Integer Factorization in PT (Polynomial Time)

Integer Factorization in PT (Polynomial Time)

Postby Guest » Wed Jul 22, 2020 1:34 pm

Can we achieve integer factorization in polynomial time?

Relevant Reference Link:

'Integer Factorization'

https://en.wikipedia.org/wiki/Integer_factorization.

Hmm. Can we combine tools (continued fraction factorization algorithm, general number field sieve algorithm, etc.) to achieve integer factorization in polynomial time?

Dave.
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Re: Integer Factorization in PT (Polynomial Time)

Postby Guest » Wed Jul 22, 2020 1:48 pm

We are seeking proof/algorithm or disproof that integer factorization is achievable in polynomial time.

Relevant Reference Links:

'Randomness can be a useful tool for solving problems.'

https://www.math10.com/forum/viewtopic.php?f=1&t=8855&sid=ee4551598cc1fa9c3dc5a0b2f292b994&start=140;

'Number Theory and Cryptography',

https://www.math10.com/forum/viewtopic.php?f=63&t=8634.
Guest
 

Re: Integer Factorization in PT (Polynomial Time)

Postby Guest » Wed Jul 22, 2020 10:16 pm

The Problem: We are given a large positive integer, I, which is a product of two unknown odd primes, p > q.

What are p and q?

The Solution Formulation:

We generate a system of two equations with three unknowns since [tex]p - q = 2 \lambda[/tex] for some unknown positive integer, [tex]2 \lambda[/tex]:

1. [tex]p * q = I[/tex];

2. [tex]p - q = 2 \lambda[/tex] such that [tex]1 \le \lambda < \frac{log^{2} (pq)}{2}[/tex].

The Question: Can we achieve integer factorization of I in polynomial time?

The Answer: Yes! We can affirmatively achieve integer factorization of I in polynomial time.

Dave,

https://www.researchgate.net/profile/David_Cole29.

Go Blue! :D
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Re: Integer Factorization in PT (Polynomial Time)

Postby Guest » Wed Jul 22, 2020 10:55 pm

Dave wrote:The Problem: We are given a large positive integer, I, which is a product of two unknown odd primes, p > q.

What are p and q?

The Solution Formulation:

We generate a system of two equations with three unknowns since [tex]p - q = 2 \lambda[/tex] for some unknown positive integer, [tex]2 \lambda[/tex]:

1. [tex]p * q = I[/tex];

2. [tex]p - q = 2 \lambda[/tex] such that [tex]1 \le \lambda < \frac{log^{2} (pq)}{2}[/tex].

The Question: Can we achieve integer factorization of I in polynomial time?

The Answer: Yes! We can affirmatively achieve integer factorization of I in polynomial time.

Dave,

https://www.researchgate.net/profile/David_Cole29.

Go Blue! :D


"Simple seeks simplest (best) solution." :)

Moreover,

1a. [tex]q = -\lambda + \sqrt{I + \lambda^{2}}[/tex] where [tex]1 \le \lambda < \frac{log^{2} (pq)}{2}[/tex];

1b. [tex]p = \frac{I}{q}[/tex].

Dave.

Go Blue!
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Re: Integer Factorization in PT (Polynomial Time)

Postby Guest » Wed Jul 22, 2020 11:59 pm

Oops! [tex](log(p * q))^{2}[/tex] is the wrong formula! The [tex](log(p * q))^{2}[/tex] works for the largest gap between consecutive primes less than p * q.

Therefore, we could try [tex](log(p * q))^{3}[/tex] or larger.

Dave.
Guest
 

Re: Integer Factorization in PT (Polynomial Time)

Postby Guest » Thu Jul 23, 2020 12:11 am

Dave wrote:Oops! [tex](log(p * q))^{2}[/tex] is the wrong formula! The [tex](log(p * q))^{2}[/tex] works for the largest gap between consecutive primes less than p * q.

Therefore, we could try [tex](log(p * q))^{3}[/tex] or larger.

Dave.


Correction! [tex]2 \le 2 \lambda < \frac{\sqrt{I}}{2}[/tex] is probably closer to the truth.

And we do not know if we can achieve integer factorization in polynomial time.

Dave :oops:
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Re: Integer Factorization in PT (Polynomial Time)

Postby Guest » Thu Jul 23, 2020 12:26 pm

Dave wrote:Oops! [tex](log(p * q))^{2}[/tex] is the wrong formula! The [tex](log(p * q))^{2}[/tex] works for the largest gap between consecutive primes less than p * q.

Correction! [tex]2 \le 2 \lambda < \frac{\sqrt{I}}{2}[/tex] is probably closer to the truth.

Is integer factorization achievable in polynomial time? WE DO NOT KNOW!

Simple Dave :oops:


"We will know!" -- David Hilbert, a great mathematician.
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Re: Integer Factorization in PT (Polynomial Time)

Postby Guest » Fri Jul 24, 2020 10:55 am

Equation 0: [tex]\sum_{k=1}^{\infty }\frac{sin(2 \pi k \sqrt{I + \lambda^{2}})}{k} = \frac{\pi}{2}[/tex]

where [tex]\lambda[/tex] is a positive integer such that [tex]1 \le \lambda < \frac{\sqrt{I}}{4}[/tex].

Given the integral value, [tex]I = p * q[/tex], what is [tex]\lambda[/tex]?

Can the Newton's Method solve equation zero? It is worth a try.

Dave.

Relevant Reference Link:

'Newton's Method',

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

Re: Integer Factorization in PT (Polynomial Time)

Postby Guest » Fri Jul 24, 2020 2:34 pm

Dave wrote:Equation 0: [tex]\sum_{k=1}^{\infty }\frac{sin(2 \pi k \sqrt{I + \lambda^{2}})}{k} = \frac{\pi}{2}[/tex]

Update:

where [tex]\lambda[/tex] is a positive integer such that [tex]1 \le \lambda < \frac{\sqrt{I}}{2}[/tex].

Given the integral value, [tex]I = p * q[/tex], what is [tex]\lambda[/tex]?

Can the Newton's Method solve equation zero? It is worth a try.

Relevant Reference Link:

'Newton's Method',

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

Re: Integer Factorization in PT (Polynomial Time)

Postby Guest » Sat Jul 25, 2020 4:59 pm

FYI: Newton's Method may not work, but it is worth a try. We must beware of the global minimum, 0, versus many local minima in the open interval, (0, 1), oscillations, and the failure analysis associated with Newton's Method.

However, Newton's Method could be a valuable complementary tool for other tools (continued fraction factorization algorithm, general number field sieve algorithm, etc.) in regards to solving our problem.

Relevant Reference Link:

'Failure Analysis associated with Newton's Method',

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

Dave.
Guest
 

Re: Integer Factorization in PT (Polynomial Time)

Postby Guest » Mon Jul 27, 2020 9:19 am

Dave wrote:Equation 0: [tex]\sum_{k=1}^{\infty }\frac{sin(2 \pi k \sqrt{I + \lambda^{2}})}{k} = \frac{\pi}{2}[/tex]

where [tex]\lambda[/tex] is a positive integer such that [tex]1 \le \lambda \le \frac{p - 3}{2}[/tex]. (Update)

Given the integral value, [tex]I = p * q[/tex] for odd primes, p > q, what is [tex]\lambda[/tex]?

Can the Newton's Method solve equation zero? It is worth a try.

Relevant Reference Link:

'Newton's Method',

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

Re: Integer Factorization in PT (Polynomial Time)

Postby Guest » Thu Jul 30, 2020 12:17 pm

STOP!

Integer factorization in poynomial time is classified research. Please do not develop it further. You have done enough! 8)
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Re: Integer Factorization in PT (Polynomial Time)

Postby Guest » Wed Aug 05, 2020 8:54 pm

Update!

Equation 0: [tex]\sum_{k=1}^{\infty }\frac{sin(2 \pi k \sqrt{I + \lambda^{2}})}{k} = 0[/tex] if [tex]\sqrt{I + \lambda^{2}}[/tex] is a positive integer.

Our analysis is wrong! :cry: Where did we go astray?

Maybe it's time to retire... We are discouraged at this point.

And we shall focus our attention elsewhere.

We apologize for the flawed analysis. But all is not lost... And there's always hope.

And we expect no feedback from our readers (mathematicians). What a predicament! :cry:

Dave.

P.S. Yes, it's time to retire and focus our energies elsewhere. Goodbye!
Guest
 

Re: Integer Factorization in PT (Polynomial Time)

Postby Guest » Wed Aug 05, 2020 9:25 pm

Guest wrote:[tex]\sum_{k=1}^{\infty }\frac{cos(2 \pi k \sqrt{I + \lambda^{2}})}{k} = \infty[/tex] if [tex]\sqrt{I + \lambda^{2}}[/tex] is a positive integer.

The answer is the very important Harmonic Series! And that's all folks! :)
Guest
 

Re: Integer Factorization in PT (Polynomial Time)

Postby Guest » Fri Aug 07, 2020 9:49 pm

Guest wrote:Update!

Equation 0: [tex]\sum_{k=1}^{\infty }\frac{sin(2 \pi k \sqrt{I + \lambda^{2}})}{k} = 0[/tex] if [tex]\sqrt{I + \lambda^{2}}[/tex] is a positive integer.

Our analysis is wrong! :cry: Where did we go astray?

Maybe it's time to retire... We are discouraged at this point.

And we shall focus our attention elsewhere.

We apologize for the flawed analysis. But all is not lost... And there's always hope.

And we expect no feedback from our readers (mathematicians). What a predicament! :cry:

Dave.

P.S. Yes, it's time to retire and focus our energies elsewhere. Goodbye!


Hey Dave,

Your analysis is fine! :)
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