Cosmology

Oh, boy. All our questions and more are answered in this little paper. The authors give examples of sums of squares with 1, 2, 3, 4, 6 (Can't remember if 5 was there) unique expressions, or factorizations, as these authors more properly call them.

A thorn in the path to watch for is their not counting 0² in their method of calculation. This means the standard formula (which I found elsewhere) will give different values. One only needs to subtract 1 from the total number of expressions.

The standard forumla for the number of expressions of any natural number to any power is:

r_k(n)=2[d₁(n)-d₃(n)].

Now n is the sum of squares itself, k is the power, which may be higher than 2, and d₁ and d₃ signify the number of actual factors or divisors of n, prime or otherwise, which are respectively congruent to 1 or 3 (mod 4). A congruency of 2 or 0 (mod 4) is not germane, and not included in the calculations.


http://www.rowan.edu/colleges/csm/de...Submission.pdf

Now as far as the two professor problem goes, I have made a terrible, amateurish oversight. I always confess to my oafish ignorance and embarrassing oversights, especially the real whoppers, because they make such good copy.

Essentially, words two and three of the problem nail down a simple constraint I flat overlooked in my zeal to forge ahead. Our ignorance always forces learning upon us when we persist. I cannot be emabarrassed for what I have learned, but only for what I used to not know, like yesterday.

We learned a lot, and the problem will be quite different when we return to it tomorrow--with some elements ejected from the set of prospects.

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Originally Posted by desiresjab

http://www.rowan.edu/colleges/csm/de...Submission.pdf

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This article made me realize that whenever one is talking about sums of squares one should think Pythagorean theorem and circles.

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Originally Posted by YesNo

This article made me realize that whenever one is talking about sums of squares one should think Pythagorean theorem and circles.

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Quite true. Remember that is what the Martinson article did.

The two professor problem is too hard for me unless I find another line of attack. Plus, it is not a fun problem for me, but has become a minor obsession anyway. For recreation I have to do beautiful problems. At least they are beautiful to me. I wonder if anyone agrees with me over what "looks" good in mathematics. Here is one.

When is x^pq divisible by (x^p)^q? Prove.

The last problem is very easy of course. Let p^q=a, let pq=b.

From basic algebra:

x^a=x^a-b.
x^b

Problems like this are good for refreshing yourself on technique.

The answer is for all x except 0.

Here is a pretty good one. It is countable, if you figure out how to do it. (Hint): There is also a technical way to do this.

Four hundred people are standing in a circle. You tag one person, then skip k people, then tag another, skip k, and so on, continuing until you tag someone for the second time. For how many positive values of k less than 400 will every person in the circle get tagged at least once?

Have fun.

It looks there would be many ways in which at least one person would not be tagged and that would depend on k being a divisor of 400. If k is relatively prime to 400 everyone should get tagged eventually.

But what about those values of k that have some factors that also divide 400, but other factors that don't? For example 6? Just checking 6 and 10 that would skip some as well.

So I assume the value would be the number of values of k relatively prime to 400. That would be given by Euler's phi function. I searched for a way to calculate that value: https://www3.nd.edu/~sevens/13150unit10.pdf

However, this isn't a proof that the above is correct, just some comments suggesting that it should be.

Yes. Euler's phi function is the right function. It will read off the answer directly. The actual work would look like this.

Φ(400)=400Π_p|n(1-1/2)(1-1/5)=400(4/10)=160.

Obviously, divisors of 400 cannot do the job. But every k relatively prime to 400 will circle the table leaving a different remainder (mod Φ(400)) at the end of each revolution.

I was thinking more about gravity lately.

It occurred to me that finding something in the universe with a dark concentration of gravity larger than three solar masses would trigger the black hole portion of Einstein's gravitational theory. If one finds something like that then the theory says all the matter in that region collapses to a point of no radius. There is no counter source of energy, such as fusion in a star, able to keep the radius larger than zero. It vanishes into a singularity of the theory, a place that effectively looks like the theory is dividing by zero.

What can one conclude if one finds such concentrations of dark gravity? I think the most reasonable thing to conclude is that the theory is wrong. Just because the theory has a singularity doesn't mean that reality has singularities. I also suspect that a theory with singularities has got problems from within. Those singularities imply that the theory can reach a point where it stops serving as an explanation.

Now the theory also has problems explaining the rotation of galaxies. They move too fast to stay together without more matter (which is also dark). But one hasn't, so far, been able to find that dark matter. It looks like the theory fits the universe of the early 20th century well. That universe didn't have galaxies, dark gravitational sources (aka black holes) nor was it believed to be expanding.

I can´t follow your complex calculations but I wonder if this article or the cosmology journal as such might be interesting for this thread:
http://journalofcosmology.com/JOC26/...26CONTENTS.htm

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Originally Posted by Danik 2016

I can´t follow your complex calculations but I wonder if this article or the cosmology journal as such might be interesting for this thread:
http://journalofcosmology.com/JOC26/...26CONTENTS.htm

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Yes/No and I will send our observations right over to those people.

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Originally Posted by Danik 2016

I can´t follow your complex calculations but I wonder if this article or the cosmology journal as such might be interesting for this thread:
http://journalofcosmology.com/JOC26/...26CONTENTS.htm

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I do find the idea that "black holes" are more like MECOs (magnetospheric eternally collapsing objects) to be interesting. https://en.wikipedia.org/wiki/Magnet...lapsing_object

MECOs solve one problem with black holes: there is no mass for a zero radius object since the mass has been radiated away throughout eternity. A black hole has a large amount of mass but its radius is zero. It also resolves the problem of a black hole collapsing faster than the speed of light to that point. I liked how such a massive point (black hole) was called a "mathematical myth".

Supposedly there is an effort to build an earth based set of radio telescopes on each side of the globe, called an Event Horizon Telescope, making the resolution power of such telescopes larger that what currently exists (https://en.wikipedia.org/wiki/Event_Horizon_Telescope) They would be able to point to the radio source at the center of our galaxy known as Sagittarius A*. Some believe this to be a black hole, but it failed to "eat" a cloud of matter passing by it a year ago as predicted, if it were the kind of black hole they imagined it to be.

John Moffat's theory does away with black holes entirely, but I don't know how this is done. Besides black holes, there is also the anomaly (based on Einstein's theory) of a too rapid rotation of galaxies as well as the singularity (like a black hole) at the beginning of the universe. Moffat's theory does not merge gravity with the other three forces found at the atomic level (electromagnetic, strong and weak nuclear forces). However, that might be a plus for his theory.

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Originally Posted by desiresjab

Yes/No and I will send our observations right over to those people.

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Seems a good idea!

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Originally Posted by YesNo

I do find the idea that "black holes" are more like MECOs (magnetospheric eternally collapsing objects) to be interesting. https://en.wikipedia.org/wiki/Magnet...lapsing_object

MECOs solve one problem with black holes: there is no mass for a zero radius object since the mass has been radiated away throughout eternity. A black hole has a large amount of mass but its radius is zero. It also resolves the problem of a black hole collapsing faster than the speed of light to that point. I liked how such a massive point (black hole) was called a "mathematical myth".

Supposedly there is an effort to build an earth based set of radio telescopes on each side of the globe, called an Event Horizon Telescope, making the resolution power of such telescopes larger that what currently exists (https://en.wikipedia.org/wiki/Event_Horizon_Telescope) They would be able to point to the radio source at the center of our galaxy known as Sagittarius A*. Some believe this to be a black hole, but it failed to "eat" a cloud of matter passing by it a year ago as predicted, if it were the kind of black hole they imagined it to be.

John Moffat's theory does away with black holes entirely, but I don't know how this is done. Besides black holes, there is also the anomaly (based on Einstein's theory) of a too rapid rotation of galaxies as well as the singularity (like a black hole) at the beginning of the universe. Moffat's theory does not merge gravity with the other three forces found at the atomic level (electromagnetic, strong and weak nuclear forces). However, that might be a plus for his theory.

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What fascinates me is that theories today have to change very quickly to keep pace with the high amount of new discoveries.

Prove that for n ε N, (n!)! is divisible by n!^(n-1)!.

Here is a compact little gem. Divisibility is a fundamental concept in number theory, so it is always good to practice. At the moment, I do not see an answer, but there should be a fairly simple way of showing that the denominator will always be smaller than or equal to the denominator and is a whole number, which is what this problem asks for in plain English. Probably a factorization which allows something to be cancelled on the bottom. It has to be put into such a form first.