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Computational Number Theory - Science topic
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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?
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 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?
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.
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.
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.
Is it possible to use heuristic methods?
How to compute three sums involving binomial coefficients (combinatorial numbers)? See pictures with this question. Thank a lot.



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 ?
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.
Given the wide attention it has received from the math community, what are the practical uses of Fermat's Last Theorem?
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?

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?

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
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 ?
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.
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 ?
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?
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?
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.
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)
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?
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.
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.
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.
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 need to know if there is such that relation in the case of real quadratic fields.
I would like to know the real history of mathematics and the first result on mathematics.
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?
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.