# Randomness can be a useful tool for solving problems.

Algebra

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

An Update: ($$\sqrt{I}$$)

The Minimum Square Greater Than I = p * q: Generally, our integer factorization problem, I = p * q, is an integer optimization problem. We seek to find the minimum positive integer, $$1 \le\lambda < \sqrt{I}$$, such that

$$\sqrt{I + \lambda^{2}}$$ is a positive integer. Remark: $$p - q = 2\lambda$$ where p and q are unknown odd primes such that p > q and such that I = p * q where I is known.

Dave.
Guest

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

Dave wrote:An Update: ($$\sqrt{I}$$)

The Minimum Square Greater Than I = p * q: Generally, our integer factorization problem, I = p * q, is an integer optimization problem. We seek the minimum positive integer, $$1 \le\lambda < \sqrt{I}$$, such that

$$\sqrt{I + \lambda^{2}}$$ is a positive integer. Remark: $$p - q = 2\lambda$$ where p and q are unknown odd primes such that p > q and such that I = p * q where I is known.

Dave.

Does fast convergence (polynomial time) from $$\sqrt{I}$$, exclusively, to $$\lambda$$ exists? Why? How?

Does fast convergence (polynomial time) from 1, inclusively, to $$\lambda$$ exists? Why? How?
Attachments Figure-D1-Polynomial-time-complexity-classes.png (15.35 KiB) Viewed 88 times
Guest

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

Remarks: $$\lambda$$ exists and is unique since the prime factor, q, exists and is unique.

Moreover, $$3 \le q < \sqrt{I} = \sqrt{p * q} < p$$.
Guest

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

An Update: ( $$\sqrt{I} / 2$$ )

The Minimum Square Greater Than I = p * q: Generally, our integer factorization problem, I = p * q, is an integer optimization problem. We seek the minimum positive integer, $$1 \le\lambda < \sqrt{I} / 2$$, such that

$$\sqrt{I + \lambda^{2}}$$ is a positive integer. Remark: $$p - q = 2\lambda$$ where p and q are unknown odd primes such that p > q and such that I = p * q where I is known.

Dave.

Does fast convergence (polynomial time) from $$\sqrt{I} / 2$$, exclusively, to $$\lambda$$ exists? Why? How?

Does fast convergence (polynomial time) from 1, inclusively, to $$\lambda$$ exists? Why? How?
Guest

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

Guest wrote:An Update: ( $$\sqrt{I} / 2$$ )

The Minimum Square Greater Than I = p * q: Generally, our integer factorization problem, I = p * q, is an integer optimization problem. We seek the unique positive integer, $$1 \le\lambda < \sqrt{I} / 2$$, such that

$$\sqrt{I + \lambda^{2}}$$ is a positive integer. Remark: $$p - q = 2\lambda$$ where p and q are unknown odd primes such that p > q and such that I = p * q where I is known.

Dave.

Does fast convergence (polynomial time) from $$\sqrt{I} / 2$$, exclusively, to $$\lambda$$ exists? Why? How?

Does fast convergence (polynomial time) from 1, inclusively, to $$\lambda$$ exists? Why? How?
Guest

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

An Update for $$I = p * q = 8.671... * 10^{999}$$:

Guest wrote:Now suppose we have access to a worldwide network of one billion personal computers for solving our integer factorization problem.

We program computer one to search randomly for a solution in the range:

$$1 \le \lambda\ < 3.09 * 10^{489}$$;

We program computer two to search randomly for a solution in the range:

$$3.09 * 10^{489} \le \lambda < 2 * 3.09 * 10^{489}$$;

We program computer three to search randomly for a solution in the range:

$$2 * 3.09 * 10^{489} \le \delta < 3 * 3.09 * 10^{489}$$;
...
...
...

We program the billionth computer to search randomly for a solution in the range:

$$(10^{9} - 1) * 3.09 * 10^{489} \le \lambda < 3.09 * 10^{498}$$.

We
Guest

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

Guest wrote:An Update for $$I = p * q = 8.671... * 10^{999}$$:

Guest wrote:Now suppose we have access to a worldwide network of one billion personal computers for solving our integer factorization problem.

We program computer one to search randomly for a solution in the range:

$$1 \le \lambda\ < 3.09 * 10^{489}$$;

We program computer two to search randomly for a solution in the range:

$$3.09 * 10^{489} \le \lambda < 2 * 3.09 * 10^{489}$$;

We program computer three to search randomly for a solution in the range:

$$2 * 3.09 * 10^{489} \le \lambda < 3 * 3.09 * 10^{489}$$;
...
...
...

We program the billionth computer to search randomly for a solution in the range:

$$(10^{9} - 1) * 3.09 * 10^{489} \le \lambda < 3.09 * 10^{498}$$.
Guest

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

Guest wrote:
Guest wrote:An Update: ( $$\sqrt{I} / 2$$ )

The Appropriate Square Greater Than I = p * q: Generally, our integer factorization problem, I = p * q, is an integer optimization problem. We seek the unique positive integer, $$1 \le\lambda < \sqrt{I} / 2$$, such that

$$\sqrt{I + \lambda^{2}}$$ is a positive integer. Remark: $$p - q = 2\lambda$$ where p and q are unknown odd primes such that p > q and such that I = p * q where I is known.

Dave.

Does fast convergence (polynomial time) from $$\sqrt{I} / 2$$, exclusively, to $$\lambda$$ exists? Why? How?

Does fast convergence (polynomial time) from 1, inclusively, to $$\lambda$$ exists? Why? How?

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.
Guest

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

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, $$\sqrt{I + \lambda^{2}}$$ ?
Guest

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

Guest 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, $$\sqrt{I + \lambda^{2}}$$ ?

Remark: The power series expansion of $$\sqrt{I + \lambda^{2}}$$ may or may not help us here.
Guest

Guest

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

Remark: We are seeking either a proof/algorithm or a disproof that our integer factorization problem can be solved in polynomial time.
Guest

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

Guest wrote:Remark: We are seeking either a proof/algorithm or a disproof that our integer factorization problem can be solved in polynomial time.

'Integer factorization',

https://en.wikipedia.org/wiki/Integer_factorization#:~:text=Unsolved%20problem%20in%20computer%20science%3A&text=In%20number%20theory%2C%20integer%20factorization,process%20is%20called%20prime%20factorization.&text=Not%20all%20numbers%20of%20a%20given%20length%20are%20equally%20hard%20to%20factor.;

'Why isn't integer factorization in complexity P, when you can factorize n in O(√n) steps?',

https://math.stackexchange.com/questions/1624390/why-isnt-integer-factorization-in-complexity-p-when-you-can-factorize-n-in-o%E2%88%9A.
Attachments Integer Factorization.jpg (67.57 KiB) Viewed 59 times
Guest

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

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

Good Luck! Guest

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

Guest wrote:

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

Good Luck! Guest

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

What is the nearest perfect square to I = p * q such p and q are primes?[/b]

Generally, that question is difficult to answer. Here are some simple examples:

If I = 91, then 100 is the nearest perfect square such that p and q are primes ($$10 \pm \lambda$$), and therefore, $$\lambda = 3$$.

If I = 32,639, then 36,864 is the nearest perfect square such that p and q are primes ($$192 \pm \lambda$$), and therefore, $$\lambda = 65$$.
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 to solve $$\lambda$$.
Guest

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

\ge
Guest wrote:
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 to solve $$\lambda$$.

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 $$0 < \beta(I) \le 1$$.
Guest

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

Guest wrote:\ge
Guest wrote:
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 $$0 < \beta(I) \le 1$$.
Guest

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

Remarks: Both cases, one and two, are equally challenging. And each case requires further subintervals or subsequences... We hope that further developments will achieve a solution.
Guest

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