Another misleading title, but it is fun to write it down so why not?

In this post (8 pictures long) we have two parts:
Part 1: The relation between the modulo row’s and the modular arithmetic groups Z/jZ.
Part 2: A proposal (or schematic outline) of an important part of the algorithm that brings you from one stratum to the other.

I think this is my last post on this subject of modulo row’s.

Lately websites using RSA encryption methods (that is why we look at large prime numbers made of two factors N = p*q) have gone from a 1048 bit long key to 2096 binary digits long keys. The idea is that it makes life just so much more safe; but the important part of the algorithm for transport over the strata is remarkable resilient towards such moves…
Furthermore, doubling the length of the encryption keys (squaring the size so to say) will in general also increase the size of the Jente basin as found just before the largest prime factor q.

I do not claim to know a lot about encryption, but as far as I know there is zero point zero use of idea’s like the Jente basin. People use a lot of so called ‘trial division’ but even that is not a real division but mostly just taking N modulo something.
For example; want to know if the number 73 is a factor of N?
They simply calculate N mod 73 and if the outcome is 0 they say that 73 is a factor of N, otherwise it is not. The use of idea’s like a Jente basin so you can scrap a lot of trial numbers in the region you are in is, as far as I know, not used.

To be honest, I also do not know how they factor large hundreds of decimal digits long numbers anyway; so it might very well be they use similar idea’s. But if that were the case why is everybody else only talking about taking a huge number of trial divisions without any strategy behind that?

The numbering of the pictures is a continuation of the previous post.
Here are the 8 pictures, have fun reading it.

At this point in time the so called quantum computers are going from the lab to the field, anyway a lot of people claim this. But since after my humble opinion electrons have one electric charge and one of two possible magnetic charges, it will be a long long time before we have a working quantum computer based on electron spin.
Just like IBM with their racetrack technology for 3D memory; the idea is ok but at IBM too they think electron spin a like a vector and not like a charge. And voila; year in year out you never hear from it again…

Ok, for the time being this is what I had to say. See you in the next post.

Ok ok, I admit instantly that the title of this post is way over the top but for once I allow myself a catchy title that has only limited resemblance to what this post is about. In this post, if I write the word computer I always mean a classical computer so not a quantum version of it.

In the previous post there is a video in from the ‘Infinite Series’ that serves as an introduction to the Shor algorithm; if this algorithm could be implemented into a quantum computer that would likely break internet security for a short while. Beside the fact that large prime numbers are used in standard classical encryption, it can also be done with elliptic curves.

This post is about the principle of Jente, with a bit of luck you can find factors of large numbers using the principle of Jente. Counter intuitively the largest (prime) factor will be the easiest to find.
Now how did Jente find the principle of Jente?
Back in the time, end 1997 or begin 1998, we lived in a house without a garden and since I still smoked a lot of tobacco I always had a window open in my working room. Since this work room was next to the entry of the house, very often when the door to my room opened papers would fly from my desk because of wind going through the room.

There was this cute baby crawling around and one day she brought me back a piece of paper that had flown off my desk. And on that piece of paper was a little cute formula that read
m_{j+1}  = m_j – d_j. So that is how this got the name the principle of Jente.

Lately Jente turned 21 years of age, she now lives temporary in Australia, and I decided to write this old stuff down as a kind of present for her. The principle of Jente is extremely easy to understand, but as far as I know mathematical reality this principle has not been exhausted very much by the entire math community over centuries of time.

What is missing in this post is a way to converge fast with high speed to one of the factors of one of those huge composite numbers the software engineers use for internet security. My gut feeling says that it should not be that hard but until now I have never found it. It might very well be that inside things like Diophante equations somewhere the solution to this problem of fast finding the largest prime factor is solved without the person who has done that being aware of it…

I tried to keep this post as short as possible so I scrapped a whole lot of stuff but it is still 15 pictures long (picture size as usual 550 x 775 pixels). A feature that I like very much is that I am using so called Harry Potter beans in order to explain as why the Jente principle works. I feel a bit proud on that because it is so simple you could explain that to elementary pupils in their highest years.

For myself speaking I also like this approach to finding prime factors because it is so different from all other ways, yet it has that underlying undeniable thing in it named the Jente principle. The most important detail in this post is the table with the diagonals in it.
If you understand that table and, for example, you can find another algorithm for quantum computers that solves that problem, you have found an alternative to the Shor algorithm…

Have fun reading it, take your time because it is not meant to be grasped in five minutes or so.

I hope you understand the fundamental problem still open after almost two decades:

You start with some number j, calculate m_j = N mod j and d_j = N div j.
Having these, the Jente principle guarantees you can find (j + k) mod N for all k > 0.

But, how oh how, do you converge towards a solution of
m_{j+k} = 0 mod (j+k) ?????

__________

The Shor algorithm: In the world of quantum computing we have the theoretical side where people just write down all kinds of elaborate scheme’s like the Shor algorithm and just as easy they throw in a lot of Hadamard gates that supposedly will bring a giant bunch of quantum bits into super position.

On the other hand you have people that actually try to build quantum computers.

As far as I know stuff, there is no way of bringing a lot of qbits into a nice super position or, for that matter, entangle them into a good initialization state in order to run your quantum software.

More info:

Hadamard transform
https://en.wikipedia.org/wiki/Hadamard_transform

Shor’s famous algorithm: Shor’s algorithm
https://en.wikipedia.org/wiki/Shor%27s_algorithm

Elliptic Curve Cryptography: a gentle introduction
http://andrea.corbellini.name/2015/05/17/elliptic-curve-cryptography-a-gentle-introduction/

__________

Ok, that was it. Don’f forget to pop open a few beers. Don’t believe all that nonsense that doctors are telling you like drinking less = good.
As far as I know reality, all people in my social environment that drink far too little beer always get killed in extremely violent events… 😉

Till updates.

A few months back suddenly there was a new video channel about math and it goes under the cute name Infinite Series. About two months back the channel posted a way to destroy internet security if you could only find that factorization of two giant prime numbers.

Most of present internet security hangs around the difficulty of observing a giant number N of, let’s say, one hundred digits and our incapability to factorize large numbers like that into their prime factor numbers.

Of course, since the Infinity Channel is USA based, it is completely impossible that fresh math will come from that space. Here is the video and indeed only ancient math is around:

How to Break Cryptography | Infinite Series

The idea’s as expressed in the video are very interesting, but is just does not use the Jente principle that ensures you can find weakness in the integers surrounding the prime numbers that make up the factorization of the stuff you want to encrypt.

In the next two pictures you see that a prime number is extremely weak in avoiding detection using the Jente principle if you are close enough to that prime number.

And if a prime number is detected, in principle you could break down the security of the communication channel.

Let’s leave it with that, after all talking about a basin around a prime number that shouts out ‘the prime number is here’ is one hundred percent outlandish to those overpaid USA math professors…

End of this teaser post, I hope I have some more next week so see you around!

In the previous post we had some stuff on the pull back map but also those links to complicated theorems. Therefore I tried to explain the inner workings of the pull back map that pull higher dimensional complex & circular numbers back to the complex plane in as easy to understand chunks as I could.

In the next post I only use advanced high school math (for my own country that would be the VWO education line, the classes 5 and 6) and for the rest any university student that has followed an elementary crash course on the complex plane.

I am very glad I could find such simple ways to pull back higher dimensional exponential circles and curves back to that goodie good old Euler formula that says stuff like e^it is related to the cosine and the sine functions.

All stuff is boiled down to things you can see in the teaser picture below, no new advanced 20th century math ideas, only using century old well known trigonometric equations and that is all…

Once more: Higher dimensional complex number systems are just there, it is a natural thing like the natural numbers like 1, 2, 3,  4, 5 etc are. Where the complex plane is something like a fish bowl, the higher dimensional complex & circular numbers are a big ocean.
But if you as a so called professional math professor can only swim your circles inside the fish bowl, can you survive the currents in this giant ocean?

No of course you can’t, so good luck with your future life inside the complex plane.

After this blah blah blah (remark the math professors are also extremely smart if you look at how much salary they suck in let alone the ‘research money’ they get to form global research groups that use at best two dimensional complex numbers) it is time for the teaser picture:

At last I would like to remark that the pull back map is on equal footage with the modified Dirichtlet kernels for my individual emotional system; I am glad I am still alive and can find stuff like this.

Till updates.

Originally I planned to upload tonight the new post on the integrals related to the number tau for the circular multiplication. But I found this very cute result from some other math professors, I believe these are two brothers Borwein & Borwein.

Beside that I also had more time to spend on a very important hobby: Brewing beer…;)

Four years back when I for the first time derived integrals like this with the cosine and sine stuff to the power three in it, I just had no clue whatsoever how to find analytical stuff for their value. These kind of integrals cannot be solved by throwing in some simple primitive or so.

At present I have two independent proofs for their value.
Back in the time I knew there was some internet website that contains a whole lot of integer sequences so if I could find that I would have at least some analytical clue about that nasty problem. Only a long time later I found that website, but is said ‘we do not know’.
Or ‘unknown integer sequence’ or whatever what.

But yesterday when I tried more or less to get a negative result my luck changed for the better: the website with the integer sequences in it actually returned an answer.

And for my few pounds of human brain tissue the answer was completely crazy.
Therefore I decided to put the result of this Borwein function on top in the teaser picture and my own idea’s at the bottom. Here it is:

I have absolutely no clue as why these two things should be the same, but four years back I had absolutely no clue as what this numerical value like 1.2092 actually meant…

The link to what might be the Borwein & Borwein function

A248897 Decimal expansion of Sum_{i >= 0} (i!)^2/(2*i+1)!.
https://oeis.org/A248897

Let’s leave it with that, see ya in the next post.

I know I know, not posted much around here lately.
And on top of that I found some serious typo’s in the pictures from just two posts back: the post upon the factorization on the Laplacian operator.

These typo’s are rather serious; we just cannot have all those wrong and misleading differentiations going not repaired.

But reparation of old faults is time consuming.

On the other hand, the curl operator has a lot of fresh new insights in simple number systems like the 2D complex plane or my own hobby; the 3D complex number system.

For myself speaking I am still wondering as shall I include the ‘Theorem without Vodka’ in the new post or just leave it out? I do not know what I will do in the future but here is the Google translate version of the Theorem without Vodka:

May be I just leave it out, after all this is supposed to be an update on the differential curl operator.

End of this post, till updates.