Randomness can be a useful tool for solving problems.

[Guest]

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).

Relevant Reference Link:

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

[Guest]

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).

Relevant Reference Link:

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

[Guest]

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]

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

[Guest]

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]

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/.

[Guest]

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.

Relevant Reference Link:

https://www.wolframalpha.com/input/?i=9687003978288689673*8951244110267773397.

[Guest]

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

[Guest]

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 [tex]1 \le i \le n[/tex]:

Data Table: [tex]( (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 }) )[/tex]

where [tex]\delta_{i1} > 0[/tex] and [tex]\delta_{i2} < 0[/tex].


We seek [tex]p_{ik }[/tex] and [tex]q_{ik }[/tex] such that [tex]\delta_{ik} = 0[/tex].

[Guest]

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 [tex]1 \le i \le n[/tex]:

Data Table: [tex]( (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 }) )[/tex]

where [tex]\delta_{i1} > 0[/tex] and [tex]\delta_{i2} < 0[/tex].


We seek [tex]p_{ik }[/tex] and [tex]q_{ik }[/tex] such that [tex]\delta_{ik} = 0[/tex].

[Guest]

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 [tex]1 \le i \le n[/tex]:

Data Table: [tex]( (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 }) )[/tex]

where [tex]\delta_{i1} > 0[/tex] and [tex]\delta_{i2} < 0[/tex].


We seek [tex]p_{ik }[/tex] and [tex]q_{ik }[/tex] such that [tex]\delta_{ik} = 0[/tex].

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

[Guest]

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]

Important Remark:

[tex]8.7101... * 10^{499} \le q \le 9.311820446... * 10^{499}[/tex].

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

[Guest]

Important Remark:

[tex]8.7101... * 10^{499} \le q \le 9.311820446... * 10^{499}[/tex].

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

Remark: Our proposed range is approximate.

[Guest]

Important Remark:

[tex]8.7101... * 10^{499} \le q \le 9.311820446... * 10^{499}[/tex].

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]

Here's more homework!

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

where [tex]\delta[/tex] is a positive even integer such that

[tex]2 \le \delta < 6.018 * 10^{498}[/tex].

Please select the correct [tex]\delta[/tex] to solve q.

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

[Guest]

Here's more homework!

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

where [tex]\delta[/tex] is a positive even integer such that

[tex]2 \le \delta < 6.018 * 10^{498}[/tex].

Please select the correct [tex]\delta[/tex] to solve q.

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

Good luck!

[Guest]

Remark: [tex]p = q + \delta[/tex]

or [tex]p = \frac{I}{q}[/tex].

[Guest]

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:

[tex]2 \le \delta \ < 6.018 * 10^{489}[/tex];

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

[tex]6.018 * 10^{489} \le \delta < 2 * 6.018 * 10^{489}[/tex];

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

[tex]2 * 6.018 * 10^{489} \le \delta < 3 * 6.018 * 10^{489}[/tex];
...
...
...

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

[tex](10^{9} - 1) * 6.018 * 10^{489} \le \delta < 6.018 * 10^{498}[/tex].

[Guest]

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, [tex]1 \le\lambda < I[/tex], such that

[tex]\sqrt{I + \lambda^{2}}[/tex] is a positive integer.

Remark: [tex]p - q = 2\lambda[/tex] where p and q are unknown odd primes such that p > q and such that I = p * q where I is known.

Dave.