. Well. See what you think. Where I am saying "print", in the outline, I just mean add. You get the idea. That will be described in the formula. Here I want the flow of the logic. Did I get it?

* * * * *

We can use the technique of adding bits to conquer 2, because it happens to be 2, and computer bits are binary. Only a general formula for all primes might conquer primes other than 2, however.

So we need a technique that will write (add) each bit for any prime. If a power of x is factorialized:

1 Write (x-1)...1's, then the next integer. This loop will be used on every print command throughout the algorithm.

2 Do that entire process x-1 times, then on the x^th time write 1...x-1 times then jump to the next integer, which for the prime 2 (our example) simply means to now write down 1 3, since we had already done our process x-1 times the first time we did it. With the prime 3, we would have had to have written 1 1 2..1 1 2..1 1 3. We would write that x-1 times before we came to a 1 1 4 in the Ruler Sequence for the prime 3. But here 2 is our example, because from it we must get the pattern. For 3 we wrote down 1 1 2 ....(x-1) times (twice), before jumping to the next integer 3. Like this 1 1 2..1 1 2..1 1 3..1 1 2..1 1 2..1 1 4....Before we get to 5, we have to do exactly what we just did, except for the last number, which increases by one.

3 We are back with 2 as our example. Write down everything you have again, except jump to the next integer on the last entry, which means write down the 1 2 1 3 you already have, except for the last entry, where 4 appears for the first time, like so: 1 2 1 3..1 2 1 4

4 Can you guess what to do next? Once more write down everything you have written down, but jump to the next integer on the last entry.

5 When you reach the correct power, add all your entries up.

The above would take care of values which are pure powers of any prime. We are using 2 as our prime in this exegesis, just as a familiar example. For values of Q, if Q were small, we would choose a simple algorithm that merely walked up to that value without any recourse to powers, perhaps. If Q were still uncomfortably large, we would instead use the method of descent with that prime, which I showed before for 2's.

6 Add the value for Q to the sum you already have, and that sum is the power of x for that factorial.

So the infinity model of the DeuceHound would then pull all factors of x exclusively from numbers of the type x^k!+Q, with the goal of later enlarging its scope to track down the factors of any number regardless of the form it is presented in, whether this involves finding the nearest power of x to the value one is dealing with, or some other technique later to be discovered.

* * * * *

In a nutshell: it adds 1...x-1 times, then adds the next integer. Now it must loop back and add exactly what it just added to the total except for the last number, before advancing to add the next integer, as in a virgin power. You know how Q is handled. So that's it. In the old Procedural platform I could have done it. There would have been a lot of ghastly plumbing and loops, for sure. I understand the OOP is much cleaner, easier, faster.

The DeuceHound may need a trademark and patent or copyright before long.


* * * * *

I do not think anyone else reads this thread. I am amazed you have followed the reasoning in detail. If it is not too personal a question, may I ask your profession before you became an old man like mysef? I am officially retired, but I have spent a total of at least thirty thousand hours in a chair or standing, doing each of the following to get by: playing poker, playing and teaching guitar; and doing the following for fun or ambition probably an equal ampunt of time: reading prose and poetry, writing prose and poetry, and doing amateur mathematics. My intersts continually pass the torch around to each other. Right now math is dominating again, sometime in the future it will be one of the other interests that dominates for a good spell. Since my interests dominate my life, they are about all I do besides kiss my family a lot, eat and sleep and watch movies on my computer. When a subject dominates me it really does, and I let it gladly. There is no resistance or regret.

* * * * *

What I want to gaze at right now is a number line with powers of numbers highlighted, the first power will not be highlighted, because that would mean highlighting every number. I know what I want, but it is hard to describe. I want to get a sense of how close any power comes to another. I do not know of a rule or a law for that. All numbers are on the line, and first powers are merely black as usual, 2nd powers and beyond would be red. I need the ability to look at long stretches far out the number line to see how powers interact. I did compare powers on 2 and 3 for a ways, and after a while they are not particularly neighborly, sometimes with thousands between a power on 2 and the nearest value that was a power on 3. I have a thinking there must have been powers of other numbers that were closer than that, which fell between those big gaps. I am trying to get a sense of how large we can expect Q to be when powers of all numbers are involved way, way out the nuumber line.

[YesNo]

The piecemeal method is like the descent method, a kind of looping or recursive or fractal or mathematical induction process. An explicit formula would be something like n(n+1)/2 for the nth triangular number. The piecemeal formula just starts adding from 1 to n. However, all the iterating can take time after a while.

With OOP one could create a class and build a set of methods supporting that class inheriting standard methods from the parent class. I didn't have to create a class and methods. I just used what Python methods were available in classes already built. However, I was thinking of building a class for a Sierpinski covering, if I ever get to that.

For the other primes there must be more involved than the base used to represent the number n, but maybe it is not as complicated as it looks at the moment.

As far as my profession, what I do could be called software engineering, computer science, data science, or database development. I have degrees in mathematics, but I do not work as an academic and I don't publish papers.

[desiresjab]

Degrees in math? No wonder you can follow this stuff. I was beginning to get suspicious. Only a technically trained person could or even would follow this thread in such detail. Vectorizing components was a pretty good clue. Maybe you should be writing the thread and I should be the one responding.

Anyway, I am still after the formula for DeuceHound (mod ∞). It is in my brain and will not straighten itself out yet. Perhaps a sigma with double indices might take care of the main routine. As for double sigmas, I truthfully must look up how they are used again. I used to know and forgot if it is simple nesting or something else.

[desiresjab]

Looking at the Ruler Sequence for 3, I worked out the explicit formula for the Tre-Tracker mode of the DeuceHound (mod ∞):

F₃(3^K!)=3^k-1.
................2

Sorry for all the dots. They were the only way the 2 would line up where it needs to be.

Will the formula for every odd prime be as simple as this and furthermore be its mimic, establishing a general formula? It seems intuitive that this should be, yet there may be a difference for powers of 4n+1 and 4n+3 primes factorialized, for instance. There may be two separate formulas.

[YesNo]

So, it is (3^k-1)/2? I will check that later today.

When you get your new computer we could exchange jupyter notebooks. We can format using mathjax.

[desiresjab]

I do not know anything about Jupyter and Python right now. For a long time I have desired Mathematica. It seems like it might be the ultimate in math software for home users and is not as expensive anymore. There are other packages available from different vendors. I have not had time to study them either. What do you know about all these math packages? I suppose some of them are good for a ways and then show glaring limitations.

We will definitely exchange some notes via internet channels once we can.

I believe I am close to a formula for all odd primes. The formula for 3 is so compact it still might be useful. Beautiful mathematical objects usually fit somewhere. The general formula may be an ugly girl you avoid when the pretty one is at hand.

When one inputs his prime p and power k, the computer checks whether p is 2 or is odd. If 2, the computer performs the by now familiar DeuceHound (mod 2) algorithm. If odd, it makes sure the input p is a prime. Then it does its thing with its alternative algorithm for odd primes.

I think I can get there within a day or two, but I might be fooling myself. First I may have to figure out the rule for what looks to be quite a cranky sequence I have come across.

Math man, my experience has been it is pretty easy to see and apply the rule of certain kinds of sequences, yet often a heavy challenge to find the formula that exactly describes that easy rule. The Fiobinacci sequence is easy to apply, but finding the formula that gives you any element by address in the sequence did not look particularly easy to me when my eyes hastily traveled over it, with √5 mysteriously appearing and all. I believe it is the same thing with the General Ruler Sequence--applying it is easy, but finding and correctly formulating the general rule requires the brain to do heavy reps. Maybe it is just my brain. More experienced mathematicians would probably struggle far less with these same concepts--which might even be described elsewhere. But investigating them seems like a personal job. I feel it is important to know the number mechanics of what is going on at all levels as thoroughly as you can, the way we investigated Quadratic Reciprocity, when one researches mathematical objects. I do not mind getting there without using a life line if I am able. The challenge is what I am into. Doing what it takes to construct such a function will enrich me along the way.

The 2 machine of the DeuceHound works. I also think the name works. Now we need its odd prime machine to work as handily. Once the formula is cracked, programming it might be easier than cracking the formula. I really do not know. I know in programming you get to work with algorithms. Instead of talking so much I should be trancing, or I have a thinking I will never get the plans for the DeuceHound's main engine drawn.

[YesNo]

Regarding the closed form of Fibonnacci numbers, I don't know how to derive it, but then I would just look it up and try to figure it out from there. Here is a source you might already be familiar with: http://www.maths.surrey.ac.uk/hosted...ibFormula.html

I like to use math.stackexchange. Here is something about that closed form there: http://math.stackexchange.com/questi...onacci-numbers It is a good place to get practice asking questions and using mathjax which is rather simple to use once you do it a few times.

One of the problems with Mathematica is that it is proprietary software. Not only does that mean that it costs money but I don't think the code implementing whatever you are using can be seen by you. I prefer open source software for mathematics. Python is free and you can download enough modules to get started through Anaconda's distribution. You can also add your own modules. Also, there are thousands of packages available which means a lot of people are using the language.

[desiresjab]

What I mean to say is, the pattern for both 3 and 5 are easy to define recursively, since each value depends on the last, and that one on the last before it. In the case of 3, an explicit formula was easy to find. 5 seems more stubborn, or different, or maybe I am having number blindness. At 7 I have not looked yet. It may be that 4n+1 types and 4n+3 types are different in this context, which makes sense as it is about powers, and that is why 5 will not be nice. Unfortunately, the next 4n+1 prime to test this hypothesis on is 13. The actual values for these tests grow huge fast and therefore harder for a human to spot a pattern or constant by eye. The DeuceHound may end up with three distinct main engines.

[desiresjab]

Jimeney Christmas! I got it!

When you trod the long way around, reasoning everything out for yourself, it takes a while. Which is the long way of saying I know I have seen this formula before. Whether it was used specifically for this or for some other purpose, I do not know (beautiful mathematical objects often have diverse applicability), but one does not forget such beauty even when there is no understanding of it.

F_p(p^k!)=

p^k-1
p-1

And this beauty even works for 2. I would have seen it earlier had I listened to my own instincts about the explicit formula for 3 being a template for all the other odd primes (little did I know it would work for the even prime, as well), but as usual I had to bull around in the china shop for a while before seeing the truth.

Well, the DeuceHound (mod ∞) is about ready to go. I suppose it has been for a couple of centuries. Sometimes it is so nice being ignorant, because playing discoverer is a great way to learn. It is infinitely better than finding the formula in a book and trying to figure out why it is so. I know why this one is so.

[Dreamwoven]

Congratulations, desiresjab!

[desiresjab]

Thanks, Dreamwoven. I only managed to labor my way into what has to be a well known fact after much sweat. That is the kind of nifty minor object Fermat was knocking out regularly 350 years ago. I am amazed I was able to find it. If the general case for 3 had not been pretty easy to spot, I am likely to have missed it altogether. Now I am really curious what else that formula might be used for. Just because I saw it before does not mean it was used for this. Something like that can be hard to find. Is there a formula hound on the internet?

[YesNo]

Here is an update pdf of the notebook: https://drive.google.com/file/d/0B96...ew?usp=sharing

This passes the initial test for 2 and 3, but not for 5 and 7. It might be a problem with the way I coded it.