Randomness can be a useful tool for solving problems.

Algebra

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

Postby Guest » Fri Jun 19, 2020 1:40 pm

Equation: [tex]\sum_{k=1}^{\infty }\frac{sin(2 \pi k \sqrt{I + \lambda^{2}})}{k} = \frac{\pi}{2}[/tex].

That equation is quite beautiful!

And it may be quite important too! :)

Go Blue! :D
Guest
 

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

Postby Guest » Fri Jun 19, 2020 4:15 pm

Guest wrote:Equation: [tex]\sum_{k=1}^{\infty }\frac{sin(2 \pi k \sqrt{I + \lambda^{2}})}{k} = \frac{\pi}{2}[/tex].

That equation is quite beautiful!

And it may be quite important too! :)

Go Blue! :D


We recall two important and beautiful equations from the analysis of the complex Riemann Zeta Function:

1A. [tex]\ \ \ \sum_{k=2}^{\infty }\frac{cos((b_{n} \pm \triangle b_{n}) * log(k))}{k^{\frac{1}{2}}}= -1[/tex]; (Real Part)

1B. [tex]\ \ \ \sum_{k=2}^{\infty }\frac{sin((b_{n} \pm \triangle b_{n}) * log(k))}{k^{\frac{1}{2}}}= 0[/tex]. (Imaginary Part)

where [tex]b_{n+1} = b_{n} \pm \triangle b_{n}[/tex] for some nonzero real values, [tex]b_{n + 1}[/tex], [tex]b_{n}[/tex], and [tex]\triangle b_{n}[/tex].
Attachments
Beauty by the Numbers.jpg
Beauty by the Numbers.jpg (37.99 KiB) Viewed 136 times
Guest
 

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

Postby Guest » Sat Jun 20, 2020 7:49 pm

Guest wrote:Tentative Remark: Yes! The power series expansion is not appropriate here. But the Fourier series expansion is appropriate here.

[tex]\triangle n(I, \lambda) = \sqrt{I + \lambda^{2}} - Floor(\sqrt{I + \lambda^{2}})[/tex].

Moreover, [tex]\triangle n(I, \lambda) = \frac{1}{2} - \frac{1}{\pi} * \sum_{k=1}^{\infty }\frac{sin( 2 \pi k \sqrt{I + \lambda^{2}} )}{k}[/tex]

where [tex]1 \le \lambda < \frac{\sqrt{I}}{2}[/tex] for a given I such that I = p * q.

Tentative Remark: If our explicit definition (Fourier series expansion) of [tex]\triangle n(I, \lambda)[/tex] holds, then we have almost solved our integer factorization problem!

Question: What is the solution of [tex]\triangle n(I, \lambda) = 0[/tex] ?

Relevant Reference Link:

'Continuity and series expansions',

https://en.wikipedia.org/wiki/Floor_and_ceiling_functions#Continuity_and_series_expansions.

Dave.


Solving the equation, [tex]\triangle n(I, \lambda) = 0[/tex], may be quite technical.

Please refer to the paper, 'Fast algorithms for multiple evaluations of the Riemann zeta function' by Prof. A. M. Odlyzko for some ideas/algorithms on how to solve [tex]\triangle n(I, \lambda) = 0[/tex].

Relevant Reference Link:

http://www.dtc.umn.edu/~odlyzko/doc/arch/fast.zeta.eval.pdf.
Guest
 

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

Postby Guest » Sat Jun 20, 2020 9:59 pm

Remark: Hopefully, we are on track to solve the integer factorization problem in polynomial time.
Attachments
Prime Time.jpg
Please keep the faith. And please keep up the hard work. Good luck! :-)
Prime Time.jpg (6.06 KiB) Viewed 105 times
Guest
 

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

Postby Guest » Sun Jun 21, 2020 3:07 pm

Solving the equation, [tex]\triangle n(I, \lambda) = 0[/tex], may be quite technical.

Please refer to the paper, 'Fast algorithms for multiple evaluations of the Riemann zeta function' by Prof. A. M. Odlyzko and Prof. A. Schönhage for some ideas/algorithms on how to solve [tex]\triangle n(I, \lambda) = 0[/tex].

Relevant Reference Link:

http://www.dtc.umn.edu/~odlyzko/doc/arch/fast.zeta.eval.pdf;


Remark: The above paper may be over the top for our needs here. But we should appreciate the great effort by its authors and the beautiful mathematics required to compute the nontrivial zeros of the Riemann Zeta Function accurately and fast. And our efforts to understand their excellent work is worthwhile. :)

Relevant Reference Link:

'THE SINE PRODUCT FORMULA AND THE GAMMA
FUNCTION
'
by Prof. ERICA CHAN
DECEMBER 12, 2006.

"Abstract. The function of sin x is very important in mathematics and has many applications. In addition to its series expansion, it can also be written as an infinite product. The infinite product of sin x can be used to prove certain values of ζ(s), such as ζ(2) and ζ(4). The gamma function is related directly to the sin x function
and can be used to prove the infinite product expansion. Also used are Weierstrass’s product formula and Legendre’s relation."

https://ocw.mit.edu/courses/mathematics/18-104-seminar-in-analysis-applications-to-number-theory-fall-2006/projects/chan.pdf.
Attachments
Inverse Fourier Transform.jpg
Inverse Fourier Transform.jpg (63.17 KiB) Viewed 95 times
Guest
 

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

Postby Guest » Sun Jun 21, 2020 8:37 pm

Relevant Reference Links:

'Twin Prime Conjecture',

https://mathworld.wolfram.com/TwinPrimeConjecture.html;

'de Polignac's Conjecture (Polignac Conjecture)'

https://mathworld.wolfram.com/dePolignacsConjecture.html.
Guest
 

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

Postby Guest » Wed Jun 24, 2020 3:39 pm

Davet wrote:Here's more homework! :)

[tex]q = \frac{-\delta + \sqrt{\delta^{2} + 4 * I}}{2} = - \lambda+ \sqrt{\lambda^{2} + I}[/tex]

where [tex]\delta[/tex] is a positive even integer such that [tex]p - q = \delta = 2 * \lambda[/tex] and such that

[tex]2 \le \delta < 6.018 * 10^{498}[/tex]... where the known I = p * q for some unknown primes, p and q.

Good luck! :)


Remark: The composite, I, is shown below in the attached file. Please compute its prime factors, p and q.

Hint: The prime factors, p > q, have 500 decimal digits each.

Thank you! :)
Attachments
I is a 1000-Digit Integer that is a product of two unknown primes, p and q, each with 500 digits..gif
I is a 1000-Digit Integer that is a product of two unknown primes, p and q, each with 500 digits..gif (134.73 KiB) Viewed 71 times
Guest
 

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

Postby Guest » Wed Jun 24, 2020 10:42 pm

Remark: [tex]\triangle n(I, \lambda) = \frac{1}{2} - \frac{1}{\pi} * \sum_{k=1}^{\infty }\frac{sin( 2 \pi k \sqrt{I + \lambda^{2}} )}{k} = 0[/tex].

Please solve that equation for [tex]\lambda[/tex] with our known value, [tex]I[/tex]. Please see the previous post. Thank you! :)
Guest
 

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

Postby Guest » Fri Jun 26, 2020 5:27 pm

Suppose we let [tex]\sum_{k=1}^{\infty }\frac{sin( 2 \pi k \sqrt{I + \lambda^{2}} )}{k}

= \sum_{n=1}^{\infty }\frac{sin( 2 \pi (2n-1) \sqrt{I + \lambda^{2}} )}{2n-1} + \sum_{n=1}^{\infty }\frac{sin( 4 \pi n \sqrt{I + \lambda^{2}} )}{2n}[/tex].

Furthermore, suppose [tex]\sum_{n=1}^{\infty }\frac{sin( 2 \pi (2n-1) \sqrt{I + \lambda^{2}} )}{2n-1} = 2* (1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - ... ) = \frac{\pi}{2}[/tex].

And suppose, [tex]\sum_{n=1}^{\infty }\frac{sin( 4 \pi n \sqrt{I + \lambda^{2}} )}{2n} = \sum_{n=1}^{\infty }\frac{sin( 4 \pi n \sqrt{I + \lambda^{2}} )}{n} = 0[/tex].

Hmm. What's next?

Relevant Reference Link:

'Leibniz formula for π',

https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80,
Guest
 

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

Postby Guest » Fri Jun 26, 2020 5:47 pm

We do not know!

Now suppose [tex]\sum_{n=1}^{\infty }\frac{sin( 2 \pi (2n-1) \sqrt{I + \lambda^{2}} )}{2n-1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - ... = \frac{\pi}{a}[/tex].

And suppose, [tex]\sum_{n=1}^{\infty }\frac{sin( 4 \pi n \sqrt{I + \lambda^{2}} )}{2n} = \sum_{n=1}^{\infty }\frac{sin( 4 \pi n \sqrt{I + \lambda^{2}} )}{n} = \frac{\pi}{b}[/tex].

We assume a and b are nonzero rationals.

Hmm. What's next?
Guest
 

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

Postby Guest » Fri Jun 26, 2020 5:54 pm

An Update: We correct the previous post.

Guest wrote:We do not know!

Now suppose [tex]\sum_{n=1}^{\infty }\frac{sin( 2 \pi (2n-1) \sqrt{I + \lambda^{2}} )}{2n-1} = \frac{\pi}{a}[/tex].

And suppose, [tex]\sum_{n=1}^{\infty }\frac{sin( 4 \pi n \sqrt{I + \lambda^{2}} )}{2n} = \sum_{n=1}^{\infty }\frac{sin( 4 \pi n \sqrt{I + \lambda^{2}} )}{n} = \frac{\pi}{b}[/tex].

We assume a and b are nonzero rationals.

Hmm. What's next?
Guest
 

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

Postby Guest » Fri Jun 26, 2020 10:39 pm

Guest wrote:An Update: We correct the previous post.

Guest wrote:We do not know!

Now suppose [tex]\sum_{n=1}^{\infty }\frac{sin( 2 \pi (2n-1) \sqrt{I + \lambda^{2}} )}{2n-1} = \frac{\pi}{a}[/tex].

And suppose, [tex]\sum_{n=1}^{\infty }\frac{sin( 4 \pi n \sqrt{I + \lambda^{2}} )}{2n} = \frac{\pi}{b}[/tex].

We assume a and b are nonzero rationals.

Hmm. What's next?
Guest
 

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

Postby Guest » Fri Jun 26, 2020 10:55 pm

Guest wrote:An Update: We correct the previous post.

Guest wrote:We do not know!

Now suppose (1) [tex]\sum_{n=1}^{\infty }\frac{sin( 2 \pi (2n-1) \sqrt{I + \lambda^{2}} )}{2n-1} = \frac{\pi}{a}[/tex].

And suppose, (2) [tex]\sum_{n=1}^{\infty }\frac{sin( 4 \pi n \sqrt{I + \lambda^{2}} )}{2n} = \frac{\pi}{b}[/tex].

We assume a and b are nonzero rationals.

Hmm. What's next?


Let [tex]I = 257 * 127[/tex]. We compute [tex]\lambda = 65[/tex]. Let's compute equations one and two. What are the values for a and b?
Guest
 

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

Postby Guest » Sun Jun 28, 2020 9:36 pm

Guest wrote:
Guest wrote:An Update: We correct the previous post.

Guest wrote:We do not know!

Now suppose (1) [tex]\sum_{n=1}^{\infty }\frac{sin( 2 \pi (2n-1) \sqrt{I + \lambda^{2}} )}{2n-1} = \frac{\pi}{a}[/tex].

And suppose, (2) [tex]\sum_{n=1}^{\infty }\frac{sin( 4 \pi n \sqrt{I + \lambda^{2}} )}{2n} = \frac{\pi}{b}[/tex].

We assume a and b are nonzero rationals.

Hmm. What's next?


Let [tex]I = 257 * 127[/tex]. We compute [tex]\lambda = 65[/tex]. Let's compute equations one and two. What are the values for a and b?


Remark: We recommend Netwon's Method for solving our immediate problem.

Relevant Reference Link:

'Netwon's Method',

https://en.wikipedia.org/wiki/Newton%27s_method
Guest
 

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

Postby Guest » Tue Jun 30, 2020 2:55 pm

Dave wrote:An Update: We correct the previous post.

"We do not know!" :?

Now suppose (1) [tex]\sum_{n=1}^{\infty }\frac{sin( 2 \pi (2n-1) \sqrt{I + \lambda^{2}} )}{2n-1} = \frac{\pi}{a}[/tex].

And suppose, (2) [tex]\sum_{n=1}^{\infty }\frac{sin( 4 \pi n \sqrt{I + \lambda^{2}} )}{2n} = \frac{\pi}{b}[/tex].

We assume a and b are nonzero real numbers.

Hmm. What's next?



Remark: We recommend Netwon's Method for solving our immediate problem.

Relevant Reference Link:

'Netwon's Method',

https://en.wikipedia.org/wiki/Newton%27s_method[/quote]

Question: What values of a and b are consistent with the truth of the Twin Prime Conjecture ([tex]\lambda = 1[/tex])?
Guest
 

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

Postby Guest » Tue Jun 30, 2020 6:09 pm

"We do not know!" :?

Remarks: We had assumed [tex]\frac{\pi}{a} +\frac{\pi}{b} = \frac{\pi}{2}[/tex] where a and b are nonzero real numbers.

And hopefully, we shall soon post some results for our previous problems. And we encourage our readers (mathematicians) to post comments/results regarding our work here. Thank you! :)

" ... We will know!" -- David Hilbert, a great mathematician.
Guest
 

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

Postby Guest » Thu Jul 02, 2020 11:37 am

Guest wrote:Remark: (1) [tex]\triangle n(I, \lambda) = \frac{1}{2} - \frac{1}{\pi} * \sum_{k=1}^{\infty }\frac{sin( 2 \pi k \sqrt{I + \lambda^{2}} )}{k} = 0[/tex].

Please solve that equation for [tex]\lambda[/tex] with our known value, [tex]I[/tex]. Please see the previous post. Thank you! :)


Relevant Reference Link:

'Montgomery's Pair Correlation Conjecture',

https://en.wikipedia.org/wiki/Montgomery%27s_pair_correlation_conjecture.
Guest
 

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

Postby Guest » Fri Jul 03, 2020 6:18 pm

Remark: Montgomery's Pair Correlation Conjecture is true!

Go Blue! :D
Guest
 

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

Postby Guest » Fri Jul 03, 2020 11:09 pm

Guest wrote:Remark: Montgomery's Pair Correlation Conjecture is true!

Go Blue! :D


Hah! Please prove Montgomery's Pair Correlation Conjecture. A claim is not proof :!:
Guest
 

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

Postby Guest » Sat Jul 04, 2020 2:51 pm

Guest wrote:
Guest wrote:Remark: Montgomery's Pair Correlation Conjecture is true!

Go Blue! :D


Hah! Please prove Montgomery's Pair Correlation Conjecture. A claim is not proof :!:


Our claim is not baseless. Please review our recent posts. And it should become clear with some effort on your part that Montgomery's Pair Correlation Conjecture is true!

Go Blue! :D
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