Cosmology

[desiresjab]

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.

I don't know what could be wrong. The programming language is so compact I cannot see anything. The explicit formula should give the value of the sequence up to a certain value for n for one prime. Your program appears to want the sum of all primes up to a certain n. It will do that individually to the primes. Adding the numbers of all the primes together will not give you anything, if that is what I am seeing. Am I looking at the program incorrectly? I just got up. I need time to wake up and examine it more closely.

* * * * *

Now we have the ability to sum the Ruler Sequence for any prime. That is with each prime having the separate metric of itself. I guess what we need for Brocard is one metric to judge all primes against. That is what we do not have. We need the abilty to see how the ordered primes are stuffed into a factorial, how many together we need to meet the conditions of 2n(2n+2), if any amount together can.

If we had a way to judge all primes just from the value of the formula for one prime,like 2, then we would have somethibng that might assist with Brocard. We would not have to change the value of p each time we wanted the facts on another prime.

Meanwhile, I have not been able to even find the expression 2^p-1/p-1 anywhere else again. It will turn up. It will not be as handy as some equation like d/r=t, used for everything from baking cookies to electronics, but will have quite a few disparate uses, I am wagering.

[YesNo]

I think I got the description of your algorithm wrong. It isn't the sum over the primes but the sum over the powers of a particular prime dividing n. I'll try to get a correction tomorrow.

Edit: In looking it over it seems I implemented the wrong algorithm.

Edit: I updated the pdf file with what I think is the correct algorithm, but I am still getting discrepancies when I try n = 95 for primes 5 and 7. I printed out intermediate results. https://drive.google.com/file/d/0B96...ew?usp=sharing

[desiresjab]

95 is not a power of 5 or 7, which means you will have some value of Q tagging along. The nearest power of 5 is 125, giving us the opportunity to use subtraction on Q for the first time rather than addition and illustrate another symmetry of the Ruler Sequence. The measures on either side of a virgin power are symmetrical, meaning that measure for meausre the sequence will look the same on either die of the virgin power until, of course, the measure containing the next virgin. This means that even when we are subtracting Q instead of adding it, all we have to do is go to the beginning of the Ruler Sequence and add those values up to and including Q, then subtract them from F_p(5^k!). No nasty backwards subtractions in the sequence. We can calculate -Q precisely as if we are calculating Q. An example below with p as 5 and k as 3, goes as follows when I simply add up Q by sight rather than instituting the repetitive breakdown on Q as a computer would do and a complete formula designate (which we call the Descent or Reduction method):

F₅(95!)=F₅(125!)-F₅(30!)=

[(5³-1)/4]-7=24

If that is not right, cancel the party hats while there is time, lads!

* * * * *

By a common metric for all primes I mean this: plug in the value 2, for instance, for p, and something else for k, and the DeuceHound spits back the F_p for any prime of your choosing up to the value 2^k+Q. 2 might be the right metric, too, because it is unique among primes and has the same relationship to every prime (mod 2). Now, that kind of formula is something we are not even close to yet, do not know is frankly possible, and doubt our toolkit in the search for such a master key, none of which shall stop us but only give us pause to whine a little before we proceed. We can acquire or invent new tools and perspectives where they are missing. Our toolbox is light for those hikes far out the number line.

[YesNo]

If we allow subtraction, then we will need a rule to tell when to use subtraction. I think the number of factors of 5 in 95! is 22. It was part of the result of running old_way(95) in the jupyter notebook. Of course, I might have programmed that wrong.

I get a remainder of 45299530029296875 when I run 95! mod 5²⁴. That should be 0 if there are 24 factors of 5 in 95!

[desiresjab]

Subtraction is no different from addition. To subtract, you just add. One could, in fact, change the negative sign in front of Q and obtain the same results, which is the whole idea. I am not sure what might be wrong with the program. Let me physically illustrate that the number of factors of 5 in and below 5³ is 31, and in and below 30 it is 7 factors of 5.

5.10.15 2025 30
1..1..1..1..2....1..1..1..1..2....1..1..1..1..2... .1..1..1..1..2....1..1..1..1..3

Sorry the multiples of 5 are kind of crowded above the sequence. As you can see, counting backwards from 3 remains the same as counting forward from 1 all the way to just before 3. That is why we can start at the beginning of the Ruler Sequence when subtracting instead of counting backwards from 3. Same thing.

The first 3 in the sequence takes us up to 5³. The sixth element in the sequence takes us up to 30. The value of the sequence (added together ) up to 30 is 7. Subtract that from 31 and we have 24, the answer. Notice we could have counted backwards from 5³ for the same answer. This means we could also have counted forward six elements from 3 to obtain an identical answer, as well. Hope that helps.

[desiresjab]

One of my beliefs is that mathematics will continue to play a significant role in our comprehension as we go about unfolding the universe for ourselves and exposing deeper and deeper levels of its reality. No one can predict what future mathematics will look like any more than Newton or Fermat could have told you squat about vectors, matrices, topology or complex analysis.

Will this future mathematics consist to a large degree in making headway in classifying and analyzing the set of transcendental numbers? And I have to ask myself, why not? We know almost nothing about them, yet the few we do have cognizance of are of extraordinary importance in the grappling contest with nature. Pi and e are primary to our understanding of nature. Besides those two, which were both finally proven to be transcendental after much effort from great mathematicians, there are a host of other important constants waiting in the wings to be proven transcendental or not. Almost surely every one of them is, but each proof for any individual number suspected to be transcendental is titantically difficult.

How many more important constants await our discovery which are the lynchpins of phenomena we have barely or yet begun to study, like consciousness and quantum reality?

There are more transcendentals in the interval between (1, 2) than there are rational numbers in the universe. In fact, there are more transcendentals in the interval between (1, 1000001/1000000) than there are natural numbers. Make the interval as small as you want, there will always be more transcendental numbers in that interval than there are fractions in the universe.

Why would one not wonder if many great secrets lie unsuspected within this set? It is more numerous than all other sets combined, and all other sets have yielded up great truths about reality.

The unapproachability of this set in general is where the difficulty lies. Liouville finally managed to very cleverly construct some artificial transcendentals. As far as I know, this particular class of transcendental has not proved useful outside of mathematics, though it still could I suppose. No one ever said every transcendental might be useful. In fact, there is no such thing as every transcendental. Their numerosity cannot be counted, or even called numerosity, or even classified beyond Cantor's description of them as having the suspected power of the continuum. Cantor does not know if any set lies between the "countable," infinities and the uncountable continuum or not.

This why I cannot help but feel great paradigm shifts lie ahead which will directly correspond to our increasing understanding of this set of numbers, after the appropriate human lag in time to understand what mathematicians have discovered. This lag is always part of our reality. Leibniz was three centuries ahead of digital computers, but understood already the basic concept of such computing machines. Boole was a century and a half ahead of his time; Archimedes played around with or very near to the fundamental ideas of calculus. Then there was Liouville who managed to artificially construct a transcendental number never seen before. How many centuries will it be until someone goes through the door Liouville could only open a tiny crack?

[YesNo]

I wonder how Liouville constructed that artificial transcendental number. He would have to make sure it could not be the root of a polynomial with rational coefficients. Here is one paper that looks promising but I haven't read it: http://deanlm.com/transcendental/con...tal_number.pdf

[desiresjab]

I wonder how Liouville constructed that artificial transcendental number. He would have to make sure it could not be the root of a polynomial with rational coefficients. Here is one paper that looks promising but I haven't read it: http://deanlm.com/transcendental/con...tal_number.pdf

I have not read the link yet either. Here is how I have read Liouville constructed his number. In every digital place of his number he put a 0, unless that digital place was the value of a factorial. He put 1's in the 1^rst, 2^nd, 6th and 120th digital places, etc. His number then looked like this:

11000100....1000000000...00000001... ...

....................↑ 24th digital place .......↑... 120th digital place.

He had to undertake to prove that this was transcendental. This may be intuitively clear, but I cannot quite make it out. More likely it is a very involved monster to prove his number is a tranny. But wait, he only constructed this number in the first place because he knew beforehand it would be a tranny. He had a concept and carried it out, then. If that was intuitively clear to him, then it must be possible for the same concept to be intuitively clear to us, I would think. We need to reverse engineer it. I believe that is what he must have done, once he realized the concept mentally.

Those occasional 1's mean a remainder of 1 when the Liouville number is divided by powers of 10. Any partial representation of the Liouville number ending in a 1 and divided by powers of 10 smaller than itself would also leave a remainder of 1. Don't ask me what this means. I only see it, I don't know how it fits into his proof. I would say he was a clever man.

[YesNo]

That construction would guarantee it is not a rational number, since the digits do not repeat, but how to show it is not algebraic?

One can come up with infinitely many transcendental numbers by us of the Gelfond-Schneider theorem. If a and b are algebraic numbers with a not equal to 0 or 1 and b not a rational number, then a^b is transcendental.: http://sprott.physics.wisc.edu/pickoveR/trans.html I don't know how that theorem was proven either.

[desiresjab]

I wonder how Liouville constructed that artificial transcendental number. He would have to make sure it could not be the root of a polynomial with rational coefficients. Here is one paper that looks promising but I haven't read it: http://deanlm.com/transcendental/con...tal_number.pdf

After having read the link, I see they point out what intutively makes this number irrational--its repetend never repeats. Repetends and the pattern of digits recurring after a decimal point are essentially the same thing. That makes it at least an irrational. The rest of the proof proves that it is of the transcendental variety of irrational number. Besides a little set theory notation, the proof mainly involves algebraic integers and several bounding theorems from calculus.

[desiresjab]

That construction would guarantee it is not a rational number, since the digits do not repeat, but how to show it is not algebraic?

One can come up with infinitely many transcendental numbers by us of the Gelfond-Schneider theorem. If a and b are algebraic numbers with a not equal to 0 or 1 and b not a rational number, then a^b is transcendental.: http://sprott.physics.wisc.edu/pickoveR/trans.html I don't know how that theorem was proven either.

I have not read this link yet. But I have seen this result before, and I must say it always surprised me that the class jump could be made so easily. Somewhere down in the the mechanical process of multiplication which a power is, the numbers are doing exactly what they must when a strictly algebraic irrational is used as an exponent on another algebraic number whether it be rational or not (is the way I took it). These number mechanics I believe we can only see from a higher level of abstraction, we cannot see the clicking of individual numbers in the process and what turns the result into a transcendental, the way such a vision of number mechanics might enable a complete mechanical understanding QR (notice I say might).

That might mean learning the theory of algebraic integers better. In number theory this is where polynomial equations are used to perform the operations of arithmetic upon each other instead of regular numbers doing it to regular numbers. If they are algebraic integers, then I believe (but don't know) they obey all the laws of integers in some form. We should remember this result and stay aware of it whenever we encounter the term.

I believe I remember reading that algebraic integers even have their own version of prime numbers. These may be more than something trivial like √(x+1)=7. I do not know a great deal about the theory of algebraic integers--big ol' polynomials you can treat just like integers in your calculations, is how I think of them.

[desiresjab]

Things which are likely shoo-ins still must be formally proven in mathematics before they can be accepted. One smiles to see that we are free to assume 2π and 2^π are transcendental, but we cannot assume π^π is.

[desiresjab]

That weird symbol that did not turn out well in the last post is pi.

[YesNo]

I was looking more at Liouville numbers. This Wikipedia link seems to contain the basic information along with a proof that these numbers are transcendental: https://en.wikipedia.org/wiki/Liouville_number

The proof depends on the concept of the "irrationality measure" of a number which is a measure of how close the number can be approximated by rational numbers. Rational numbers have an irrationality measure of 1. Basically, they are not irrational. Algebraic numbers that are not rational have irrationality measure of 2. Transcendental numbers have an irrationality measure of 2 or greater. Liouville transcendental numbers have infinite irrationality measure. They are kind of extreme as far as this measure goes. It looks like they were constructed to make sure they were as far away from being algebraic as possible which allowed them to be more easily proven to be transcendental.

The algebraic numbers are roots of polynomials with integer coefficients. They include the rationals which are roots of linear polynomials f(x) = rx + s where r and s are integers. Algebraic "integers", a subset of algebraic numbers are roots of such polynomials where the coefficient of the highest power of x is 1, that is, they are roots of "monic" polynomials, such as, x - 7 would have the root 7, an integer.

[desiresjab]

I was looking more at Liouville numbers. This Wikipedia link seems to contain the basic information along with a proof that these numbers are transcendental: https://en.wikipedia.org/wiki/Liouville_number

The proof depends on the concept of the "irrationality measure" of a number which is a measure of how close the number can be approximated by rational numbers. Rational numbers have an irrationality measure of 1. Basically, they are not irrational. Algebraic numbers that are not rational have irrationality measure of 2. Transcendental numbers have an irrationality measure of 2 or greater. Liouville transcendental numbers have infinite irrationality measure. They are kind of extreme as far as this measure goes. It looks like they were constructed to make sure they were as far away from being algebraic as possible which allowed them to be more easily proven to be transcendental.

The algebraic numbers are roots of polynomials with integer coefficients. They include the rationals which are roots of linear polynomials f(x) = rx + s where r and s are integers. Algebraic "integers", a subset of algebraic numbers are roots of such polynomials where the coefficient of the highest power of x is 1, that is, they are roots of "monic" polynomials, such as, x - 7 would have the root 7, an integer.

I thought Liouville numbers could be more closely approximated by rational rumbers than pi and e. It looks like I could get pretty close with Liouville numbers. Just those occasional pesky 1's are in the way of the exact value.