JSH: Factoring problem solution, latest objections

Some posters have STILL been attacking the simple solution to the factoring problem so I will answer their claims and first point out a crucial step that I caught one of them doing which was removing m_1 and m_2.

You'll notice in what follows that m_1 = floor(f_1*f_2/p_1) and m_2 = floor(g_1*g_2/p_2) where all the variables will be explained. What's important here is that the floor() function creates an integer requirement, and is key to solving the factoring problem.

Without it, there is no solution.

So a poster sneakily taking out m_1 and m_2 is removing the very basis for the solution.

With that said there IS a situation when you can not find specific solutions which I need to address, so it's worth addressing that as well and I'll do so below.

The full system again is

(f_1 + c_1*p_1)(f_2 + c_2*p_1) = T = r_1 + k_1*p_1

(g_1 + d_1*p_2)(g_2 + d_2*p_2) = T = r_2 + k_2*p_2

set T to your target, and pick two primes p_1 and p_2.

Now f_1, f_2, g_1 and g_2 are residues where

f_1*f_2 = T mod p_1 and g_1*g_2 = T mod p_2.

And it can be shown that solving for the factors reduces to finding integer solutions to a family of 4 equations and 4 unknowns, which are

c_1, c_2, d_1 and d_2:

k_1 = c_1*f_2 + c_2*f_1 + c_1*c_2*p_1 + m_1

k_2 = d_1*g_2 + d_2*g_1 + d_1*d_2*p_2 + m_2

(f_1 + c_1*p_1) = (g_1 + d_1*p_2)
(f_2 + c_2*p_1) = (g_2 + d_2*p_2)

where m_1 = floor(f_1*f_2/p_1) and m_2 = floor(g_1*g_2/p_2).

However, if the primes lack enough residues for all the solutions of T, then the math has no choice but to kind of loop on itself, and not allow you to solve for the c's or d's, BUT you can still solve in that case:

c_1 = (k_1 - m_1 - c_2*f_1)/(f_2 + c_2*p_1)

and

d_1 = (k_2 - m_2 - d_2*g_1)/(g_2 + d_2*p_2)
c_1 = (k_1 - m_1 - c_2*f_1)/(f_2 + c_2*p_1)

and

d_1 = (k_2 - m_2 - d_2*g_1)/(g_2 + d_2*p_2)

Those are just incredible equations, as they give you something you
can't have without the floor() function as otherwise the m's are
fractions.

Stupid simple is what the answer to the factoring problem is.

Stupid simple.
where you can factor the numerators in each case to find integers that will work, which is an approach that cannot work, you'll notice with the original equations as you are then just factoring T itself!

Here it works because

m_1 = floor(f_1*f_2/p_1)

and

m_2 = floor(g_1*g_2/p_2)

though if the f's and g's are wrong when you substitute back in before, you won't get T.

So m_1 and m_2 are crucial to the solution as floor() is a discrete function.

If you have selected the correct f's and g's then you will get integer solutions, otherwise you will not, so then you'd shift to another set, so there are

(p_1 - 1)(p_2 - 1)

MAXIMUM total checks without regard to the size of T.

Modern mathematicians unfortunately lie. Their system collapses if they are caught so they CANNOT celebrate a fantastically simple proof as I'm telling them that I'll inform the world that they lie all the time.

So it's a fascinating situation. I'm curious how long the world will let them lie about the factoring problem since they can collapse civilization with the lie.

Seems a few days already, which makes you wonder how much rich people really value their billions.

Maybe they don't really care if they end up being poor, soon.
What continues to amaze me though are the students.

Getting taught crap ideas from people who really are kind of dumb would upset me.

My guess is that it's about not wanting to accept having wasted so much of your lives with junk ideas, so you take in more junk.

It's like after being told you were eating crap for caviar you stuff yourself with more out of denial.

They did it on purpose.

I just want you to know that they had to know they were wasting your time, your life, and your mind, and they did so because they are parasitic.

Just con artists masquerading as mathematicians.

Nothing more.

[A reply to someone who told James that it was time for him starting to delete posts.]

Let's see how confident you are tomorrow.

The math is in the post.

The m's would be fractions without the floor() function.

Your objections all centered on removing the floor() function.

Sigh. If I'm wrong, then so what? It's just another mistake of mine.

But if I'm right, then the destabilization of society is just around the corner.

And then it won't matter what dreams you used to have as the civilization necessary to realize them, will no longer exist.