Confirmed already! How? Because it was trivial after all. If one 4n+1 prime overlaps another by 4, four is a square, so they will obviously both have to be residues of the other, since one of them is at a glance known to be.
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Confirmed already! How? Because it was trivial after all. If one 4n+1 prime overlaps another by 4, four is a square, so they will obviously both have to be residues of the other, since one of them is at a glance known to be.
By the way, a previous post did not post, and the previous conjecture was easily solved, since the two primes in question overlapped by 4, a square number.
Which leads us to the next question, actually observation. Yeah, it is like a took a math pill tonight.
Suppose we have two super even 8n+1 primes. One is relatively close to zero, such as 17, for instance; the other is tremendously far out the number line. As long as the distance between them, their difference, the overlap number, is an obvious square number that we can see, we are guaranteed positive reciprocity positively gained, that is, by multiplying (1)(1) instead of (-1)(-1).
So in the case of this hugely eccentric rectangle we are guaranteed positive reciprocity positively gained. But do we know if the quadrant rectangle WAXY in Eisenstein's diagram on Wikipeja will be divided by the diagonal into two equal sets? Unfortunately, we do not. We only can guarantee that both sets will contain a positive number of lattice points, not that they will be equal. We cannot even say the difference between lattice points of the two sets cannot exceed two. We believe this is the case, but cannot prove or demonstrate it to our own satisfaction yet. We is me, apparently.
I am glad to see you back doing number theory. I am off and on thinking of the Sierpinski sequences and whether I can form coverings of them. I have been thinking about using Python to generate a cover, but I keep getting distracted.
That gasquet is often used to illustrate similarity across scale in books on fractal geometry. I have seen it, and about all I know is that it has fractal properties.Quote:
Originally Posted by YesNo
It is good to be back on QR, especially since I am making progress. My current attempt is to find something in the behavior of highly even numbers that distinguishes them. Only 8n type numbers are super even (more than two factors of 2).
I am beginning to suspect there might be no defining behavior that sets them apart other than what I have already stated about the role of "degree of evenness", otherwise I would already have found it in the literature. This likely why 4n types are the only ones used in the formal definitions.
Something I consider quite important that I learned last night from graphing is that the difference between lattice point sets can exceed 2. For the primes (5, 41), one set has 16 points and the other 24. Now we know any number can be this difference, depending only on the eccentricity of the rectangle. For me this is a huge breakthrough.
The thing about 8n numbers being the only ones with super evenness I should have realized long ago. I have been in possession of the ruler function for about a year and only just now have put the thoughts together.
The question might be asked, "Why explore this in a cosmology thread?" Cosmology is what I think it is. I am trying to look into the mind of the creator, to quote an idea found in Peter Martinson's paper on QR. Every time one understands something about math they previously did not, it amounts to looking into the creator's mind and methods. The deeper the proposition, the deeper one must look into the creator to understand it
By the way, if anyone looks at the Martinson paper be aware that there are two mistakes in it. Specifically, it once lists 2 as a quadratic residue of 19, and it once lists 67 as a 4n+1 number. These mistakes can stall an amateur, as they did me.
I don´t understand anything about number theory, but I´m glad that this scientific thread is alive again.
I am, too, Danik!
I have a dear friend who regularly likes to blast science and technology, and even math. Sometimes I fight back, but more often I let it go. It is useless to argue with anyone about what something is. But I find that most people who rail against science have misconceptions about what it is and what it is supposed to do. In short, one might call it the art of numerical observation. Poets and novelists are keen observers, too, but not normally numeric observers. Scientific observation is tied to numerics because of the world around us--the world and things in it quantify naturally, once humanity taught itself the knack. Objects fall the same speed every time, so through repeated experiment men were able to quantify that speed and finally find a formula for it.
Some people, honestly, expect way too much of mathematics and science, but when asked for suggestions they come up with the same old criticisms. Do they expect all scientists to drop what they are doing and go look for a ghost or proof that aliens built the earth's ancient pyramids?
The legitimate mathematicians plug along, as they always have, noting patterns in innocent numbers. Brains like Fermat, Euler, Gauss, Eisenstein and Reimann and many others, had built up quite a cache of these number patterns in three hundred years. Lo and behold, almost every one of them has a reflection in nature or a direct expression taken from a pine cone, a seashell or a flower stalk, or at least has a very strong application. It is a fact that much of the action in our everyday world of man and nature can be compressed into a simple forumla, a number pattern. These patterns were there, someone had to find them, someone had to eventually realize their applicability to some corner of our universe. It is a cause not for blame but celebration.
Will numbers prove as useful in the study of so called spiritual phenomena, dark matter, time travel, astral travel, dream awareness, consciousness itself? Will it be able to handle what physicists dig up? I suspect it will be useful for the things it is now useful for and some of what science uncovers. We may discover another tool. Math is very fatalistic. Math is a grand tautology.
I feel the biggest laws of the universe are yet undiscovered, even barely suspected. I think this has to be the case when our observations are only impeccable concerning 4% of the stuff in the universe, yet almost totally ignorant of the other 94%. The fruit does not hang so low anymore.
How gigantic was it when Newton discovered the laws of 4% of our stuff, and then later when Einstein replaced the model? An actual scientific breakthrough in any of the fields mentioned above would be huge. I expect something odd when someone finally lays a finger on these mysteries. We may find what is holding together those galactic clusters which are moving too fast is a form of consciousness. Sometime in the future the discovery of a consciousness particle would not shock me. I expect the strange out of the universe.
The problem with that 94% of the supposed missing stuff is that it may not be there. All that we may need to do is reformulate the mathematical gravitation theory and do away with the need to find dark matter. And since we haven't found any, so far, maybe it doesn't exist at all.Quote:
Originally Posted by desiresjab
Einstein did something like this in the early 20th century. At that time astronomers were looking for a planet they called Vulcan near Mercury that should exist if Newton's laws were correct which would explain the orbit of Mercury. Einstein's modification of Newton's gravitational theory made the search for Vulcan unnecessary.
I'm getting that account of Vulcan from John Moffat's "Reinventing Gravity". He has a new theory of gravity that should make dark matter and black holes unnecessary. Of course, if someone finds dark matter that would shoot down his new theory.
Reformulate? Hmmm. Not so sure about that. Whatever dark matter & dark energy turn out to be, they represent new phenomnena. I believe theories are reformulated when they are pretty close but off. Any theory of gravity does not even get us close to understanding the phenomena we are observing. But, yes, it could even turn out you are right. I am doubtful we will do away with theories of gravity altogether, and since it would need serious modification to fit today's observations, what we will have around is a modified one. Of course it could also turn out that our threries of gravity are essentially correct, that DM and DE are a new type of phenomena requiring a new structure piled on top of our theories of gravity.Quote:
Originally Posted by YesNo
Maybe a reformulated theory of gravity would occur if we discovered new features of gravity that could account for the phenomena. There may be types of neighborhoods where gravity behaves differently. There are not supposed to be, the way the theory is formulated, but the universe is full of surprises and I believe it will continue to be. Maybe there is more than one type of gravity--a Higgs Boson stock split of sorts.
I think we need to keep some distance from media reports about what is or is not real in the universe when it comes to dark matter, dark energy, black holes or a singularity at the big bang.Quote:
Originally Posted by desiresjab
Regarding the current need for dark stuff, the following seems to be true: the current evidence from viewing the rotation of galaxies has falsified Einstein's theory of gravity in a way so big that measurement inaccuracies do not account for the discrepancy in the prediction and the observations.
There are two ways around the problem and, from what I understand from reading Moffat's book, many people are pursuing both approaches:
1) Einstein's theory is correct. That means there exists dark stuff, but we cannot detect it. As people look they eliminate possible candidates for this dark stuff and these negative results are valuable.
2) Einstein's theory is not correct. We need a new theory of gravity. However, that theory of gravity is not easy to come by. Moffat mentioned some of the notable failures. He does think his version is sound and fits the observations.
You cannot be a phyicist without a thorough knowledge of calculus. What you have to know if you are a number theorist is modular arithemetic. Many people do not know what that is. If they look it up, they are told it is "clock arithemetic," and so it is, as far as that goes. If I say to you, "It is twenty-five o'clock," you will easily figure out it is one o'clock.
Mathematicians call it conguence theory. That is what Gauss named it. The notation looks like this 16≡4 (mod 12). Translated into English that means 4 is the remainder when 16 is divided by 12. Just as in normal arithemetic, this is equivalent to 16-4≡0 mod(12). However, we could not say (16/4)≡1 (Mod12), as in normal arithemetic, since the remainder when 4 is divided by 12 is 4. Twelve is called the modulus, because Gauss knew Latin.
Usually, the modulus is a prime number, but it does not have to be. There are a few more traps to watch out for and exceptions to know when dealing with composites, the theory a little more extended. We will keep it prime.
To show its usefulness, let us consider an easy problem.
1 Use mod notation find the last digit of 340. (Hint): In other words, the remainder when 340 is divided by 10.
To solve this with mod notation we first have to know a simple law:
If ar≡b (mod m), then ars≡bs. We merely need to factorize the exponent and use this law.
34≡1 (mod 10). Then 34(10)≡110 (mod 10).
The answer is one
****
Let's look at one slightly harder.
2 What is the last digit in 720?
74≡1 (mod 10). Therefore 74(5)≡15 (mod 10).
The answer again is one.
* * * * *
Maybe someone can solve this next one.
3 Find the last digit in 79?
The calculational difficulties grow fast with only a little increase in the base. This number is probably too large to find the last digit on your calculator.
What is the last digit in 1920?
Factor the way easiest for calculation.
192≡1 (mod 10), 192(10)≡1 (mod 10)
The answer is one.