Science topics: MathematicsNumber Theory
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# Number Theory - Science topic

The study of properties of integers and prime numbers.
Questions related to Number Theory
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Hello everyone! I am actively looking forward to collaborating on a Number Theory research project this summer.
I am interested.@
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A tunable clock source will consist of a PLL circuit like the Si5319, configured by a microcontroller. The input frequency is fixed, e.g. 100 MHz. The user selects an output frequency with a resolution of, say, 1 Hz. The output frequency will always be lower than the input frequency.
The problem: The two registers of the PLL circuit which determine the ratio "output frequency/input frequency" are only 23 bit wide, i.e. the upper limit of both numerator and denominator is 8,388,607. As a consequence, when the user sets the frequency to x, the rational number x/108 has to be reduced or approximated.
If the greatest common divider (GCD) of x and 108 >= 12 then the solution is obvious. If not, the task is to find the element in the Farey sequence F8388607 that is closest to x/108. This can be done by descending from the root along the left half of the Stern-Brocot tree. However, this tree, with all elements beyond F8388607 pruned away, is far from balanced, resulting in a maximum number of descending steps in excess of 4 million; no problem on a desktop computer but a bit slow on an ordinary microcontroller.
F8388607 has about 21*1012 elements, so a balanced binary tree with these elements as leaves would have a depth of about 45. But since such a tree cannot be stored in the memory of a microcontroller, numerator and denominator of the searched Farey element have to be calculated somehow during the descent. This task is basically simple in the Stern-Brocot tree but I don't know of any solution in any other tree.
Do you know of a fast algorithm for this problem, maybe working along entirely different lines?
Many thanks in advance for any suggestions!
Now I tested the idea of jumping downwards in the Stern-Brocot tree. This idea is based on the observation that long paths from the root to a Farey element always seem to contain long sequences of steps directed exclusively to the left or exclusively to the right. As long as the direction is constant, the values of numerator and denominator which are added to the current node are also constant, obviously. Therefore, the products of numerator and a certain jump width resp. denominator and a this jump width can be added to the node.
In order to determine the largest possible jump width, bitwise successive approximation is used in this first approach. The result is quite satisfactory:
With an input frequency of 100 MHz, and the output frequency in the range 1 Hz to 100 MHz - 1 Hz (at the extrema, the approximation by Farey elements is poor, of course), the sum of the passes through the outer loop (movement through the tree) and through the inner loop (determining the maximum jump width) never exceeds 386. Attached is my C source. Compared to the maximum number of single steps, this is an improvement of 4 orders of magnitude.
While this approach solves my practical problem, I would still be interested in other solutions because sometimes it's amazing how a problem can be tackled in different ways.
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Dear fellow mathematicians,
Using a computational engine such as Wolfram Alpha, I am able to obtain a numerical expression. However, I need a symbol expression. How can I do that?
I need the expression of the coefficients of this series.
x^2*csc(x)*csch(x)
where csc: cosecant (1/sin), and csch: hyperbolic cosecant.
Thank you for your help.
Today I spared time to get an accurate answer to this question, which is showed in the picture uploaded here.
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Dear Researchers,
Do researchers/universities value students/researchers having published sequences to the OEIS?
Dear Marco Ripà ,
I have done both. I cited my work in the sequences and the sequences in my work.
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Why is a Proof to Fermat's Last Theorem so Important?
I have been observing an obsession in mathematicians. logicians and number theorists with providing a "Proof for Fermat's Last Theorem". Many intend to publish these papers in peer reviewed journal. Publishing your findings is good but the problem is that a lot of the papers aimed at providing a proof for Fermat's Last Theorem are erroneous and the authors don't seem to realize that.
So
Why is the Proof of Fermat's Last Theorem so much important that a huge chunk of mathematicians are obsessed with providing the proof and failing miserably?
What are the practical application's of this theorem?
Note: I am not against the theorem or the research that is going on the theorem but it seems to be an addiction. That is why I thought of asking this question.
Muneeb Faiq , the situation has changed and there should be no fear, when staying before the FLT.
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Hi, I'm an undergraduate math student. I've already passed linear and abstract algebra and recently passed a course in elementary number theory and I really enjoyed it.
now I wanna self-study more books this semester and I'm searching for good books with exercises.
my most interest is in solving diophantine equations and their applications.
tnx for your suggestions btw and sry for my bad English :)
You may consult the Number Theory section of MAA Reviews
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=====================================
See also my list of links to my other RG documents:
=====================================
Question posted on May 20, 2018
Are there other pieces of information about “Victory Road” to FLT?
I have posted a research project on Research Gate on the history of the construction of “Victory Road” to the proof of Fermat Last Theorem (FLT).
Are there pieces of information that are missing in this history?
I will add to this history any new references, pieces of information, and good comment about this history, with full credit to the first who finds it.
You gave beautiful reasoning. But, my reasoning remains valid. I will give details a little later.
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Conjecture 1. [November, 2021]
The Fermat number $F_n=2^{2^n}+1$.
Discuss the irrationality and transcendence of $\sum\limits_{n=0}^{\infty} \frac{1}{F_n}$.
Conjecture 2. [November, 2021]
Discuss the irrationality and transcendence of $\sum\limits_{n=1}^{\infty} \frac{1}{2^{p_n}-1}$.
Here $p_n$ is the nth prime.
Conjecture 3. [June, 2019]
Give a sequence $(a_1, a_2, a_3, ...)$, $a_n \in \{-1, 0, 1\}$.
Discuss the irrationality and transcendence of $\sum\limits_{n=1}^{\infty} \frac{a_n}{n^{2}}$.
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Dear Colleagues
I need an inequality for the ratio of two Bernoulli numbers, see attached picture. Could you please help me to find it? Thank you very much.
Best regards
Feng Qi (F. Qi)
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This paper is a project to build a new function. I will propose a form of this function and I let people help me to develop the idea of this project, and in the same time we will try to applied this function in other sciences as quantum mechanics, probability, electronics …
Are you sure you have defined your function correctly?
1. Usually z=x+iy. But in your function z is in the limit, thus being in both the arguments and what the integral is computed against. If z is not x+iy, the function is not a function of (x,y).
2. What do you mean by limit? Do you want to compute the case when z->0?
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Leibniz’s formula for the n-th derivative of a product is an extremely important and useful formula. Is an explicit formula for the n-th derivative of a quotient also very important?
Issam Kaddoura Nop, that is cheating, that doesn't work.
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Here we discuss about one of the famous unsolved problems in mathematics, the Riemann hypothesis. We construct a vision from a student about this hypothesis, we ask a questions maybe it will give a help for researchers and scientist.
I put together a solution of the RH myself. While it can't be considered a complete proof while not vetted by experts, it presents various strong arguments and a real breakthrough, which is the inversion formula for Dirichlet series. Given any Dirichlet F(s), you know a(n) from F(s). Unfortunately, it's impossible to have an integral representation for a(n) usually, it's a Taylor power series. Please head to my page for the paper.
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I’m defining a number system where numbers form a polynomial ring with cyclic convolution as multiplication.
My DSP math is a bit rusty so I’m asking when does inverse circular convolution exist? You can easily calculate it using FFT but I’m uncertain when does it exist? I would like to have a full number system where each number only has a single well defined inverse. Another part of my problem is derivation. Let c be number in my number system C[X] where coefficients are complex numbers. Linear functions can be derivated easily but I’m struggling to minimize mean squared error (i = 0..degree(C[X]), s_i(c) selects i:th coefficient of the number, s_i(x): C[x]->C):
error(W) = Exy{sum(i)(0.5|s_i(Wx - y)|^2)}
I can solve problem in case of complex numbers W E C but not in case of W E C[X] where multiplication is circular convolution. In practice my linear neural network code diverges to infinity when I try to minimize squared error.
Pointing any online materials that would help would be great.
Have you tried Mathematica? It's my favorite software for Maths.
Better than Python (easier to use).
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In my modest of opinion, this paper provides a very simple proof of Fermat's Last Theorem using methods that were available in Fermat's days. If correct, it follows that this proof squashes any criticism against Fermat's claims.
It took 358 years (1637-1994) to get Professor Wiles' complicated proof, but this paper shows that one could achieve this in a simpler manner. I am sure this proof is flawless as I have gone through it very thoroughly.
I think Fermat had the proof.
It looks like we have climbed a high mountain to look for something that is right at the foot of the mountain.
Fermat's claim: there does no exist integers x,y,z greater than unity for any (n>2) for which the equation:
xn+yn=zn.
has a solution. He went on to say:
"I have discovered a truly marvelous proof of this, which this margin is too narrow to contain."
Good job,
Cheers
Andrea Rossi
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Do you know the web sites of the journals Ars Combinatoria and Utilitas Mathematica? Are these two journals ceased? One of my manuscripts was accepted a long time ago for publishing in the first journal, but now I cannot contact any editor, I cannot find its website, and I cannot get any message of these two journals. So I wonder if these two journals have been ceased.
Dear Dr.,
The information above is provided for possible search. That is, the corresponding transitions to other sites needed, that are possible from the links provided by me.
Unfortunately, I don't have anything more yet.
Sincerely,
Sergey Klykov
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There are many outstanding articles written in the last 20 years in Number Theory. What do you think what is the most interesting and groundbreaking article in Number Theory?
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For the COOP function I get a message that indicates the program cannot find my Group.txt file even though I have systematically placed a copy in every folder within the Gaussum directory. Can you tell me how I can create the Group.txt file. I have attached the Gaussian output file also below. Secondly, I like to know that, from the DOS image in gausssum, how can I understand one peak is belongs to which orbital state. I also like to know how I draw the DOS in origin from the Gausssum?
The first line needs to be either "atoms", "orbitals", "allatoms" or "allorbitals". If it is "allatoms" or "allorbitals", then no further input is required and a separate group will be made for each atom or each orbital. You need to describe which atoms or which orbitals are in each group. An easy way to obtain these for Gaussian calculations is to open the output file in GaussView and turn on the labels. Groups.txt needs to obey the following rules:
1. Every atom in the molecule must be listed. 2. No atom may be listed more than once.
A single point calculation should be done with the following keywords: (Gaussian) pop=full iop(3/33=1,3/36=-1), (GAMESS) NPRINT=3. This creates a large log file containing information on the overlap matrix among other things.
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I have confirmed that the Hessenberg determinant whose elements are the Bernoulli numbers $B_{2r}$ is negative. See the picture uploaded here. My question is: What is the accurate value of the Hessenberg determinant in the equation (10) in the picture? Can one find a simple formula for the Hessenberg determinant in the equation (10) in the picture? Perhaps it is easy for you, but right now it is difficult for me.
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What lessons and topics are prerequisites for algebraic number theory and analytic number theory?
Please tell me the exact topic of each lesson.
It would help if you studied advanced abstract algebra, topology, mathematical analysis besides the introductory courses in general number theory.
Regards
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Dear researchers, I am very interested in making more progress in the direction of the functional analysis and number theory. After the second stage of my private research regarding the Riemann Zeta function and Xi-function, I have been able to validate one fascinating consequence of the Riemann Hypothesis, probably never seen before, and it is related to the set of the Taylor even coefficients a_2n and Jensen C_n (and the Turán moments b_m or b_n calculated by professors Csordas, Norfolk and Varga, almost 36 years ago). I have validated these b_m, and their impact in the definition of the famous coefficients of the Jensen polynomials with shift N=0 and Taylor coefficients a_2n which evidently (or undoubtedly) lets formulate a representation by a series for the well-known and important Euler-Mascheroni constant, and also the Lugo's. I do not have any contract with any institution (university) nor studying any master/PhD, but I considere that I could show an significant set of results that would open the door for future researchs not only in mathematics, but also in physics. As I live in Europe, electronics engineer graduate from UPB, several years ago (2007) and my passion for searching and validating models ( I am good at Matlab and other programms), I can work in a research project if I had such priceless opportunity. The reason for investigating in functional analysis and number theory when being an electronics engineer is because I like mathematics being applied to several fields, I believe in the possibility to determine new formulas that impact in exact science and engineering. Nowadays, I feel alone in my own research and I have tried to contact several magazines for showing these golden results and to be able to coming back to the college world, unfortunately, time and some circumstances during my years as engineer trying to pursuit some chance in research have affected my way to do what I would love to do the best: researching and completing my PhD. That is why I post this question about who could help me to come back to an institution and be able to build an interesting path of results and findings I know I can give.
For now, I have reached a tremendous data and fascinating equations that only could be exposed once finished my 10 or 13 pages of current article, as I am aware of the formal stages of a publication. I would like to contact not only magazines, but also prestigious universities whose groups or members could get inmersed in my ideas and results.
Let me know if there were good information and contacts via inbox or replying to this post.
Carlos López
Electronics engineer
Graduate from UPB, Medellín, Colombia
Technical representative, current city: Szczecin, Poland
Firstly, I think you should publish your results in a good reputable journal, and next, your request gets easier.
I wish you good luck in your endeavor.
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More precisely, if the Orlik-Solomon algebras A(A_1) and A(A_2) are isomorphic in such a way that the standard generators in degree 1, associated to the hyperplanes, correspond to each other, does this imply that the corresponding Milnor fibers $F(A_1)$ and $F(A_2)$ have the same Betti numbers ?
When A_1 and A_2 are in C^3 and the corresponding line arrangements in P^2 have only double and triple points, the answer seems to be positive by the results of Papadima and Suciu.
See also Example 6.3 in A. Suciu's survey in Rev. Roumaine Math. Pures Appl. 62 (2017), 191-215.
Your question is very interesting,
Regards and the best wishes,
Mirjana
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The aim of this Conference was more far-reaching than the presentation of the latest scientific results. It consisted of finding connections between this fundamental theoretical branch of mathematics and other fields of mathematics, applied mathematics, and science in general, as well as the introduction of top scientists with paragraded structures, which would lead to the connection and cooperation of scientists working in various fields of abstract algebra and algebraic theory of numbers, ultrametric and p-adic analysis, as well as in graph theory and mathematical logic.
SARAJEVO JOURNAL OF MATHEMATICS, Vol. 12 (25), No.2-Suppl.
This issue is dedicated to the memory of Professor Marc Krasner, Officier des Palmes de l'Academie des Sciences de Paris on the occasion of the 30th anniversary of his death.
All manuscripts of this issue were presented at International Scientific Conference "Graded Structures in Algebra and their Applications" held in Inter University Center, Dubrovnik Croatia, September 22-24, 2016
Contents of Vol. 12, No. 2-Suppl.                                                   DOI: 10.5644/SJM.12.2.00
Professor Marc Krasner - photos                                                                                                   DOI: 10.5644/SJM.12.3.01
Mirjana Vuković, Remembering Professor Marc Krasner                                                               DOI: 10.5644/SJM.12.3.02
Alain Escassut, Works involving Marc Karsner and French mathematicians                                 DOI: 10.5644/SJM.12.3.03
Emil Ilić-Georgijević, Mirjana Vuković, A note on radicals of paragraded rings                           DOI: 10.5644/SJM.12.3.04                                                                                                                 Emil Ilić-Georgijević, Mirjana Vuković, A note on general radicals of paragraded rings
DOI: 10.5644/SJM.12.3.05
Mirna Džamonja, Paragraded structures inspired by mathematical logic                                       DOI: 10.5644/SJM.12.3.06
Vlastimil Dlab, Towers of semisimple algebras, their graphs and Jones index                               DOI: 10.5644/SJM.12.3.07
Elena Igorevna Bunina, Aleksander Vasilevich Mikhalev, Elementary equivalence of linear groups over graded rings with finite number of central idempotents                                                 DOI: 10.5644/SJM.12.3.08
Nadiya Gubareni, Tensor algebras of bimodules and their representations                                    DOI: 10.5644/SJM.12.3.09                                                                                                               Dušan Pagon, On codimension growth of graded PI-algebras                                                       DOI: 10.5644/SJM.12.3.10
Smiljana Jakšić, Stevan Pilipović, Bojan Prangoski, Spaces of ultradistributions of Beurling type over ℝd+ through Laguerre expansions                                                              DOI: 10.5644/SJM.12.3.11
Alexei Panchishkin, Graded structures and differential operators on nearly holomorphic and quasimodular forms on classical groups                                                                                           DOI: 10.5644/SJM.12.3.12
Siegfried Böcherer, Quasimodular Siegel modular forms as p-adic modular forms                       DOI: 10.5644/SJM.12.3.13
Alain Escassut, Kamal Boussaf, Abdelbaki Boutabaa, Order, type and cotype of growth for p-adic entire functions                                                                                                                         DOI: 10.5644/SJM.12.3.14
Dear Sajda,
A good conference ... please offer other conferences.
I 'll invite you to the next conference if you can fit in.
All the best for you,
Sincerely Mirjana
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I was wondering is there any
• model theory of number theory ,hence are there model theorists working in number theory
• the development of arithmatic geometry ,does it have anything to do with questions in logic;and is there any group studying this interaction.
• Anyone is welcome and up for collaboration
• I am interested in finding interaction between algerraic and arithmatic number theory with logic,and to study it to answer logical questions about Arithmatic
As far as I know, in the entire history of mankind, only two philosophers have seriously dealt with logic, this is Aristotle and Hegel. Of these, only Hegel did mathematics. Nobody else dealt with this problem.
Sincerely, Alexander
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Dear fellow researchers,
In your expert opinion, do you think the Riemann Hypothesis is true? The first billion zeros have been computed and they all verify this hypothesis. However, we have previously seen patterns hold until a very large number than break (There is a conjecture that holds for n<10^40). Do you think there is any reason to believe that it might be false?
If anyone proposes a theory and no one can make it wrong, at least it is not wrong, but it remains a mystery, and that is why it is called a mystery .. This does not mean that the theory is wrong.
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How many more years do you predict it will take before the Riemann Hypothesis is solved?
Do you think we are close or does it seem that we are still very far?
I agree with you. Although year after year we are getting closer. A few year ago, it was proven that at least 40% of zeros have to be on the critical line. So a lot of progress has been done. Many conjecture have also been presented, which if proven, they would imply the Riemann Hypothesis. So have gotten some important result, however, we have not quite solved it yet.
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I have written two articles about a generalization of Multiple zeta values and Multiple zeta star values. I also presented applications for this generalization including partition identities, polynomial identities, a generalization of the Faulhaber formula, as well as MZV identities. If you are intrested check them out on my profile and give me your opinion.
The Riemann Z function finds applications in physics Dear Roudy El Haddad, I just cannot remember all of them, but the theory of phase transitions, Planck radiation, statistical sums, we use the identities without too much thinking for several partition functions.
Also, some applications are related to the most general class of Z functions with multiple Z values.
The Riemann Hypothesis is somehow connected to some real eigenvalues of a hermitian “Riemann Operator” so that these zeroes are spectral, according to some authors (Lyon, 2008).
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I find that multiple sums are a very interesting concept. However, an even more interesting concept for me is the sum of such sums (the sum of multiple sums). The sum of multiple sums can be turned into a simple product by the formula in the first attached image.
What mathematical applications could this formula have? Where could it be useful?
I have found 2 interesting applications:
1- sum of multiple zeta values (see image 2).
2- sum of multiple power sums (see image 3).
Could you suggest any applications for these 2 particular cases? Could you suggest additional particular cases that would be of interest to mathematicians or physicists?
Why are you asking?
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Does Faulhaber’s formula for the sum of powers have any useful mathematical or real life applications?
Does generalizing this formula to calculate the multiple sum of power have useful applications?
Yes dear Dr Roudy
I just have seem them.
Really good.
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Do you think the "New Whole Numbers Classification" exactly describes the organization of set N ?
Whole numbers are subdivided into these two categories:
- ultimates: an ultimate number not admits any non-trivial divisor (whole number) being less than it.
- non-ultimates: a non-ultimate number admits at least one non-trivial divisor (whole number) being less than it.
Non-ultimate numbers are subdivided into these two categories:
- raiseds: a raised number is a non-ultimate number, power of an ultimate number.
- composites: a composite number is a non-ultimate and not raised number admitting at least two different divisors.
Composite numbers are subdivided into these two categories:
- pure composites: a pure composite number is a non-ultimate and not raised number admitting no raised number as divisor.
- mixed composites: a mixed composite number is a non-ultimate and not raised number admitting at least a raised number as divisor.
From the paper:
very interesting....
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Is it possible to use heuristic methods?
Agreed with dear Neil J Calkin
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Could you prove that for any big number N there exists a composite Fermat number F_k such
that F_k > N?
Fermat numbers are odd numbers.
I think you are confused with Fibonacci numbers where F3n even numbers.
Best wishes
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Number theory is among cryptography foundations, but sometimes it is hard for students to understand the theory, mostly due the lack of previous skills and knowledge of that mathematical theory by students.
Have you dealt with that problem? Have you faced other problems while teaching number theory? How to overcome them?
This is a good question! I was trying to figure that out a couple years ago.
The best approach I found was Coursera's course from USC San Diego - Intro to Discrete Math for CS Specialization with the 4th course being Number Theory and Crypto. The first two courses give enough grounding for those students that don't have enough of the correct mathematical background for Number Theory. I think they need something pretty accessible to not be overwhelmed. People often have such a love of crypto that they slog through with just a really basic and algorithm specific understanding of number theory - which really won't help them in the long run if they are going to do cryptanalysis etc. Dan Boneh's refresher on Probability is really short and helpful as well.
Please post here if you find a good solution.
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To whom might be interested,
I have been thinking about prime numbers and how they might fit into our world, and this idea came to me that the primes might be constructed in a similar way to how elements fuse in stars.
To demonstrate my idea I wrote a short php script showing the construction of the first 25 primes. I have never seen anything like this before and I believe it is an original idea.
Would love to have some feedback on this from someone in number theory who have studied the primes.
Steven
.
They orbit around some composite numbers like the electrons around the nucleo
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I am a bachelor's student in mathematics.
Hello Amirali: I am physician and for that I studied math deeply. One day the teacher tell us that all maths are algebra and analisys. I dont want to generalize because there are branches of math that we dont study in physics. And we dont have to proof math conjectures. But Physician have a great instinct to solve problems with the general math. Study deeply Algebra and Analisys was on the best things I did in my live. FIrst is to understand very good these general branches and after you could focus in what you want.
Best regards
Hugo
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Can we apply the theoretical computer science for proofs of theorems in Math?
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The following formally published papers are related to this question:
[1] Feng Qi, Diagonal recurrence relations for the Stirling numbers of the first kind, Contributions to Discrete Mathematics 11 (2016), no. 1, 22--30; available online at https://doi.org/10.11575/cdm.v11i1.62389
[2] Feng Qi and Jiao-Lian Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, Bulletin of the Korean Mathematical Society 55 (2018), no. 6, 1909--1920; available online at https://doi.org/10.4134/bkms.b180039
[3] Feng Qi and Bai-Ni Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Applicable Analysis and Discrete Mathematics 12 (2018), no. 1, 153--165; available online at https://doi.org/10.2298/AADM170405004Q
[4] Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean Journal of Mathematics 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1
[5] Feng Qi, Da-Wei Niu, Dongkyu Lim, and Yong-Hong Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, Journal of Mathematical Analysis and Applications 491 (2020), no. 2, Paper No. 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382
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Where to find the answer?
The following formally published papers are related to this question:
[1] Feng Qi, Diagonal recurrence relations for the Stirling numbers of the first kind, Contributions to Discrete Mathematics 11 (2016), no. 1, 22--30; available online at https://doi.org/10.11575/cdm.v11i1.62389
[2] Feng Qi and Jiao-Lian Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, Bulletin of the Korean Mathematical Society 55 (2018), no. 6, 1909--1920; available online at https://doi.org/10.4134/bkms.b180039
[3] Feng Qi and Bai-Ni Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Applicable Analysis and Discrete Mathematics 12 (2018), no. 1, 153--165; available online at https://doi.org/10.2298/AADM170405004Q
[4] Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean Journal of Mathematics 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1
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Bell polynomials of the second kind Bn,k(x1,x2,...,xn-k+1) are also called the partial Bell polynomials, where n and k are positive integers. It is known that Bn,k(1,1,...,1) equals Stirling numbers of the second kind S(n,k).
What are the values of the special Bell polynomials of the second kind Bn,k(0,1,0,1,0,1,0,...) and Bn,k(1,0,1,0,1,0,...)? Where can I find answers to Bn,k(0,1,0,1,0,1,0,...) and Bn,k(1,0,1,0,1,0,...)? Do they exist somewhere?
The following formally published papers are related to this question:
[1] Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean Journal of Mathematics 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1
[2] Feng Qi, Da-Wei Niu, Dongkyu Lim, and Yong-Hong Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, Journal of Mathematical Analysis and Applications 491 (2020), no. 2, Paper No. 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382
[3] Feng Qi, Da-Wei Niu, Dongkyu Lim, and Bai-Ni Guo, Closed formulas and identities for the Bell polynomials and falling factorials, Contributions to Discrete Mathematics 15 (2020), no. 1, 163--174; available online at https://doi.org/10.11575/cdm.v15i1.68111
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What are the special values for the Bell polynomials of the second kind $B_{n,k}(1,1,3!!,5!!,7!!,\dotsc,(2(n-k)-1)!!)$? See the picture below. Please help me find a reference to cite. Thank a lot.
The following formally published papers are related to this question:
[1] Feng Qi, Xiao-Ting Shi, Fang-Fang Liu, and Dmitry V. Kruchinin, Several formulas for special values of the Bell polynomials of the second kind and applications, Journal of Applied Analysis and Computation (2017), no. 3, 857--871; Available online at http://dx.doi.org/10.11948/2017054
[2] Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean Journal of Mathematics 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1
[3] Feng Qi, Da-Wei Niu, Dongkyu Lim, and Yong-Hong Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, Journal of Mathematical Analysis and Applications 491 (2020), no. 2, Paper No. 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382
[4] Feng Qi, Da-Wei Niu, Dongkyu Lim, and Bai-Ni Guo, Closed formulas and identities for the Bell polynomials and falling factorials, Contributions to Discrete Mathematics 15 (2020), no. 1, 163--174; available online at https://doi.org/10.11575/cdm.v15i1.68111
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Trial division: To test if n is prime, one can check for every k≤ sqrt (n) if k divides n. If no divisor is found, then n is prime.
Or 6k+/-1
Primality test and easy factorization
This is an algorithm that test if one number is prime and if the number it´s not prime returns the biggest factor of that number.
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Do you think then that there is a universality of this particular ratio?
I, for my part, highlighted this ratio in these different areas:
The most famous ratio is the golden value. Anyway, this value generated by the famous Fibonacci recurrence Fn+1 = Fn + Fn-1 ,
{0,1,1,2,3,5,8,13,21,....} Fibonacci numbers
that is
lim Fn+1 / Fn = the golden value = (1 +sqrt5)/2
Also, one can see that the following sequence of numbers
Gn+1 = 4Gn - 3Gn-1
{ o,1, 4, 13, 40,....}
converges to the value 3/2,
that is lim Gn+1 / Gn = 3/2
I hope you can connect this sequence with your observations in nature and science.
Regards
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I have tried to link some topics in mathematics which included the word " Rational" , I have got many references which used " Rational" in Group theory and Probability and number theory and algebraic geometry and Topology,Chaos theory and so on , Now I'm confused and I have asked my self many times why that "Rational" occurs so much in all topics of mathematics probably informatic and physics ? Why this word interesting in mathematics ? According to the below linked reference I ask why always we investigate to get things in mathematics to be rational ?What is the special of that word rational in mathematics ?
**List of Linked reference include word " Rational"**:
[Regularization of Rational Group Actions](https://arxiv.org/abs/1808.08729)
[Rational Points on Rational Curves](https://arxiv.org/abs/1911.12551)
[Automatic sets of rational numbers](https://arxiv.org/abs/1110.2382)
[Rational homology 3-spheres and simply connected definite bounding](https://arxiv.org/abs/1808.09135)
[A Rational Informatics-enabled approach to the Standardised Naming of Contours and Volumes in Radiation Oncology Planning](https://www.academia.edu/7430350/A_Rational_Informatics-enabled_approach_to_the_Standardised_Naming_of_Contours_and_Volumes_in_Radiation_Oncology_Planning)
[A note on p-rational fields and the abc-conjecture](https://arxiv.org/abs/1903.11271)
[Remarks on rational vector fields on CP1](https://arxiv.org/abs/1909.09439)
[A trace formula for the distribution of rational G-orbits in ramified covers, adapted to representation stability](https://arxiv.org/abs/1703.01710)
[Rational cobordisms and integral homology](https://arxiv.org/abs/1811.01433)
[Conditioned invariant subspaces, disturbance decoupling and solutions of rational matrix equations](https://www.tandfonline.com/doi/abs/10.1080/00207178608933450)
[Why study unirational and rational varieties?](https://mathoverflow.net/q/287364/51189)
[Degree of rational maps via specialization](https://arxiv.org/abs/1901.06599)
[Minimum rational entropy fault tolerant control for non-Gaussian singular stochastic distribution control systems using T-S fuzzy modelling](https://www.tandfonline.com/doi/abs/10.1080/00207721.2018.1526984)
[On the generic nonexistence of rational geodesic foliations in the torus, Mather sets and Gromov hyperbolic spaces](https://link.springer.com/article/10.1007/BF01377597)
[Irrationality Measure of Pi](https://arxiv.org/abs/1902.08817)
[Rational Unified Process](https://arxiv.org/abs/1609.07350)
[Rational Computations of the Topological K-Theory of Classifying Spaces of Discrete Groups](https://arxiv.org/abs/math/0507237)
[On Periodic and Chaotic Orbits in a Rational Planar System](https://arxiv.org/abs/1405.3124)
**Note** I do not investigate about the meaning of the word "Rational" in each topic but I want to know why its were dominated why it is interesting whatever the kind of its meaning ?
In elliptic curves theory Rational is derived from the firleld which the curve is built over
It is very impotant to study the rational points (rational coordinates) since the rational field has many algebraic extensions and this is an important property
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Dear Researchers,
Kindly let me know about 5 top most research problems in number theory which are concerning to
Double series and / or combinatorics
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How to prove or where to find a proof of the lower Hessenberg determinant showed by two pictures uploaded here?
Dear All Colleagues
The final version has been accepted for publication in the Mathematica Slovaca. For details, please click at the website:
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The origin of gravitation, the origin of electric charge and the fundamental structure of physical reality are resolved, but these facts are not yet added to common knowledge. Also the structure of photons is resolved and the begin of the universe is explained. A proper definition of what a field is and how a field behaves have been given. These facts are explained in .
This model still leaves some open questions. The model does not explain the role of massive bosons. It does not explain the existence of generations of fermions. The HBM also does not provide an explanation for the fine details of the emission and absorption of photons. The model does not give a reason for the existence of the stochastic processes that generate the hopping paths of elementary particles. The model does not explain in detail how color confinement works. It also does not explain how neutral elementary particles can produce deformation. The referenced booklet treats many of its open questions in sections that carry this title.
The model suggests that we live in a purely mathematical model. This raises deep philosophical questions.
With other words, the Hilbert Book Model Project is far from complete. The target of the project was not to deliver a theory of everything. Its target was to dive deeper into the crypts of physical reality and to correct flaws that got adapted into accepted physical theories. Examples of these flaws are the Big Bang theory, the explanation of black holes, the description of the structure of photons, and the description of the binding of elementary particles into higher order modules.
The biggest discovery of the HBM project is the fact that it appears possible to generate a self-creating model of physical reality that after a series of steps shows astonishing resemblance to the structure and the behavior of observed physical reality.
A major result is also that all elementary particles and their conglomerates are recurrently regenerated at a very fast rate. This means that apart from black holes, all massive objects are continuously regenerated. This conclusion attacks the roots of all currently accepted physical theories. Another result is that the generation and binding of all massive particles are controlled by stochastic processes that own a characteristic function. Consequently the Hilbert Book Model does not rely on weak and strong forces that current field theories apply.
The HBM explains gravity at the level of quantum physics and thus bridges the gap between quantum theory and current gravitation theories.
The Hilbert Book Model shows that mathematicians can play a crucial role in the further development of theoretical physics. The HBM hardly affects applied physics. It does not change much in the way that observations of physical phenomena will be described.
Lee Smolin has an opinion about “Which fundamental questions about physical reality are still open?”:
1. The problem of quantum gravity: Combine general relativity and quantum theory into a single theory that can claim to be the complete theory of nature.
2. The foundational problems of quantum mechanics: Resolve the problems in the foundations of quantum mechanics, either by making sense of the theory as it stands or by inventing a new theory that does make sense.
3. The unification of particles and forces: Determine whether or not the various particles and forces can be unified in a theory that explains them all as manifestations of a single, fundamental entity.
4. The tuning problem: Explain how the values of the free constants in the standard model of particle physics are chosen in nature.
5. The problem of cosmological mysteries: Explain dark matter and dark energy. Or, if they don't exist, determine how and why gravity is modified on large scales. More generally, explain why the constants of the standard model of cosmology, including the dark energy, have the values they do.
With kind regards, Sydney
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Because "the first generation of infinite set theory" (Cantor’s set theory) is based on present classical infinite theory system, it is bound to be unable to get rid of the confusion of "potential infinite and actual infinite" contents, and people have to take the quantity of elements in an infinite set as the mixed number forms of "potential infinite number and actual infinite number" derived from those concepts of "potential infinite and actual infinite". This inevitably leads to two contradictory cognitive behaviors: On the one hand people deny the necessary relationship between "element” and “set" (not knowing at all that the existence of "elements with different properties" leads to the existence of different infinite sets?), firmly deny the characteristic differences of elements contained in different infinite sets (the unique existing meaning, unique existing form and unique existing condition as well as unique relationship between), and construct a kind of “cardinal number theory” by "double abstraction" which is conflict with the nature of infinite set. Through the "double abstraction", elements contained in many infinite sets are turned into piles of “geometric points" without any differences of “unique existing meaning, unique existing form and unique existing condition as well as unique relationship between”, to ensure that they all have the same "cardinal number" (to ensure many different infinite sets have the same quantity of elements). After "double abstraction", elements contained in different infinite sets have lost their original unique properties (including the property of number) and become "endless infinite geometric point"--------"endless" becomes the only numerical property for all the elements contained in many infinite sets (as we know, all the points on the lines are just piles of "endless abstract things" without any differences of “unique existing meaning, unique existing form, unique numerical property and unique existing condition as well as unique relationship between”. All the points on the line segments are with the same "cardinal number" and heir quantities are uniformly "infinite"). So, in present classical set theory (Cantor’s set theory), after "double abstraction", many subsets and their original sets contain same amount of elements -------- the elements contained in many infinite sets have the same "cardinal number" and "endless" becomes the only numerical property for all the elements contained in many infinite sets. On the other hand, all in a sudden, people recognized the necessary relationship between "elements and sets", firmly recognized the importance of the essential differences in the manifestation, nature, existing conditions and relationship among the elements in infinite sets, recognized that it is the existence of "elements with different characteristics" that leads to the existence of different infinite sets; all in a sudden, people denied the "double abstraction theory”, suddenly decided not to apply "double abstraction theory” in the cognitive process for elements contained in the infinite set, so as to ensure the applying of T = {x|x📷x}theory which has nothing to do with "double abstracted” to find some elements still with their special original features (not being "double abstracted”) and to complete some proofs that some infinite sets contain more infinite elements than other sets (for example, the infinite elements contained in Real Number Set are more than the infinite elements contained in Natural Number Set, the infinite elements contained in any infinite set are less than the infinite elements contained in its power set -------- Infinite Real Number Set is more infinite than Infinite Natural Set Number Set, and any infinite set is less infinite than its power set,... For hundreds of years, people have been trying so hard to study and fabricate various "infinite concepts", various formal logic, formal languages and "assembly line" operations related to those "contradictory and colorful concepts of infinite". However, the fundamental defects revealed by these two problems have determined the impossibility of scientific, effective and systematic qualitative and quantitative studies to elements contained in infinite sets, paradoxes are inevitably produced -------- because it is impossible to know at all what the infinite related elements contained in infinite sets are (the abstract things that are both potential infinite and actual infinite: the "ghost" disappearing and reappearing at any time?). Therefore, we draw an important conclusion: "the third mathematical crisis" is another manifestation of "the second mathematical crisis" in set theory. They are "twins". Studies have shown that, the unavoidable conceptual confusion of "potential infinite, actual infinite" in present classical infinite theory system determines Cantor's theory and operations of “cardinal number” and "double abstraction" are not self-justification at all and lack of scientific and systematic, which inevitably results in the impossibility of scientific, effective and systematic qualitative and quantitative studies to elements contained in infinite sets.
Nothing wrong. All correct. Wrong are your attempts to find errors in this theory. Get busy, buddy!
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Its clear that there is about 15 years (2004-2019) after the publication of AKS primality testing in 2002 and its modifications in 2003-2004. AS result, is there any development happened in this theory after 2004 ?
Many results appeared every day; all that you need is to followup on the published articles. That required access that allows you to see all the updates.
As of November 2017, the largest discovered prime was in November 2017
using the ECPP ( elliptic curves primality tests ) method. It has 34,987 digits.
It takes 756 days to test the primality of 2116224- 15905.
See
Caldwell, Chris. The Top Twenty: Elliptic Curve Primality Proof from the Prime Pages.
Best wishes
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The following is calculation plot from MATLAB, which I considered a modified version of Euler totient function.
can this be analytically proved from number theory?
P.S- The legend of the attached graph is not correct
Yes, the prime counting function π(x), counts all primes not exceeding x,
it includes x if x is prime.
Example: we have 2, 3, 5, 7, 11 are prime numbers not exceeding 11
therefore, π(11) = 5.
Best regards
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If corresponding factorials and triangular numbers are added, the results form the sequence of numbers, {2, 5, 12, 34, 135, 741, 5068, 40356, ...}, which I call factoriangular numbers. In the list of the first few factoriangular numbers, I found three Fibonacci numbers: 2, 5 and 34. Aside from these three, are there other Fibonacci numbers in the sequence of factoriangular numbers?
Now it is a a complete useful answer.
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Kurt Mahler classified the complex numbers into fours categories. These are A-numbers, S-numbers, U-numbers and T-numbers. It is known that A-numbers are precisely the algebraic numbers. Also, Liouville's numbers are contained in the set of U-numbers. It is also known that all the set of numbers A, U, T are set of measure zero and hence S-numbers form a set of full measure. Does any one know example of S-number or T-number?
You can read the complete proof on one of my favourite books on trnascendental number theory, called “Making transcendence transparent: an intuitive approach to classical transcendental number theory“, by Edward Burger and Robert Tubbs, Section 1.6.
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Given that N is a non cubic integer and x, y and z are integers, what is the general solution of x^3 + Ny^3 = z^3?
Using Fermat's proof, it is well- known that, if N is non zero cubic integer, then
x^3 + N^3y^3 = z^3 has no solutions over Q.
A trivial example: for N = 9 is a complete square
we have (-2)3 + 9(1)3 =(1)3 .
(-2,1,1), (2,-1,-1) are primitive solutions.
Also, the equation of the form x^3 - d y^3 = 1 has a solution for which y= 3 if and only if d = u(27 u^2 + 9 u + 1) for some integer u, for example; the equation x^3 - 37 y^3 = 1 has a solution in nonzero integers which is (10, 3).
Thus the equation x^3 - d y^3 = 1 is nontrivially solvable for certain values of d.
Rewriting this equation as x^3 + (_1)^3 = d y^3, then equation of the form x^3 + Z^3 = d y^3 are studied in [*] Chapter The Cubic Analogue of Pell’s Equation .
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and apply this to the Riemann Zeta Function
Tapas Chatterjee We display results in separation of concerns: The first parr comes from:
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We are proposing the following theorems (along with proofs) (See attached Images) as a new technique for use in ascertaining convergence of non-monotone series with smooth differentiable functions. We will be using this in another important proposal later on.
I am of an Actuarial background as as such I will kindly ignore all references to my notation rather than trying to see the simple series of integrals for what they are.
We put our ideas on this platform for constructive criticism.
Dear Issam Kaddoura you are completely correct, I seem (in the very first image) to have missed a portion of the sentence and this is a fault on my part.
"This is a wrong response, your claim was
the sequence Σ(f)is convergent if and only if ∫x(df/dx )dx exists."
I kindly have attached the image displaying the proof and reasoning in the first set of images when beginning the discussion. My fault again for not presenting this in a manner conducive. I have taken the images and put these into a simple document attached here.
Many thanks and best wishes
Rajah
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The equation may be of the form:
Let (x,y,z) be the unknowns.
Ax+By+Cz=p, A,B,C, p\in \mathbb{Z} and gcd(A,B,C)=1. Does this system exist an integer solution? If exist, how to prove?
So there are techniques different from Euclid's Algorithm that determine the solutions of the Linear Diophantine equation. I have published a note before (13 years ) that shows a closed formula for a general solution of ax+by = c. See the attached note.
Best regards
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I've seen incorrect proofs for some cases of m, and I've seen it claimed to be proven by Tout, Dabboucy, and Howalla, but cannot access their paper.
Yes, the graph is prime. It is can be shown easily. There is a conjecture due to Seoud and Youssef in 1999 (in Congrussus Numerantium ) states that every unicyclic graph is prime
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Question posted on August 12, 2018
We consider the product P of the first N primes.
We split this product into two factors A and B
so that P = AB and A > B.
We consider the distance d = A – B between A and B.
What is the splitting that gives the smallest distance d?
Is there a pattern?
in the following document:
Minimum distance from splitting the product of the first N primes
With best regards, Jean-Claude
Onto Jean_Claude & Romeo thread
• Let $P_n$ denote the set of the first $n$ odd primes. Thus, $P_n={3, 5...p_n}$.
• Let $C$ be the ceiling of the square root of the product of the elements of $P_n$. That is. $C=ceil(sqrt(3.5....p_n))$.
• Choose $B$ to represent the largest product less than $C$ of distinct primes in $P_n$. It is clear that $B$ exists.
• Set $A=(3.5.....p_n)/B$.
Lemma 1: $A$ is not a prime for n>2. (proof: this is straightforward.)
Lemma 2: $A$ is the smallest product greater than $C$ of distinct primes in $P_n$. (proof: by contradiction is also simple).
The 3rd dot and Lemma 2 imply that $A-B$ is minimal.
If Romeo conjecture holds, then the partition above will satisfy it.
Although the argument is based on odd primes, the partition should hold if the prime $2$ is included.
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Is there a difference between pure and applied mathematics?
In Wikipédia, we can find the following definition :
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers. But Number Theory is mostly applided, for example, in modern Data Encryption Techniques (cryptography).
So, Is there really a difference between pure and applied mathematics?
The division of “Mathematics” into two “kinds” is fictitious and misleading. A particular mathematical technique may turn out to be useful in practical situations and, conversely, the application of mathematics to the solution of a practical problem may lead to new mathematical discoveries. In either case, the adjectives “pure” and “applied” are not descriptors of the technique itself, but to its context.
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My question 18:
How to find central pairs of complementary divisors of n?
Posted on September 10, 2018
======================
Because of difficulties of typing and formatting the description of my question here, I have posted a description in the following document:
Methods to find central pairs of complementary divisors of n
My questions are:
1. What are the elementary methods that work in special cases.
2. Have my methods already been published?
With best regards, Jean-Claude
I have tried to find solutions using the integer part of the square root (r), of the number n=AB for examble for 66, r=8 for 77, r=8 for 105, r=10 etc. So, if we say that S= 4r+3+N and N is an even number not smaller than -2 e.g. -2, 0, 2, 4, 6, etc then A and B can be given by the equations:
[S-sqrt(S^2 - 16n)]/2 and
[S+sqrt(S^2 - 16n)]/8
For examble for n=77=8^2+13 if we put 4 for N we have S=4.8+3+4=39 and
(S^2 - 16n) = 17^2, so (39-17)/2 = 11 and (39+17)/8 = 7
A possible research on the item should specify the parameters of N, e.g. for n=77 N=4 in terms of 13.
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Question posted on August 22, 2018
A primorial is a product of the first N primes.
For example, the third primorial is the product
of the first three primes, 2, 3, and 5.
So, 30 is the third primorial.
The first primorials are 2, 6, 30, 210, 2310, 30030, 510510, …
My question is: What are the pairs (n, n + 1)
of consecutive positive integers
whose product n(n + 1) is a primorial?
When n = 1, we get that n(n + 1) = 2 is the first primorial.
When n = 2, we get that n(n + 1) = 6 is the second primorial.
When n = 5, we get that n(n + 1) = 30 is the third primorial.
When n = 14, we get that n(n + 1) = 210 is the fourth primorial.
When n = 714, we get that n(n + 1) = 510,510 is the seventh primorial,
that is, 510,510 is the product of the first
seven primes 2, 3, 5, 7, 11, 13, and 17.
Is there another one?
Since a primorial is a product of distinct primes,
it does not have any repeated factors.
In other words, primorials are square free.
Therefore, n(n + 1) must be square free.
Consequently, both n and n + 1 must be square free.
Therefore, n and n + 1 must lie strictly between multiples of 4,
strictly between multiples of 9, 25, 49, 121, 169, ….
Of course, being square free is necessary,
but far to be sufficient to give a primorial of the form n(n + 1).
A way to find n for a given primorial P
is to split P into a product of two equal factors
in the set of real numbers (that are not necessarily integers):
P = (SQRT(P)) times (SQRT(P)).
Then n is the integer just below SQRT(P)
and n + 1 is the integer just above SQRT(P),
that is
n = FLOOR[SQRT(P)],
n + 1 = CEILING[SQRT(P)].
For example, the fifth primorial is 2310.
SQRT(2310) = 48.062…
We try n = FLOOR(48.062…) = 48,
n + 1 = 48 + 1 = 49.
We observe that 48 and 49 are not square free.
So, we already know that it will not work.
We check: 48 times 49 = 2352 > 2310.
Indeed, it does not work.
We have 2310 = 42 times 55,
but 42 and 55 are not consecutive integers.
Note that positive integers of the form n(n + 1)
are at the same time the most square
and the most un-square integers.
They are the most square, because nobody can see
the difference between a rectangle with sides
of length 1000 and 1001, compared with a square
with sides of length 1000.
They are the most un-square, because they are
so allergic to squares that they always lie as far
as possible from squares: Integers of the form n(n + 1)
always lie almost exactly in the middle between
two consecutive squares.
For example, 5 times 6 = 30 lies almost exactly
in the middle between 25 and 36
Related questions:
What is the splitting P = AB of the product P of the first N primes
that gives the smallest distance between A and B?
Is there a published improvement of Euclid’s Theorem?
Related documents:
Minimum distance from splitting the product of the first N primes
Splitting in Euclid's proof to find the next prime
With best regards, Jean-Claude
Dear Jean,
Test for any potential solution for the product of 2 consecutive integers is Promorial:
Theorem: P1P2P3.....Pn = m(m+1) if and only if
4P1P2P3.....Pn +1 is a complete square .
where P1 ,P2 , P3.....Pn are consecutive primes , m is an integer.
Examples:
4(2)+1 = 32, Done
4(2x3)+1=52, Done
4(2x3x5) +1=(11)2 ,Done
4(2x3x5x7) +1=(29)2 . Done 2x3x5x7=14 x 15 Done
4(2x3x5x7x11) +1= 9241 not complete square, Failed.
4(2x3x5x7x11x13) +1=120121 not complete square, Failed.
4(2x3x5x7x11x13x17) +1=(1429)2 Done ,2x3x5x7x11x13x17=714x715.
The proof of the theorem is direct .
You can construct a simple program to detect the next solutions.
Best regards
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The answer may take the form of a comment or a full paper submitted here
Dear Michel,
Your question is about symmetry in general.
Concerning factorization and prime testing, RSA provides an amazing
algorithm but nowadays the use of ECC is more efficient for cryptography purposes.
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For n>=6 there are some random intervals with [n/3] - 1 many perfect squares between two consecutive Fibonacci numbers. Can anyone give some idea how the number of ?
Is there any formula for number of squares between two consecutive Fibonacci numbers?
@ Tarit,
Yes, there are many closed formulas for Fn such as Binet's closed formula also I have two papers showing new closed formulas using matrix approach.
Wish you good luck.
See the following papers:
(1) On a new improved unifying closed formula for all Fibonacci-type sequences and some applications, Journal Of Number Theory 182(2018)271–283, Science Direct, ELSEVIER, https://doi.org/10.1016/j.jnt.2017.06.014.
(2) I. Kaddoura, B. Mourad, On a new closed formula for the solution of second order linear difference equations and applications, preprint, arXiv:1707.00179, 2017.
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There are a number of such theories (Freud and Jung to name early ones) that sprung originally from German and Austrian universities, some concerned with understanding the mind others with altering personality. To your mind, which ones are the most persuasive?
Since you're limiting the question to Germans and Austrians, my vote is for the three founders of Gestalt psychology: Max Wertheimer, Kurt Koffka, and Wolfgang Köhler. As for psychoanalytical theories, my vote goes to Viktor Frankl's existential psychology.
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Is it a variant of Vandermonde convolution formula for falling factorials? What is the answer? See the picture.
See the picture uploaded here.
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People usually say that the number greater than any assignable quantity is infinity and probably same in the case of -ve ∞.
We are dealing with infinity ∞ in our mathematical or statistical calculations, sometimes we assume, sometimes we come up with it. But whats the physical significance of infinity.
Or
Anyone with some philosophical comments?
Mathematicians have a precise definition of infinity that can be used to prove theorems about it. A set S has infinite size if it is possible to create a 1-to-1 correspondence of the elements of S with the elements of a proper subset of S. For example, the positive integers S = {1,2,3,...} is an infinite set because there is a 1-to-1 correspondence between S and the even integers in S: 1 <-> 2, 2 <-> 4, 3 <-> 6, 4 <-> 8, ... Such a 1-to-1 correspondence is impossible for finite sets.
• asked a question related to Number Theory
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The number 1 is neither prime nor composite. why?
@peter Kepp
You said
[There are no units -i and +i ] and
[The square root of minus one is solved by me. It´s the reform of mathematics.]
What? Are you sure?
Do you know what Integral domains are? Euclidean domains?Unique factorization domains? Modular arithmetic? Elliptic curves? Gaussian Integers? Units? unitary divisors? Generalizations of Euler function?
P-adic integers? Cyclomatic integers?? Unique factorization domains?
Do you hear the proof of last Fermat's theorem? Etc.
Let me tell you something about the Euclidean domain of Gaussian integers:
Z[i]={ a + ib, a,b in Z }
The nunits are 1 , -1 , i , -i .
Primes in Z are not necessary primes in Z[i]
5 is prime in Z but 5 = (1-2i)(1+2i) is not prime in Z[i]
7 is a prime in Z and it is prime in Z[i].
I invite you to new horizons in analytic number theory.
Any introductory textbook in this field can help.
Best wishes
• asked a question related to Number Theory
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The emergence of new infinite system (new 'infinite' idea, new number system and new limit theory) determines the production of "new mathematical analysis". The mathematical analysis based on the classical infinite system is called "classical mathematical analysis", and the mathematical analysis based on the new infinite system is called "new mathematical analysis (the fourth generation of mathematical analysis)".
Three major differences between new and classical mathematical analysis in treating "X ----> 0 infinite mathematical things"
1 Different "infinite ideas" and different "treating objects"
In the foundation of classical mathematical analysis, "infinite" is composed by two concepts of "potential infinite and actual infinite" which can not be well defined and are inevitably confused. So, it is impossible to know clearly what on earth those "X----> 0 infinite mathematical things" are for people to treat (numbers or non-numbers? potential infinite mathematical things or actual infinite mathematical things? potential infinite numbers or actual infinite numbers? ...?)------they are non-number "variables" that are forever in change.
In new mathematical analysis, "infinite" is composed by two concepts of "abstract infinite law" and "the carriers of 'abstract infinite law'", which can be well defined and cannot be confused. According to the new number system, all of those "X----> 0 infinite mathematical things" for people to treat are clearly known as "the mathematical carriers of 'abstract infinite law'" ------they are numbers.
2 Different "infinite number theories and infinite number forms"
In classical mathematical analysis, people can not get rid of the confusion and bondage of "potential infinite number and actual infinite number" and have had to define all the "infinite numbers" participating in any quantitative calculations as "non-number variables of both potential infinite number and actual infinite number that are forever in change" mixed by the "potential infinite number and actual infinite number" derived by the confused "potential infinite and actual infinite" concepts. But, there is never such "variables" in existing classical number system at all; so, whenever the "infinite numbers" are calculated mistakenly as "potential infinite numbers in forever change", paradoxes have been inevitably produced (as a members of zeno's paradox family, harmonic series paradox is the most typical case).
In new mathematical analysis, all the "infinite numbers" participating in any quantitative calculations are "mathematical carriers of abstract infinite law" with characteristics of number sense representing the existence of abstract infinite law, which are clearly defined as numbers with quantitative nature and within the new number system. And, "abstract infinite law in forever change" will never be taken as number in any quantitative calculations. In new number system, those "X----> 0 infinite mathematical things" belong to “inter--number” and is called “intersimal”
3 Different processing theory and operations to "infinite mathematical things"
In classical mathematical analysis, the defects of existing classic number system unable people to understand the nature of "variables", making its quantitative cognitive theory of "infinite mathematical things" can be nothing to do with existing classical number system and nothing to do with the specific quantitative properties of "infinite mathematical things". So, theoraticaly or operationaly, it is impossible to established essential rules of "mathematical analysis" to know exactly what "X----> 0 infinite mathematical things" being processed are and what quantitative properties they have (because there is no such number form in the existing classic number system), some kinds of formal languages and pipeline operations are ok (such as the completely equivalent languages and pipeline operations in three generations of classical mathematical analysis: "let infinitesimal be 0" in pre--standard analysis, "take the limit" in standard analysis and "take the standard number" in non-standard analysis) [1]. In many situations, some unified pipelining operation theories can be used for the processing of all the "infinite mathematical things" (the most typical situation is: whether in calculus or in harmonic series, all the "X----> 0 infinite mathematical things" are with exactly the same quantitative nature without any differences). And such pipeline operation theories, processes and results in the "potential infinite--actual infinite" based three generations of classical mathematical analysis (pre--standard analysis, standard analysis, non-standard analysis) are completely equivalent, inevitably producing exactly the same "infinitesimal--infinity quantitative cognition paradox families".
In new mathematical analysis, the quantitative cognitive theory of "infinite mathematical things" is a targeted operation theory closely related to the new number system as well as the specific quantitative properties of "infinite mathematical cariers". Before calculating, all the "X----> 0 infinite mathematical things" participating in any quantitative calculations must be truly "mathematical analyzed" according to the "theory of infinite law carrier" to know what quantitative meanings and properties they have so as to perform the targeted operations to them. And, such targeted operation theories, processes and results are nonequivalent to any of the three generations of classical mathematical analysis in many situations--------it is impossible to have those suspended "infinitesimal--infinity related paradox families"