# Randomness can be a useful tool for solving problems.

Algebra

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

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

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

Now only one of the two following cases is true!

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

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

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.

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

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

Now only one of the two following cases is true!

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

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

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.

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

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

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.

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 $$\lambda$$, is $$I + \lambda^{2}$$ 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.

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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 $$\lambda$$, is $$I + \lambda^{2}$$ becoming a perfect square?

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

We recall: $$n(\lambda) + \triangle n(\lambda) = \sqrt{I + \lambda^{2}}$$.

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

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

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.

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.

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

A "Redundant" Question: For some $$1 \le \lambda < \frac{\sqrt{I}}{2}$$, the perfect square, $$I + \lambda^{2}$$, guarantees that the factors, $$\sqrt{I + \lambda^{2}} \pm \lambda$$, 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.

"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 (11.9 KiB) Viewed 45 times
Guest

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

'Congruence of Squares',

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

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

'General number field sieve',

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

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.

'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.

Dave wrote:FYI: 'Continued fraction',

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

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

Good Luck! A Tentative Update: $$\beta(I) * (\frac{1 + \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}}{2}) \le \lambda < \sqrt{ \frac{(\sqrt{2} - 1) * I}{2}}$$ where $$0 < \beta(I) \le 1$$.

Remark: We want to construct a convergent sequence that solves $$\lambda$$.

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

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

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

'Continued fraction factorization',

https://en.wikipedia.org/wiki/Continued_fraction_factorization.
Attachments Continued Fractions Factoring method.pdf
Guest

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

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
Guest

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

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.

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

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

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

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

'Power series',

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

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

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

$$n(\lambda) = \left \lfloor{\sqrt{I + \lambda^{2}}} \right \rfloor$$;

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

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

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

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

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

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

'Power series',

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

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

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

$$n(\lambda) = \left \lfloor{\sqrt{I + \lambda^{2}}} \right \rfloor$$;

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

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

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

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

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