# Randomness can be a useful tool for solving problems.

Algebra

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

Tentatively, we have generated the first twelve digits of p (968 700 397 828) and q (895 124 411 925) based on I.

Their product is 867 107 373 065 471 689 253 700.

So far, things look good, and we have room for improvement...

And we are confident we can crack our current problem by April 30, 2021 in honor of the great Gauss (https://en.m.wikipedia.org/wiki/Carl_Friedrich_Gauss).

https://www.wolframalpha.com/input/?i=968700397828*895124411025.
Guest

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

Guest wrote:Tentatively, we have generated the first twelve digits of p (968 700 397 828) and q (895 124 411 025) based on I.

Their product is 867 107 373 065 471 689 253 700.

So far, things look good, and we have room for improvement...

And we are confident we can crack our current problem by April 30, 2021 in honor of the great Gauss (https://en.m.wikipedia.org/wiki/Carl_Friedrich_Gauss).

https://www.wolframalpha.com/input/?i=968700397828*895124411025.
Guest

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

Remark: Our current problem is great fun (difficult, very interesting, and hard work)! We shall expect the unexpected (a few failures, etc.) too!

The solution to our current problem is not easy/certain nor straightforward...
Guest

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

Remark: The possibilities (the permutations of digits and the complexity) makes the outcome the more difficult.
Guest

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

Remarks: Our current problem should be an ideal problem for Artificial Intelligence or dynamic/evolutionary learning algorithms.

Thus far, our work seems a bit chaotic. However, that's deceptive since the given outputs (the digits of I) determine the unknown inputs (the digits of p and q).
We must adapt and have some foresight too.

We are like wasps building their hives that seems initially chaotic to an onlooker until the hives take form.

There's purposeful order in our chaos (our efforts to crack our problem). And time will tell if we are successful. We are confident!
Guest

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

Please post your correct answers, p and q, here. Otherwise, keep up the good work. And the results will come! Thank you!!

FYI: 'A Tour of Machine Learning Algorithms',

https://machinelearningmastery.com/a-tour-of-machine-learning-algorithms/.
Attachments
What are the prime factors, p and q, for I?
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 49 times
Guest

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

Guest wrote:Tentatively, we have generated the first twelve digits of p (968 700 397 828) and q (895 124 411 025) based on I.

Their product is 867 107 373 065 471 689 253 700.

So far, things look good, and we have room for improvement...

And we are confident we can crack our current problem by April 30, 2021 in honor of the great Gauss (https://en.m.wikipedia.org/wiki/Carl_Friedrich_Gauss).

...

An Update:

Tentatively, we have generated the first 19 digits of p (9 687 003 978 288 689 673) and q (8 951 244 110 267 773 397) based on I.

Their product is 86 710 737 306 797 123 277 077 952 710 418 029 181.

So far, things look good, and we have room for improvement...

Remark: We may miss our target many times at the end of this process. And that's okay because we have learned something useful each time. And the knowledge gained will advance our efforts to solve our problem eventually.

https://www.wolframalpha.com/input/?i=9687003978288689673*8951244110267773397.
Attachments
What are the prime factors, p and q, for I?
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 46 times
Guest

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

Remark: We should be able to write a short program to process our data until a solution is found.
Guest

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

Updated Remark: We should be able to write a short program to gather data until a solution is found.

List of Possible Outcomes (data to be analyzed) for all $$1 \le i \le n$$:

Data Table: $$( (p_{i1 }, q_{i1 }, \delta_{i1} = I - p_{i1 } * q_{i1 }), (p_{i2 }, q_{i2 }, \delta_{i2} = I - p_{i2 } * q_{i2 } ) ), ..., ( (p_{n1}, q_{n1 }, \delta_{n1} = I - p_{n1 } * q_{n1 }), (p_{n2 }, q_{n2 }, \delta_{i2} = I - p_{n2 } * q_{n2 }) )$$

where $$\delta_{i1} > 0$$ and $$\delta_{i2} < 0$$.

We seek $$p_{ik }$$ and $$q_{ik }$$ such that $$\delta_{ik} = 0$$.
Guest

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

Guest wrote:Updated Remark: We should be able to write a short program to gather data until a solution is found.

List of Possible Outcomes (data to be analyzed) for all $$1 \le i \le n$$:

Data Table: $$( (p_{11 }, q_{11 }, \delta_{11} = I - p_{11 } * q_{11 }), (p_{12 }, q_{12 }, \delta_{12} = I - p_{12 } * q_{12 } ) ), ..., ( (p_{n1}, q_{n1 }, \delta_{n1} = I - p_{n1 } * q_{n1 }), (p_{n2 }, q_{n2 }, \delta_{i2} = I - p_{n2 } * q_{n2 }) )$$

where $$\delta_{i1} > 0$$ and $$\delta_{i2} < 0$$.

We seek $$p_{ik }$$ and $$q_{ik }$$ such that $$\delta_{ik} = 0$$.
Guest

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

Guest wrote:
Guest wrote:Updated Remark: We should be able to write a short program to gather data until a solution is found.

List of Possible Outcomes (data to be analyzed) for all $$1 \le i \le n$$:

Data Table: $$( (p_{11 }, q_{11 }, \delta_{11} = I - p_{11 } * q_{11 }), (p_{12 }, q_{12 }, \delta_{12} = I - p_{12 } * q_{12 } ) ), ..., ( (p_{n1}, q_{n1 }, \delta_{n1} = I - p_{n1 } * q_{n1 }), (p_{n2 }, q_{n2 }, \delta_{n2} = I - p_{n2 } * q_{n2 }) )$$

where $$\delta_{i1} > 0$$ and $$\delta_{i2} < 0$$.

We seek $$p_{ik }$$ and $$q_{ik }$$ such that $$\delta_{ik} = 0$$.

Remak: Ideally, we hope with sufficient data to discover p and q such that I = p * q.
Guest

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

Guest wrote:Please post your correct answers, p and q, here. Otherwise, keep up the good work. And the results will come! Thank you!!

FYI: 'A Tour of Machine Learning Algorithms',

https://machinelearningmastery.com/a-tour-of-machine-learning-algorithms/.

FYI: "ARTIFICIAL INTELLIGENCE:
Symbolic Mathematics Finally Yields to Neural Networks
After translating some of math’s complicated equations, researchers have created an AI system that they hope will answer even bigger questions.
"

https://www.quantamagazine.org/symbolic-mathematics-finally-yields-to-neural-networks-20200520/.
Guest

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

Important Remark:

$$8.7101... * 10^{499} \le q \le 9.311820446... * 10^{499}$$.

That helpful range makes our integer factorization problem a quite difficult problem!
Guest

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

Guest wrote:Important Remark:

$$8.7101... * 10^{499} \le q \le 9.311820446... * 10^{499}$$.

That helpful range makes our integer factorization problem a quite difficult problem!

Remark: Our proposed range is approximate.
Guest

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

Guest wrote:
Guest wrote:Important Remark:

$$8.7101... * 10^{499} \le q \le 9.311820446... * 10^{499}$$.

That helpful range makes our integer factorization problem a quite difficult problem!

Remark: Our proposed range is approximate.

Remark: We are 99% certain that our computed range is correct.

Why? That's homework!
Guest

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

Here's more homework!

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

where $$\delta$$ is a positive even integer such that

$$2 \le \delta < 6.018 * 10^{498}$$.

Please select the correct $$\delta$$ to solve q.

Remark: There are almost $$3.09 * 10^{498}$$ choices for $$\delta$$.
Only one is correct.
Guest

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

Guest wrote:Here's more homework!

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

where $$\delta$$ is a positive even integer such that

$$2 \le \delta < 6.018 * 10^{498}$$.

Please select the correct $$\delta$$ to solve q.

Remark: There are almost $$3.09 * 10^{498}$$ choices for $$\delta$$.
Only one is correct.

Good luck!
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 19 times
Guest

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

Remark: $$p = q + \delta$$

or $$p = \frac{I}{q}$$.
Guest

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

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:

$$2 \le \delta \ < 6.018 * 10^{489}$$;

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

$$6.018 * 10^{489} \le \delta < 2 * 6.018 * 10^{489}$$;

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

$$2 * 6.018 * 10^{489} \le \delta < 3 * 6.018 * 10^{489}$$;
...
...
...

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

$$(10^{9} - 1) * 6.018 * 10^{489} \le \delta < 6.018 * 10^{498}$$.
Guest

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

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, $$1 \le\lambda < I$$, 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

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