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Algebraic Number Theory - Science topic
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Updated information of my thoughts and activities.
This is meant to be a one-way blog, albeit you can contribute with your recommendations and comments.
This question is now closed, reaching a YES conclusion and preventing bering hijacked by 'wolves" in ResearchGate. It serves the purpose of explanation to those interested, as an open group, from a core of 10 people who participate behind the scenes. Enjoy and pursue the new ideas with your contributions in your space.
To be objective, K=26 661 462 837 357 923.
This large base 10 number is found to be the product of two prime numbers, K=p×q.This breaks the RSA cybersecurity method for K.
The much larger number H=74 481 443.869 551 262 986 707 503 438 165 513 011 429 940 762 703 277 812 267 530 769 921 052 121 342 275 484 565 273 568 067 051 66*10^991 with the missing integer values known, albeit not shown here, reveals quantum properties of numbers, that help break RSA structurally, making it impossible to protect.
That number is found quickly to be the product of two very large prime numbers, H=m×n, where m=2189435657951002049032270810436111915218750169457857275418378508356311569473822406785779581304570826199205758922472595366415651620520158737919845877408325291052446903888118841237643411919510455053466586162432719401971139098455367272785370993456298555867193697740700037004307837589974206767840169672078462806292290321071616698672605489884455142571939854994489395944960640451323621402659861930732493697704776060676806701764916694030348199618814556251955925669188308255149429475965372748456246288242345265977897377408964665539924359287862125159674832209760295056966999272846705637471375330192483135870761254126834158601294475660114554207495899525635430682886346310849656506827715529962567908452357025521 and n= 340185579782030309029142285845485748073406778702270938755484147318382420338087834406828955714187005654640257038495796545155402280055987076251704557994637589726712709889312042801858044039590155407650471667907995888292123909278046563998441725881316702608454953284969473141146885140822683049274853701491, breaking RSA with values naively considered large enough to be "safe".
We postulate without proof here, except numerical, that this exemplifies how RSA can be quickly broken, e.g., for a 2048 bit-length number. in 2048 bits one can store a number with 617 decimal digits; and we passed that in the last example. The larger the number of digits in each prime number, the easier it is to numerically calculate them.
RSA gets weaker with large prime numbers. This is a structural weakness, much more important for cybersecurity than numerically finding prime numbers.
This shows objectively the weakness of RSA. QM is our most successful model of nature. Classically, i.e., without QM, those results are not calculable and RSA looks stronger for large numbers.
RSA seems to be broken easily by quantum computing -- more so for very large numbers. It is a hopeless case using QM, and quantum computing.
This shows the importance of periodic structures in mathematics. And we can find them using QM, and quantum computing.
What is your opinion?
Our answer is NO. Think of it: Pythagorean Triples would NOT exist if numbers are arbitrary as values. Given a and b, c is fixed or it doesn't exist.
Given 2 and 3, what is c?
Prime numbers seem not arbitrary either. Some people consider prime numbers as just some feature of Z, which does not exist for composite numbers. And, they think, there are no primes or coprimes in Z_p, p-adic numbers, except for some numbers, which end by 0; there are no negative and positive numbers; there are not even or odd numbers (I e., they may point to the number 19 underscore 31, is it even or odd?).
Instead, let's be humble and observe nature. A prime number in any place of the universe must be a prime number. Here on Earth and in the star Betelgeuse. It is not a feature defined by a human.
Dedekind (1888) was incorrect, and mathematical real-numbers an illusion, that cannot be calculated (Gisin, Gerck).
That is why a number is a semiotic quantity. Numbers can be thought of as a 1:1 mapping between a symbol and a value. Digits become a “name”, a reference, and it is clear that one can use different “names” for the same number as a value.
So, the number 1 can have a name as "1", "2/2", "3/3" and infinitely many more, but is always 1 in value.
Equality of rational numbers does not have to have the same name for each other, as "2/3=2/3".
They can also obey the rule that their cross product is equal in value, so that "2/3=4/6".
That way, equivalence extends equality in a consistent way, even though the numbers are neither equal nor divisible. This is possible because numbers are semiotic quantities, and is essential to understand quantum computing.
Numbers are not arbitrary as values, which can allow us to calculate prime numbers using periodicity.
What is your qualified opinion?
It finally has occurred to me that there is a similarity between i = √-1 and √2. They are each linearized representations of essentially quadratic values. We use the former in complex numbers and include the latter in the real number system as an irrational number. Each has proved valuable and is part of accepted mathematics. However, an irrational number does not exist as a linear value because it is indeterminate – that is what non-ending, non-repeating decimal number means: it never can exist. Perhaps we need an irrational number system as well as a complex number system to be rigorous.
The sense of this observation is that some values are essentially quadratic. An example is the Schrödinger Equation which enables use of a linearized version of a particle wave function to calculate the probability of some future particle position, but only after multiplying the result by its complex conjugate to produce a real value. Complex number space is used to derive the result which must be made real to be real, i.e., a fundamentally quadratic value has been calculated using a linearized representation of it in complex number space.
Were we to consider √-1 and √2 as similarly non-rational we may find a companion space with √2 scaling to join the complex number space with √-1 scaling along a normal axis. For example, Development of the algebraic numbers a + b√2 could include coordinate points with a stretched normal axis (Harris Hancock, Foundations of the Theory of Algebraic Numbers).
A three-space with Rational – Irrational – Imaginary axes would clarify that linearization requires a closing operation to restore the result to the Rational number axis, where reality resides.
[Note: most people do not think like I do, and almost everyone is happy about that: please read openly, exploringly, as if there might be something here. (Yes, my request is based on experience!) Tens of thousands of pages in physics and mathematics literature from popular exposition to journal article lie behind this inquiry, should you wish to consider that.]
The formula for sin(a)sin(b) is a very well know highschool formula. But is there a more general version for the product of m sine function?
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.
In general in mathematics we work with groups that includ numbers, generally we work with constants.
So for exempel: For any x from R we can choose any element from R we will find it a constant, the same thing for any complex number, for any z from C we will find it a constant.
My question say: Can we find a group or groups that includ variabels not constants for exemple, we named G a group for any x from G we will find an x' it's not a constant it's another variabel ?
and if it exisit can you give me exempels ?
Thank you !
Many proposals for solving RH have been suggested, but has it been splved? What do you
What lessons and topics are prerequisites for algebraic number theory and analytic number theory?
Please tell me the exact topic of each lesson.
In the 19 hundreds, many though that an explicit formula for the partition function was never going to be found. In 2011, finally, an explicit formula for the partition function was discovered.
For that reason, I am fascinated by how close do mathematicians think we are currently to discovering an explicit formula for prime numbers.
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
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?
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?
Does Hilbert-Polya Conjecture Infer Riemann Hypothesis by the means of spectral theory? Can anyone describe the procedure
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?
- We know ,euclidean algorthm is feasible in the set of integers.
- Taking motivation from this,we define an Euclidean Domain(E.D.) as follows:
R is an E.D. if it is a domain,andwe have a map d:R->non-zero integers,
such that, for given a,b(non-zero),
there exists q,r with
a=bq+r and d(q)<d(r).
- Now,this d function ,for any abstract ED,is the counterpart of |.| function,in case of intgers,
we have ,a=bq+r with,0 \le r < |b|.
- Now the question is:
[] It turns out (q,r) that exists for a given (a,b) in case of intgers ,is unique .
[A simple proof would be:if another (q',r') exists for same (a,b),
if q'=q,we have r'=r,and we are done.
and if,q' and q are distinct,bq+r=bq'+r' implies r' and r differ by multiplies of b,thus if 0 \le r < |b| holds,it is clear it won't hold for r',and thus we can never have (q',r') and (q,r) distinct,and we are done.]
[] But,one now would ask,is it true as well that for given (a,b) in any E.D. the existing (q,r) would be unique ? If yes,we need a proof,and clearly,the same proof does not work,as d(.) is much more generalized than |.| .Or,we need a counter eg!
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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:
Does anyone know of applications of multiple sums of a sequence?
I know of the Multiple Zeta values (which is a multiple sum of 1/N^s). This has multiple applications in quantum physics, QED, QCD, connection between knot theory and quantum physics, ...
Does anyone know of potential applications for this more general form which is a general multiple sums? I have written an article about it and about its applications including partition identities, polynomial identities. I wanted to know if anyone know of applications outside mathematics or additional applications in math.
Dears, please help me.
Could you prove that for any big number N there exists a composite Fermat number F_k such
that F_k > N?
I am a bachelor's student in mathematics.
I have a fairly large matrix (250*250) in symbolic form in MATLAB. Matrix is square invertible with size multiple of 2. I have to calculate the first two elements of the first two rows of its inverse (i.e. first 2x2 block). Matrix is very large and in symbolic form ,so, Matlab is not able to calculate the whole inverse due to time limitations. I have tried guassian elimination, LU factorization, block wise inverse technique. I have also tried the simple method of cofactors and determinent. In all the cases the problem is the same: very long time in the range of hours. Can anyone suggest some technique?
We fix a number field K=Q(a) of degree 4 that is a CM-field. For an order O, if I(O) is the monoid of fractional ideals of O and P(O) is the subgroup of principal fractional ideals, we define the Ideal Class Monoid ICM(O) as the quotient I(O)/P(O).
Given two orders O\subset O', I define the ratio r(O,O')=#ICM(O')/#ICM(O).
Set R=Z[a,\bar{a}], where \bar{a} is the complex conjugate of a, and R'=R[S(a)/p], where S is a polynomia with integer coefficients and p a prime number.
I am interested about the value of r(R,R'), or if we have some inequality as for example r(R,R')=<1/p. Do you have some ideas?
Obs: there is a bijection between ICM(Z[a]) and the conjugacy classes of matrices M_4(Z) with characteristic polynomial the minimal polynomial of a, but I dont know how that can be useful.
Thanks in advance.
Is there an algorithm to find the Lengths of period of continued fraction sqrt((an)^2+4a)) whith a,n are odd integer.And can we find the continued fraction of the previous number.
is ther any solution of the diophantine equation :
x^2-((a*n)^2+4*a)*(y^2) =4*m,-4*m
more spesific : is it true that the equation have integer solution then m>a ?
I have solved many quadratic,cubic, biquadratic, quintic, sextic, heptic and mth degree diophantine equations. I wish to know about the applications in real life as well as in other fields.
A subexponential algorithm for elliptic curves over F_{2}^{n} ?
The main question is whether or not the strategies described in the elliptic curve discrete logarithm can lead to a general subexponential discrete logarithm algorithm for elliptic curves in characteristic 2.
If we are in unison inre to the aforementioned ,then the cosmic speed is one-third the speed of light for particles of masses greater than 1kg. Please verify my publications in regard to the same.
Please can someone give me a reference on the distribution of Pisot numbers on the real line.
In imaginary quadratic fields we have:
* ELL(O_K) : = {elliptic curves E/C with End(E) ∼= O_K}/{isomorphism over C}
∼= {lattices L with End(L) ∼= OK}/{homothety}∼=ideal class group CL(K)
* #CL(K)=#ELL(O_K)
this notation at the papper :A SUMMARY OF THE CM THEORY OF ELLIPTIC CURVES
JAYCE GETZ
Since it is difficult to write mathematical formulae please consider the attached file.
Factoriangular numbers (denoted by Ft_{n}) are formed by adding corresponding factorials and triangular numbers, that is Ft_{n} = n! + T_{n}, where T_{n} is the nth triangular number. The closed form of the exponential generating function of the sequence of factoriangular numbers can be easily derived from the exponential generating functions of n! and of T_{n} (see the link below). How about the closed form of the ordinary generating function of such sequence?
We define factoriangular numbers as sum of corresponding factorials and triangular numbers, that is, n! + n(n+1)/2, n is a natural number. We listed the first few factoriangular numbers: 2, 5, 12, 34, 135, 741, 5068, 40356, 362925, 3628855, 39916866, 479001678, 6227020891, 87178291305, 1307674368120 and notice that each is either a deficient or an abundant number. Is there any such number that is a perfect number? Or, how can we prove that no factoriangular number is a perfect number?
I have been using DLS to characterize my Pt nanoparticles as well as AFM and TEM, my AFM and TEM results match up but the intensity distribution doesn't that's why I'm using the number distribution but I would like to know how does the software do the conversion.
Thanks
Digital root as successive sum of the digit of a positive integer.
Where does the digital root came from?
Is there bibliography about this history of mathematics?
Dear searchers,
The set of all algebraic numbers is countable. I wish if someone can help by giving the smallest gap between algebraic numbers in general, or between the conjugates in particular, or if that gap depends on the degree or the wight or the height of the minimal polynomial of them.
Recall that the conjugates are the roots (zeros) of a minimal polynomial
Is a bracket-preserving map between Lie algebras necessarily Linear? Usually, it is assumed to be linear in advance and the second condition of bracket-preservation males it a Lie homomorphism, by definition.
For the degree 2 or three with the third conjugate positive it is easy.
key words: Pisot number, strong Pisot number, minimal polynomial.
Let a and b be two reals such that -b<a<0<b<1, does a+b+ab always positive?
How to find the total number of normal subgroups for symmetric group S_{n }and A_{n} and is there any relation between the normal subgroups of two?
how can i prove the following statement? In other words, how can i prove existence of the following norm?
Let K be any totally disconnected local field. Then there is an integer q=p^{r}, where p is a fixed prime element of K and r is a positive integer, and a norm ∣⋅∣ on K such that for all x∈K we have ∣x∣≥0 and for each x∈K other than 0, we get ∣x∣=q^{k} for some integer k.
An infinite-countable set of infinite-tuples of 0 and 1 is countable or uncountable?
Examples:
(0,1,0,0,........)
(0,1,1,0,1,0,0.....)
Note
The number of 1 is finite for each tuple.
Since every reduced binary quadratic form with discriminant d corresponds to one class of ideals in the quadratic fields Q(d^(1/2))with d>0 , Is there an effective way to determine all reduced binary quadratic form with discriminant d or at least can we determine the number of reduced binary quadratic form with discriminant d ?
How can I calculate the radius of convergence of Pulita pi-exponential?
I am looking for the properties of Jacobson Radical of modules over noncommutative rings.
I want to determine the number of points on E in terms of æ and ß. For example:
For Koblitz curves one can compute this cardinality using a Lucas sequence but I am working in Elliptic curves over F_{2^{n}} (not necessary Koblitz curves) given by the equation Y^{2} + XY = X^{3} + æX^{2} + ß where æ and ß are elements in F_{2^{n}} and Tr(æ) = 1.
Does Gaussian include group theory? Are there any limitations?
In 1850 P.L. Chebyshev and in 1932 P. Erdos proved that [n,2n] interval contains a prime. Bachraoui proved the case k = 2 in 2006. k=3 case proved by Andy Loo in 2011.
From the other hand Legendre’s Conjecture states that for any n, there is a prime in [n^{2}, (n+1)^{2}]. For instance between 13^{2} (=169) and 14^{2} (=196) there are five primes (173, 179, 181, 191, and 193).
This interests me and I formulated two different questions. I hope that the problems are interesting for someone else
- whether there are k and n such that [kn,(k+1)n] contains odd number of primes, while [(k+1)n,(k+2)n] contains even number of primes with interval [(k+2)n,(k+3)n] having odd number of primes?
- whether there are A, B and n such that [A^{n},B^{n}] contains odd number of primes, while [A^{n+1},B^{n+1}] contains even number of primes and at the same time intervals [(A+1)^{n}, (B+1)^{n}] and [(A+1)^{n+1}, (B+1)^{n+1}] have even and odd number of primes, respectively?
I have a set of data from my experiment (120 instances with 1458 attributes), but these data is in complex numbers.
How can I represent a complex number into another real number? I found that complex number can be represented by using polar coordinates (r and theta), however what I need is a single number, so that I can further process the data using Weka.
Alternatively, is there any equation that can take 2 numbers and produce a single unique number?
Thank you very much in advance.
Is there any closed form expression for the infinite sum $\sum_{n \geq 0}q^{n(n+1)/2}(1+q)(1+q^2)\cdots(1+q^n)u^n$ where both $q$ and $n$ are variables and $n \in N \cup {0}$?
I am looking for a reference giving the decomposition of a prime p in the maximal real subfield of a cyclotomic number field Q(zeta_m). Something along these lines: If p is a prime not dividing m, then let f be the smallest positive integer such that p to the f is congruent to 1 or -1 modulo m. Write f.g=phi(m)/2. Then (p) splits into g different prime ideals of norm f (plus what happens if p divides m).
I´m currently working on my master's degree thesis and I would like to know some interesting topics on this subject.
What is the method to decide if a given number (integer) is going to have an integer square-root or not (without actually computing the square-root)?
For example, assume I have a method M that does this. Then it should behave like below:
M(16) should return true (since sq-root(16)==4 which is integer)
M(17) should return false (since sq-root(17) is not an integer)
and M should not actually compute the square-root to decide this.
Any literature or info of this?
I am looking for a complete reference on introduction and basic definitions on Selmer group and Shafarevich-Tate group. Any suggestions would be welcomed.
P ≡ 1 (mod 48),
2^(p-1)/4 ≡ −1 (mod p), and
6^(p-1)/4 ≡ 1 (mod p).
I need to know if there is such that relation in the case of real quadratic fields.
http://vixra.org/abs/1207.0071 or see the presentation on YouTube http://www.youtube.com/playlist?list=PL_G0M-Rz92yIIRK9O5ijea3vmLA31m8HE
I had began by examining multiplication modulo n with n being a primorial. The idea was to find a tool to study the differences between prime numbers. To my surprise I found I could prove the twin prime conjecture and even see how to generalize it further. This is because I could show there exists a linerar algebra of Euler products that can count gap patterns in each set. I found why the Hardy Littlewood constant needs to be multiplied by 2 and I found better, more natural formulas for gap pairs. In fact I can now describe all gap behaviors from 2-28 exactly.
The next step, the subtle step, was I needed to extend the PNT. I have to assume a single concept, the PNT operates everywhere with the log at P being the mean of a set and for each P there is a new set of possibilities and a new mean. But it must be random everywhere with a local mean.
From there RH just "fell out", and the result is more surprising than expected. In fact it took me a few months to reconcile Von Koch's famous "if RH than proof" because the answer was somewhat tighter than he found. I show by construction the worst possible error between the log integral and the prime counting function at any x is strictly less than the log integral at root x. That is, the actual worst error is (ln(P))^-3/2 smaller than Von Koch's ln(P) root (P).
It's 30 pages of what I hope is some of the most interesting math you'll read this week.
I'm confident it's correct because the paper is not a proof per se but a framework that proves many conjectures, not just RH. Also, over the last 2 years I have about 30 different programs I wrote amounting to about 100k lines of code that all support this work. I hope every idea has been tested at least a few different ways.
So I ask the local community here to please look at my paper or watch the presentation and if possible find the mistakes or validate the ideas.
This is also actually a test of crowd sourcing as I submitted the paper for peer review 24h ago. How fast can this crowd settle the validity of the paper against 3 editors. The editors promise an answer in 12 weeks, can RG crowd sourcing improve on this?
My question is: Is this paper published in a journal?