# Randomness can be a useful tool for solving problems.

Algebra

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

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

That equation is quite beautiful!

And it may be quite important too!

Go Blue!
Guest

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

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

That equation is quite beautiful!

And it may be quite important too!

Go Blue!

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

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

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

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

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

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

$$\triangle n(I, \lambda) = \sqrt{I + \lambda^{2}} - Floor(\sqrt{I + \lambda^{2}})$$.

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

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

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

Question: What is the solution of $$\triangle n(I, \lambda) = 0$$ ?

'Continuity and series expansions',

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

Dave.

Solving the equation, $$\triangle n(I, \lambda) = 0$$, 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 $$\triangle n(I, \lambda) = 0$$.

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

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

Remark: Hopefully, we are on track to solve the integer factorization problem in polynomial time.
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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.

Solving the equation, $$\triangle n(I, \lambda) = 0$$, 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 $$\triangle n(I, \lambda) = 0$$.

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.

'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.
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Inverse Fourier Transform.jpg (63.17 KiB) Viewed 95 times
Guest

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

'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.

Davet wrote:Here's more homework!

$$q = \frac{-\delta + \sqrt{\delta^{2} + 4 * I}}{2} = - \lambda+ \sqrt{\lambda^{2} + I}$$

where $$\delta$$ is a positive even integer such that $$p - q = \delta = 2 * \lambda$$ and such that

$$2 \le \delta < 6.018 * 10^{498}$$... 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 (134.73 KiB) Viewed 71 times
Guest

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

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

Please solve that equation for $$\lambda$$ with our known value, $$I$$. Please see the previous post. Thank you!
Guest

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

Suppose we let $$\sum_{k=1}^{\infty }\frac{sin( 2 \pi k \sqrt{I + \lambda^{2}} )}{k} = \sum_{n=1}^{\infty }\frac{sin( 2 \pi (2n-1) \sqrt{I + \lambda^{2}} )}{2n-1} + \sum_{n=1}^{\infty }\frac{sin( 4 \pi n \sqrt{I + \lambda^{2}} )}{2n}$$.

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

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

Hmm. What's next?

'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.

We do not know!

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

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

We assume a and b are nonzero rationals.

Hmm. What's next?
Guest

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

An Update: We correct the previous post.

Guest wrote:We do not know!

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

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

We assume a and b are nonzero rationals.

Hmm. What's next?
Guest

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

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

Guest wrote:We do not know!

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

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

We assume a and b are nonzero rationals.

Hmm. What's next?
Guest

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

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

Guest wrote:We do not know!

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

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

We assume a and b are nonzero rationals.

Hmm. What's next?

Let $$I = 257 * 127$$. We compute $$\lambda = 65$$. 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.

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

Guest wrote:We do not know!

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

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

We assume a and b are nonzero rationals.

Hmm. What's next?

Let $$I = 257 * 127$$. We compute $$\lambda = 65$$. 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.

'Netwon's Method',

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

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

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

"We do not know!"

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

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

We assume a and b are nonzero real numbers.

Hmm. What's next?

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

'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 ($$\lambda = 1$$)?
Guest

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

"We do not know!"

Remarks: We had assumed $$\frac{\pi}{a} +\frac{\pi}{b} = \frac{\pi}{2}$$ 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.

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

Please solve that equation for $$\lambda$$ with our known value, $$I$$. Please see the previous post. Thank you!

'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.

Remark: Montgomery's Pair Correlation Conjecture is true!

Go Blue!
Guest

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

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

Go Blue!

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

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

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

Go Blue!

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!
Guest

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