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Even if ECFM has demonstrated effective methods of resolution in some contexts, its completeness and improvement of performance context are still lacking relative to the scope of literature, particularly when it comes to comparison with newer or hybrid factorization algorithms. In addition, the effect of elliptic curves factorization on the security of certain cryptographic protocols remains to be established, especially as technology continues to undergo rapid changes.
The research is to be conducted in order to provide an in depth analysis of the working of ECFM, its mathematical parameters, its working efficiency and its implications for the cryptographic security, more specifically the following questions shall be raised in this context:
What are the general implications of ECFM usage in performance of other for the state of the art factorization algorithms with respect to diverse computational environments?
What are the merits and the possible demerits of using ECFM in any given scenario where cryptography is exercised?
What are the suggested methods of performance improvement and how can ECFM be modified in relation to other standard factorization performance tasks?
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dear Sureshkumar Somayajula 1. General Implications of ECFM Usage
Performance Relative to State-of-the-Art Algorithms:
  • Computational Efficiency: ECFM has been recognized for its efficiency in factoring large integers compared to classical methods like the quadratic sieve and general number field sieve, particularly for certain types of numbers. The implications of this efficiency can benefit fields that require rapid factorization, such as cryptanalysis or integer-based cryptographic algorithms.
  • Parallelism and Scalability: Considering diverse computational environments, ECFM can be parallelized effectively, which allows adaptation to multi-core and distributed computing setups, enhancing its performance significantly.
  • Hybrid Approaches: ECFM can be integrated with other factorization techniques to form hybrid approaches, possibly leveraging strengths from both ECFM and newer algorithms, thus improving overall performance.
2. Merits and Possible Demerits of Using ECFM
Merits:
  • Efficient for Certain Classes of Numbers: ECFM excels when factoring numbers with small prime factors or those that are not excessively large, making it a useful tool in specific scenarios.
  • Lower Memory Consumption: ECFM often requires less memory than some other methods, which can be advantageous in resource-constrained environments.
  • Quantum Resistance: Compared to classical algorithms, ECFM may present certain advantages in contexts where quantum algorithms could impact security, especially since traditional factoring advantages may not apply similarly to elliptic curves.
Demerits:
  • Complexity and Implementation Barriers: While ECFM is efficient, its mathematical complexity may present challenges in implementation and understanding compared to simpler methods.
  • Limited Scalability Beyond Certain Thresholds: ECFM's advantages may diminish when dealing with extremely large integers, particularly if hybrid methods are not utilized.
  • Potential Vulnerabilities: The security implications regarding advancements in quantum computing and the advent of new mathematical breakthroughs could pose risks to cryptographic protocols reliant on ECFM.
3. Suggested Methods of Performance Improvement
Modification and Efficiency Enhancement:
  • Algorithm Optimization: Analyzing the current algorithm for bottlenecks and implementing optimized arithmetic operations can help improve the performance of ECFM.
  • Hybridization with Other Algorithms: Combining ECFM with other factorization methods (such as the general number field sieve for large integers) can lead to improved efficiency and versatility.
  • Adaptive Parameter Selection: Developing adaptive techniques for the selection of elliptic curves based on the specifics of the integer being factored could enhance performance.
  • Leveraging Parallel Processing: Enhancing the implementation of ECFM for multi-threaded or distributed computing environments, thus allowing simultaneous execution of multiple instances of the algorithm.
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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.
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Use (free PDF, also in print) the book QUICKEST CALCULUS with programmed instruction. Integral is immrdiate, just the inverse of differentiation plus a constant.
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This question has been closed, reaching a YES conclusion and preventing being hijacked by 'wolves" in ResearchGate. It served 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. The question is now open again.
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?
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As published in ResearchGate, here is the solution to break RSA-2048 and even higher, fast.
In the old method, in order to #factor one would need to inchworm, prime number by prime number, to reach 308 decimal-digits.
That would take some trillion years in computation. Impossible?
Now, QC reveals that prime numbers are #eigenvalues. Must obey the Schrödinger equation for bound-states. That is a diophantine equation -- solutions ARE only integers with arbitrary-length, as published in Phys. Rev. Lett. in 1982. That's valuable, including for structural dynamicists.
One can calculate any 308 decimal-digit prime number, as one already knows and does it, and then just add or subtract even numbers and test -- in a quick search one hits the jackpot!
We have found the only one prime number that factors RSA.
Done -- RSA-2048 has been broken, by the Prime Function Theorem (Gauss, +220 years ago -- see image -- from ). The world has to change.
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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.
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Stop Hallucination.
Stop the fake fan team.
Stop disturbing and spreading wrong math education.
Show your idea in your thread and stay there.
I have hundreds of students and university colleagues worldwide, and I feel shame asking them to grant a fake recommendation that votes for wrong, bad answers.
I dare any one of your team to join a serious debate in mathematics
about FLT and its consequences.
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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.
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Dear colleagues,
I trust you are fine. As far as I know, new managing editors and editors-in-chief for Ars Combinatoria have been chosen. I remember, the last time I saw their website, they had written they would create a new website on which the papers would be appear (not only their titles with the names of authors as before). You can ask Dr. J.L. Alston, the former managing editor of Ars Combinatoria about this.
Regards,
Babak
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Recently I am doing a project on Integer factorization, but sometimes the terminologies are out of my knowledge. If anyone helps me by giving this code, It will be very helpful for me.
Thanks in advance.
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Also please have a look the very efficient Pollard P-1 algorithm implemented in PARI/GP. Because it is a "one line" algorithm it is straightforward and very easy to implement in MATLAB. You can almost use the same commands.
Effective , fast Pollard P-1 in PARI/GP:
A very good example of factoring N=(2^1003)-1. Using the built-in factorization method in PARI/GP takes a while to factor this number, however using Pollard P-1 gives the result immediately:
N=2^1003-1; x=3;s=1;while(gcd(x-1,N)==1,s=s+1;x=lift(Mod(x,N)^s));print(gcd(x-1,N))
Result: 179951
Regards,
Norbert
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Is it possible to use heuristic methods?
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Agreed with dear Neil J Calkin
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How to compute three sums involving binomial coefficients (combinatorial numbers)? See pictures with this question. Thank a lot.
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The following formally published papers are related to this question:
[1] Feng Qi, Qing Zou, and Bai-Ni Guo, The inverse of a triangular matrix and several identities of the Catalan numbers, Applicable Analysis and Discrete Mathematics 13 (2019), no. 2, 518--541; available online at https://doi.org/10.2298/AADM190118018Q
[2] 祁锋,一个三角矩阵之逆与Catalan数恒等式, 湖南理工学院学报(自然科学版),2020年第33卷第2期,第1页至第11页转第22页;available online at https://doi.org/10.16740/j.cnki.cn43-1421/n.2020.02.001
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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 ?
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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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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.
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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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Given the wide attention it has received from the math community, what are the practical uses of Fermat's Last Theorem?
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From reading about it, my understanding is that apart from its history, FLT in and of itself is not considered an important theorem by experts in the field. While there are some perhaps unimportant applications as detailed in https://brilliant.org/wiki/fermats-last-theorem/, by itself FLT is likely not a gateway to new and important mathematics. However, the mathematical techniques used to prove it and the general theorems for which FLT is a corollary, are considered important. In fact, even failed attempts to prove FLT have provided mathematics that have an abundance of applications. Ideal theory is an example.
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The famous Sylvester matrix was studied in 1854. It is defined as showed in the picture uploaded here. My question is: What and where is the inverse of the Sylvester matrix?
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Dear Dr. Cherdieu
I will read it as soon as I can and then reply you here
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I have the following graph with 2 different parameters called p and t. 
Their relationship is experimentally found. Manually by knowing (t,p), you can simply find the area number (group) of the point based on where it is located. For example, point M(t,p), locates in area 3 and belongs to group number 3. However, I would like to write a code/logical approach which automatically finds the group numbers. therefore when it reads (t,p) it will find the location of the point and give the group/Area number it belongs.
Is there any solution in Matlab for this scope? 
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Dear Bahareh, thank you very much for your reply. In fact, they are some sort of isopleths. I don't know if you are familiar with that. Therefore I have some tables containing the data points. 
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Dear colleagues,
What is special in 1109?
We can use straight lines as starting configurations in Game of Life. Then, if we check the equilibration time vs length, we will see that many lines with different lengths reach equilibrium exactly after 1109 steps. Why?
*More details in our project Mysteries of Game of Life
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Hello Rhoderick,
Thanks for your comment.
The question "What is special in 1109?" was just a way of saying. We decided to study lines as a first step to understand some facts of Life, and lines, despite being "simple starting configurations (the simplest possible?) " also present very interesting behaviour.
Now, the fact is that different lines (i.e, different values of L) reach equilibrium exactly at 1109 steps, and this special number is not the only one. the same thing happens for 1388 for example. So we can conjecture that there are infinite special numbers like these (when L goes to infinite).
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Intuitively I would say that the number of concepts is bounded by min(2|O| , 2|A|) where O and A are resp the objects and attributes sets. I get this (most likely wrong) intuition from the observaton that given X0, X1 included in O and the Y0 , Y1 included in A, for any pair of concepts (X0,Y0) and (X1,Y1) we have X0=X1 iff Y0=Y1. Thus there can not be more concepts than the number of object sets neither than the number of attribute sets appearing in the lattice. But the best upper bounds I find in some research papers are much more larger than the one I propose : for example 2|O|+|A|, or 2sqrt(|O|.|A|). And I really dont understand why... Could someone explain me where is my mistake ?
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Your upper bound min(2|O| , 2|A|) is correct for the reason you indicate: "there can not be more concepts than the number of object sets neither than the number of attribute sets." This upper bound is achieved when O = A and every element of O is related by the incidence relation to every other element except itself. In this case, (S, O\S) is a concept for every subset S of O and there are no other concepts. 
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Is anyone know where to get the copy of the original paper of Levinger theorem?
B. W. Levinger, An inequality for nonnegative matrices, Notices Amer. Math. Soc. 17:260 (1970).
The web site of Notices has only up to 1995.
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It seems that, in those days, the "papers" published by the Notices were just some kind of abstracts. They are not collected on the internet. If you really want to read it, you should look for it in an university library. Nevertheless, according to Fiedler in Numerical Range of Matrices and Levinger’s Theorem (Linear Algebra and its Applications 220, 1995), there is no proof of Levinger's theorem in the notice you are looking for; but you have one in the very same paper of Fiedler, which you can locate here:
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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.
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If a<0 then the period is apparently 12.
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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 ?
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class number  " h" :is the order of Ideal class group.
Ideal class group :In Z[sqrt(n)] the fractional ideal F is a group ,and the principal ideal P is a subgroup in it so we can take the quotient F/P.
real quadratic field: an extension of the rationals Q by the root of an irreducible quadratic equation over the rationals Q(sqrt((an)2+4a)).
the equation is :   x2- ((an)2+4a)y2 = 4m [or -4m]  with  :  a: odd integer ,n : odd integer.
m: integer .
the relation of "class number" and "real quadratic field" with the integer problem eich i put in question is: " h=2" then the equation represent a norm of ideal so we have an integer solution . now if I know when the equation have an integer solution i can determine the real quadratic fields
the solution wich i need "or wich I need the condition for exist it or not exist "should be (x,y) with " y" don't equal zero.
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Let us explore the number-theoretic question: suppose we have what we shall refer to as a Prime-Indexical Base Number System, such that the sum of any two primes is the next prime in the series of primes. So such as system would contain all and only prime numbers. For instance, 1+2=3, 3+2=5, 5+2=7; but 5+3=11, and 7+3=11 also. Furthermore, 11+13=29 and 29+31=61 ... and so on. Can we render a formula that would make such a Number System consistent?
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I don't have any proof for what i say but it feels right to me.
It is impossible to make prime indexial base number system which is consistent because there exists no closed form relation for primes i.e predicting  nth prime, So eventually we need to know the distance between the primes to make mechanism to form the system. hence my hypothesis.
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I am looking for a way in which I can add some cut-offs to the zeta function. By doing this, I want to plot this new function and look for the zeros. Analitically the expressions are very involved. Mathematica is a black box that shows me the properties of the zeta but doesn't allow me to modify the function. I am trying an implementation of zeta in C++ but I'm having some problems with the analytic continuation. What is the best way to deal numerically with special functions in the comlex plane?
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My logo is the three-dimensional graphics of the zeta function,where s=x+iy0,0<x<1.
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Can anyone prove the following equation, occurring while optimizing an n-body program?
floor(sqrt(2k) + 1/2) =?= floor(sqrt(2k-3/4)+1/2) for all k integer >= 1
floor is the integer part of the argument. It is false for some k real >=1
Extensive numerical checks seems to confirm it, except for some very large k the numerical check was finding the equation false, but this could be ascribed to round-off errors.
I would be grateful knowing what are the general methods that experts in number theory would use to prove this kind of equation.
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You finds a very simple solution of the N-body problem in my book (Chapter 4) available at Researchgate at my name. I use very simple Non Standard numbers theory to express the solution.
Sorry, but the book is in French language
Best Wishes
Thierry Bautier
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Assuming that the digits of π can be uniquely represented in the Zekendorf code; and
because the number of summands in the Zekendorf representation converges to a Gaussian as n → ∞ (please see http://arxiv.org/pdf/1008.3204.pdf & http://arxiv.org/pdf/1107.2718v1.pdf
Does it follow that:
1) π's digits are random?
2) π is not absolutely normal? (please see http://mathworld.wolfram.com/AbsolutelyNormal.html
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@Neil, @Stam
I guess that the proof is connected to question (4) in section 1.4 of  Miller & Wang's paper. 
"What can we say about the distribution of the largest gap between summands in
the Zeckendorf decomposition? Appropriately normalized, how does the distribution
of gaps between the summands behave? What is the distribution of the
largest gap? How often is there a gap of 2?"
If the gaps are distributed randomly I would argue that the Zekendorf code is random. 
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For normalized auto correlation, we normalizes the sequence so that the auto-correlations  at zero lag are identically 1.0.
So, I want to know how it will be in the case of cross correlations?
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Hello,
Assume you would like to calculate the normalised cross correlation of two sequences, x(n) and y(n), of length N. Then
Normalised_CrossCorr = 1/N * sum{ [x(n) - mean(x)]* [y(n) - mean(y)] }/ (sqrt(var(x)*var(y))
where
>the sum is taken from 1 till N.
> mean(x) is the mean of x
>var(x) is the variance of x.
>sqrt is square root
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I wrote a function, GetCumulativePrimeSpacing[i], in Mathematica to return the interference pattern created by all primes up to the ith prime. This sequence works for finding primes up to the next prime squared. I have obtained some interesting results.
Some interesting results are that the Length and Total of this sequence are related to the Primorial[i]. The count of 2s and 4s in the sequence is equal to the product of primes as well.
I have some ideas on what to do with these results but I'd like to hear thoughts from the experts first.
Any help would be appreciated.
Thank you.
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I wish i could understand it better. Am a physicist with a lifelong love for
mathematics, so i appreciate this wonderful area to work in.. Have read a lot
of stuff about euler and Ramanuthan, ,Riemann conjecture, etc. , so, hopefully
you will learn something here..Good luck! ken
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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.
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Dear Fajri Pratama
You look before the world mathematical community as the module on the complex plane with impotentny argument (forgive me for sad humour). Dear Fajri Pratama find to yourself the mistress or present to your girl or to the wife a box of chocolate truffles. Only then before you the world of natural and imaginary numbers, the main thing, in this question strong argument will open in all beauty.
Alexander
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What is the Indicator function for set membership of integers?
I have a black-box function (whose definition is not known) producing some numbers (precision is not known).
How to test if the generated number is integer or not?
I am looking for computable definition of "integerness".
Ceil, Truncate, Floor etc.. does not really fit the bill.
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Interesting. Could you please give a mathematical definition of such ceil and floor?
Would like to see how they are able to work without worrying about ranges.
Especially the "wipe off" part and how it takes care of the arbitrary precision.
Thank you.
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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?
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There are 3 main solutions to this problem.
1. Simple to implement is binary search (as already mentioned). Given a pair of values (L,U) such that L^2 < n < U^2 (where n is the number to be tested), we compute ((L+U)/2)^2 and deduce that the solution is contained in an interval of half size. This is polynomial-time (unlike some of the other solutions already stated).
2. Optimal complexity is to use Newton interation. See Chapter 9 of the book by von zur Gathen and Gerhard. In particular, Section 9.5 talks of a 3-adic Newton iteration to compute integer square roots. This leads to a running time of O( M(n)) = O( \log(n) \log\log(n) \log\log\log(n)) the same as integer multiplication. But this is harder to implement.
3. A Monte-Carlo method is to reduce n modulo small primes p and compute the Legendre symbol (n/p) (which will always be 0 or 1 if n is an integer square). One would expect to test around log(n) primes to have good confidence that n is a square. I do not know a reference for a theoretical analysis of this algorithm.
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I need to know if there is such that relation in the case of real quadratic fields.
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There is a notion of (divisor) class group 'Cl(X) ' associated to algebraic varieties X--affine and projective. (I will define this notion below for elliptic curves which are projective.). Now, if A is a Dedekind domain, (e.g. the ring of integers in a number field F (where F could be a quadratic field you are interested in) Cl(Spec(A)) coincides with the ideal class group of A as defined in Algebraic Number Theory. So, this gives you the connection between affine varieties and ideal class group. (Spec A is affine.)
I will now define Cl(X) for an elliptic curve X. . A divisor D of X is a linear combination of points x_i with coefficients k_i in Z(the integers).. The degree of D (deg D) is the sum of the k_i's. The divisors of X form a group Div(X)--the free Abelian group (Z-module) generated by the x_i's . A principal divisor is a divisor D such that the degree of D = 0.The principal divisors (P(X)) form a subgroup of Div (X) and the quotient group Div(X)/P(X) is denoted by Cl(X). Now, there is homomorphism deg: Cl(X) --> Z whose kernel is denoted by Cl^o(X). Cl^o(X) has the following interesting property:
There is a one-one correspondence between the points of x and the elements of the group Cl^o(X). Moreover by using this one-one correspondence, one can transfer the group law from Cl^o(X) to X itself.
Elliptic curves is a very active area of research probably because the curves have various deep ramifications. For example, an elliptic curve is an Abelian variety of dimension 1 i.e an irreducible projective algebraic group; it is also a curve of genus 1. etc.
for further information you may look at the following books:
1) Basic Algebraic Geometry--Varieties in Projective spaces by I. R. Shavarevich.
Springer-Verlag.
2) Algebraic Geometry-- R. Hartshorne. Springer-Verlag.
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I would like to know the real history of mathematics and the first result on mathematics.
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Here is my version of the history of mathematics.
Invention in early civilisation of summerian of early base counting system or number representation, early arithmetic operation, and land measuring systems with ropes and rods for standard unit length. Early version of pythagore theorem and early algebriaic problems expressed in prose. This knowledges is an engineering type of knowledge. A set of effective procedures but these procedures are not systematically organized into whole.
The first pre-socratic philosopher such Thales and pythagore have learned the early counting techniques and measurement techniques from their commercial patners , the Egyptians, and gradually invented axiomatic geometry whose primary axioms are the simplification of the complex land measuring technique and use simple logical combinations in such a way that the measurement procedures could be logically built. It is the unification of the implicit logic that exist in human language and cognition and to merge these with spatial concepts. So this provided a language where spatial measurement procedure could be mechanically built with a mechanic called logic. Geometrical language was invented and a new notion of truth, logical truth was created into an ideal mechanical mechanical world. The development of this language climax in the ancient world with Euclid and Archimedes around 300BC. Mathematical modeling was invented and also science in the form of the mental mechanic of Euclid and Archimedes.
Here I will go very fast on the next 2000 years that mostly took place in India and the muslin world. The decimal numerical system was invented simplifying enormously the arithmetic operations. The algebriaic notation using constants and variable allowed to create an algebriaic language of equations. Formal algebra was invented and procedures for equation solutions.
The next major break through was Galileo. He spatialized time. The biggest scientific invention of all time.
Descartes then invented analytical geometry: the merging of geometry with algebra. Geometry and the axiomatic method became subsumed under an algebraic Euclidean geometry.
This is the creation of the algebrian space time. This is the creation of the equivalence between a form and an equation. Modern science started there as the project of geometrizing the whole world.
What was missing from the Cartesian scientific method was the mechanics of Archimedes for calculating surface, volume and length in this new framework and Leibniz and Newton invented it in the forms of Calculus. Dynamic in the form of differential equations was invented and here is completed the modern mental framework of modern science. The consequence of the Newtonian paradigm will be work out in the next two hundred years.
Mechanics will be reformulated in terms of Hamiltonian mechanics in terms of trajectories of minimum action in configuration space along the initial conceptions of Leibniz, Fermat, Cauchy, Laplace.
Most of the mathematics of Engineering have been invented in this fertile period by all the names which we are familiar: Fourier, Dalembert, etc etc
The greatest innovation of this period was done by Galois. His idea will give rise to the concept of group which will become a privilege frameword structuring the whole of algebraic geometry. Physics and mathatics will be restructured based on group. The group theoretical way to express algebriaic geometry allowed it to distance itself from Euclidean geometry, in fact to realized that there exist all kind of geometry that can be invented in this language. Non-Euclidean geometries were invented. The Erlander program of Felix Klein allowed to classified all the new geometry into a systematic framework. Then the revolutionary epistemology of Kant could take new forms where the apriori of the physical science could be one of these new geometry. This has lead Gauss and Rieman , Poincarre and Lorentz and Einstein towards the new physics of GR. This has lead the creation of the new quantum physics on Hilbert Space.
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Can anybody point me to a simple program/routine (desirably in fortran, or any math library supported by fortran90) that tests whether any given number is prime or not? My max wouldn't go too high, no more than say 10^9. Is there any routine like that in "slatec", old/golden math-package? (if anybody here still heard of it...:-) Or is there a simple recipe how to write a routine?
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Libpari is part of the actual program gnu pari -- to be more portable you probably should compile gnu pari yourself and force the creation of a static libpari.a, link your fortran program against that so you wont need pari-installation to use it. If you can read c code you might be interested in looking at the sources how the sieve algorithm is implemented and port this to fortran. As for making use of c-libs in fortran, take a look at http://people.sc.fsu.edu/~jburkardt/f_src/f90_calls_c/f90_calls_c.html or whatever else you find on the net on that topic.
The simplest way to do what you want is to create a bitfield of 1's, set all bits to 0 that can be divided by 2, then all that can be divided by 3, and so on (remember to leave the primenumbers themselves at 1). And once you arrive at the squareroot of your bitfield's size, you can check primality of some number by looking up the corresponding bit. So you don't need a function, just read out the value of an array.
To speed up the initialization a bit you might consider crossing out the product between your current prime number and one of the numbers that haven't been crossed out yet (i.e. The 1s in your array), starting at the end of the array (i.e. The array size divided by the current prime, rounded down, is where you start), going backwards to avoid crossing out something you still might need. All that of course assuming you traverse the already discovered primes sequentially from lowest to highest. Parallelizing this algorithm is a roblem of its own...
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For example, binary representation of 65535 is 1111111111111111 in 16 bit representation. The number 65535 depends on the value at each bit, but can we comment on relationship among the bits if we are given a number and it is represented in binary form.
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If you mean some kind of correspondence between decimal digits and bits in binary representation of the same number, then probably no such obvious correspondence may be given. This is in contrast to octal or hexadecimal notation: here every octal (or hexadecimal) digit has always the same binary representation, for example:
6 (octal) = 110 (binary) = 0110 (hex) = 6 (decimal)
7 (octal) = 111 (binary)
C (hex) = 13 (decimal) = 1010 (binary)
More specifically, a single octal digit always is represented as 3-bit sequence, and a single octal digit - as a 4-bit sequence, where leading zeroes (zero bits), if any, may be omitted *only* at the begining of the said number.
This is fine for integer numbers. When fractions are allowed, things get more complicated as their binary (or decimal) representations need not to be unique (nor even finite ..), see: 1.0 = 0.9999.... (decimal). But that's another story, especially when you deal with finite-length binary arithmetic, as computers do. Here, even very simple cases, as 0.1 (decimal, finite length representation) often translate into infinite binary fraction.