Randomness can be a useful tool for solving problems.

Algebra

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Mon Jun 15, 2020 3:20 pm

Suppose Case 1 is true, but [tex]\lambda < \frac{1 + \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}}{2}[/tex].

Case 1.1: [tex]\beta_{11}(I) * (\frac{1 + \frac{1 + \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}}{2}}{2}) \le \lambda < \frac{1 + \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}}{2}[/tex] when [tex]0 < \beta_{11}(I) \le 1[/tex].

Now only one of the two following cases is true!

Case 1.1A: [tex]\lambda >\frac{1 + \frac{1 + \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}}{2}}{2}[/tex] when [tex]\beta_{11}(I) = 1[/tex].


Case 1.1B: [tex]\lambda = \beta_{11}(I) * (\frac{1 + \frac{1 + \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}}{2}}{2})[/tex] when [tex]0 < \beta_{11}(I) \le 1[/tex].


Remarks: The development process is ongoing, and it is quickly becoming convoluted too. We stress caution and accuracy.
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Mon Jun 15, 2020 3:32 pm

Oops! We choose the wrong case! Please consider case two instead. We have made corrections below.

Guest wrote:Suppose Case 2 is true, but [tex]\lambda < \frac{1 + \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}}{2}[/tex].

Case 2.1: [tex]\beta_{21}(I) * (\frac{1 + \frac{1 + \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}}{2}}{2}) \le \lambda < \frac{1 + \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}}{2}[/tex] when [tex]0 < \beta_{21}(I) \le 1[/tex].

Now only one of the two following cases is true!

Case 2.1A: [tex]\lambda >\frac{1 + \frac{1 + \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}}{2}}{2}[/tex] when [tex]\beta_{21}(I) = 1[/tex].


Case 2.1B: [tex]\lambda = \beta_{21}(I) * (\frac{1 + \frac{1 + \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}}{2}}{2})[/tex] when [tex]0 < \beta_{21}(I) \le 1[/tex].


Remarks: The development process is ongoing, and it is quickly becoming convoluted too. We stress caution and accuracy.
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Mon Jun 15, 2020 3:46 pm

Remarks: In either case, one or two, we must be mindful of how close we are achieving a true solution for [tex]\lambda[/tex]. Do our efforts make sense?
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Mon Jun 15, 2020 4:06 pm

Remarks: We must make sure there's no gap in our programming. All cases must be fully developed and verified. We must test and do more testing. Math is hard work! :)
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Mon Jun 15, 2020 4:32 pm

Guest wrote:Remarks: We must make sure there's no gap in our programming. All cases must be fully developed and verified. We must test and do more testing. Math is hard work! :)


For each case for [tex]\lambda[/tex], is [tex]I + \lambda^{2}[/tex] becoming a perfect square?

And we expect the two factors of I to be prime.
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Mon Jun 15, 2020 5:25 pm

[/b]
Guest wrote:
Guest wrote:Remarks: We must make sure there's no gap in our programming. All cases must be fully developed and verified. We must test and do more testing. Math is hard work! :)


For each case of [tex]\lambda[/tex], is [tex]I + \lambda^{2}[/tex] becoming a perfect square?

And we expect the two factors of I to be prime.


We recall: [tex]n(\lambda) + \triangle n(\lambda) = \sqrt{I + \lambda^{2}}[/tex].

Important Question: Does the error, [tex]0 \le \triangle n(\lambda) < 1[/tex], associated with each relevant case of [tex]\lambda[/tex], helps us to decide how to proceed with our programming?
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Mon Jun 15, 2020 5:31 pm

Remarks: We are confident that we can crack our integer factorization problem. But there's a great deal of work we must accomplish...
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Mon Jun 15, 2020 5:55 pm

Remark: Some randomness/guessing may help our programming.

Remark: We are hopeful that the integer factorization problem can be solved in polynomial time.
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Mon Jun 15, 2020 10:10 pm

Question: Can we define the function, [tex]0 < \beta(I) \le 1[/tex], more explicity in terms of I?

A "Redundant" Question: For some [tex]1 \le \lambda < \frac{\sqrt{I}}{2}[/tex], the perfect square, [tex]I + \lambda^{2}[/tex], guarantees that the factors, [tex]\sqrt{I + \lambda^{2}} \pm \lambda[/tex], of I are prime?
We had assumed the known value, I, is a product of unknown primes.
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Tue Jun 16, 2020 1:28 pm

"Point us (mathematicians) in the right direction, and we will find the right solutions to our problems." :)
Attachments
Important Steps in the Right Direction.jpg
Important Steps in the Right Direction.jpg (11.9 KiB) Viewed 45 times
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Tue Jun 16, 2020 7:27 pm

Relevant Reference Link:

'Congruence of Squares',

https://en.wikipedia.org/wiki/Congruence_of_squares.
Attachments
A congruence of squares method.jpg
A congruence of squares method.jpg (80.66 KiB) Viewed 43 times
History of congruence of  squares methods.jpg
History of congruence of squares methods.jpg (100.12 KiB) Viewed 43 times
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Tue Jun 16, 2020 7:41 pm

Relevant Reference Link:

'General number field sieve',

https://en.wikipedia.org/wiki/General_number_field_sieve.
Attachments
An Integrated Parallel  GNSF Algorithm for Integer Factorization.jpg
An Integrated Parallel GNSF Algorithm for Integer Factorization.jpg (46.33 KiB) Viewed 43 times
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Tue Jun 16, 2020 7:52 pm

Remarks: We have existing technology to factor large composites (product of two large primes, each with 250 decimal digits). What we are seeking is an edge/breakthrough that will guarantee integer factorization in polynomial time.
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Tue Jun 16, 2020 8:05 pm

Guest wrote:Relevant Reference Link:

'General number field sieve',

https://en.wikipedia.org/wiki/General_number_field_sieve.


FYI: 'An integrated parallel GNFS algorithm for integer factorization based on Linbox Montgomery block Lanczos method over GF(2)',

https://www.sciencedirect.com/science/article/pii/S0898122110000428.
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Tue Jun 16, 2020 8:48 pm

Dave wrote:FYI: 'Continued fraction',

https://en.wikipedia.org/wiki/Continued_fraction#Infinite_continued_fractions_and_convergents.

The generalized continued fraction of [tex]\sqrt{I + \lambda^{2}}[/tex] offers great promise. So between the power series expansion of [tex]\sqrt{I + \lambda^{2}}[/tex] and its generalized continued fraction representation, we hope to solve [tex]\triangle n(\lambda) = 0[/tex]...

Good Luck! :)

A Tentative Update: [tex]\beta(I) * (\frac{1 + \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}}{2}) \le \lambda < \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}[/tex] where [tex]0 < \beta(I) \le 1[/tex].

Remark: We want to construct a convergent sequence that solves [tex]\lambda[/tex].

Depending on I, only one of the following two cases is true!

Case 1: [tex]\lambda > \frac{1 + \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}}{2}[/tex] when [tex]\beta(I) = 1[/tex].

Case 2: [tex]\lambda = \beta(I) * (\frac{1 + \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}}{2})[/tex] when [tex]0 < \beta(I) \le 1[/tex].


Relevant Reference Link:

'Continued fraction factorization',

https://en.wikipedia.org/wiki/Continued_fraction_factorization.
Attachments
Continued Fractions Factoring method.pdf
(305.73 KiB) Downloaded 2 times
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Tue Jun 16, 2020 10:35 pm

We spoke very briefly on how integer factorization and integer optimization are related/connected in a past post.

Please refer to the attachment below for more details.
Attachments
Factoring as Optimization.pdf
(179.62 KiB) Downloaded 2 times
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Wed Jun 17, 2020 12:40 am

Remark: We have enough information to solve our problem either affirmatively or negatively.

Good luck! Go Blue! :-)
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Wed Jun 17, 2020 9:29 am

Dave wrote:Let's ponder the following sum of two functions, [tex]n(\lambda)[/tex] and [tex]∆n(\lambda)[/tex]:

[tex]n(\lambda) + ∆n(\lambda) = \sqrt{I + \lambda^{2}}[/tex] for some positive integers, [tex]n(\lambda)[/tex] and [tex]\lambda[/tex], where [tex]0 \le ∆n(\lambda) < 1[/tex].

Remark: [tex]1 \le \lambda < \frac{\sqrt{I}}{2}[/tex] and I = p * q where I is known.

Key Ideas: Power Series Expansion, Convergence, Rate of Convergence, etc.

Relevant Reference Link:

'Power series',

https://en.wikipedia.org/wiki/Power_series.]

Can we explicitly identify [tex]n(\lambda)[/tex] in the expression,[tex]\sqrt{I + \lambda^{2}}[/tex] ?

And can we explicitly identify [tex]∆n(\lambda)[/tex] in the expression, [tex]\sqrt{I + \lambda^{2}}[/tex] ?


[tex]n(\lambda) = \left \lfloor{\sqrt{I + \lambda^{2}}} \right \rfloor[/tex];

[tex]\triangle n(\lambda) = \sqrt{I + \lambda^{2}} - \left \lfloor{\sqrt{I + \lambda^{2}}} \right \rfloor[/tex].
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Wed Jun 17, 2020 10:00 am

Guest wrote:
Dave wrote:Let's ponder the following sum of two functions, [tex]n(\lambda)[/tex] and [tex]∆n(\lambda)[/tex]:

[tex]n(\lambda) + ∆n(\lambda) = \sqrt{I + \lambda^{2}}[/tex] for some positive integers, [tex]n(\lambda)[/tex] and [tex]\lambda[/tex], where [tex]0 \le ∆n(\lambda) < 1[/tex].

Remark: [tex]1 \le \lambda < \frac{\sqrt{I}}{2}[/tex] and I = p * q where I is known.

Key Ideas: Power Series Expansion, Convergence, Rate of Convergence, etc.

Relevant Reference Link:

'Power series',

https://en.wikipedia.org/wiki/Power_series.]

Can we explicitly identify [tex]n(\lambda)[/tex] in the expression,[tex]\sqrt{I + \lambda^{2}}[/tex] ?

And can we explicitly identify [tex]∆n(\lambda)[/tex] in the expression, [tex]\sqrt{I + \lambda^{2}}[/tex] ?


[tex]n(\lambda) = \left \lfloor{\sqrt{I + \lambda^{2}}} \right \rfloor[/tex];

[tex]\triangle n(\lambda) = \sqrt{I + \lambda^{2}} - \left \lfloor{\sqrt{I + \lambda^{2}}} \right \rfloor[/tex].


Remark: The discrete graph of [tex]\triangle n(\lambda)[/tex] for [tex]1 \le \lambda < \frac{\sqrt{I + \lambda^{2}}}{2}[/tex] is quite interesting and valuable!
Guest
 

Re: Randomness can be a useful tool for solving problems.

Postby Guest » Wed Jun 17, 2020 10:16 am

Updated Remark: The discrete graph of [tex]\triangle n(\lambda)[/tex] for [tex]1 \le \lambda < \frac{\sqrt{I }}{2}[/tex] is quite interesting and valuable!
Guest
 

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