Cyclic prime numbers

This is aplication to visualise few of mathematical regularities related to the themes:

• Cyclic numbers
• Full reptend prime numbers

And it show how cyclic prime numbers class appears, was used to write a paper:

https://arxiv.org/abs/2105.04400

https://habr.com/ru/post/555548/

https://habr.com/ru/post/556564/

https://habr.com/ru/post/672832/

History

Started as C++ console apliction, then was built as C++ with Qt GUI, later most of the code was migrated into Python. It was a part of another project, but not it reduces to this very few pages and components.

Usage

To start run 'python ath.py'

Aplication is a browser that can open qml files located in folders components and pages. You can edit any page on the go, if MD5 of page becomes different engine would reload it.

There are few prepared pages, first of them opened as default, rest could be opened using button '+'.

Description of pages and components:

• There are many pages that use digital circle component, that helps visualize periodic fractions

• PrimeScales - shows length of periods for 1/P in different Numeric Systems, useful to see many full reptend primes at once, and structure of their appearence

Clicking on each cell you can find a fraction made from 1/P in certain numeric system, then you can jump from it into different pages

• FullRepentPrime - shows some regularities of full reptend prime numbers - it may be opened from PrimeScales page

• ManyRationals - interesting animation, that represents the structure that appears in cyclic numbers, it is called so because it contains at least P-1 animations of rationals made from 1/P to P-1/P

• GeometricProgression - component for visualisation of infinite geometric progressions that 1/P can be represented

• Calculating cyclic prime numbers - used from console, run python ath.py -c P [N], where P is prime number and N is numeric system, if it is absent there would be a search for multiple numeric systems

• Cyclic primes - makes almost the same thing but not from console, it has limits for calculations, as it cannot be too long in GUI case

• NumericSystem - simple page for converting one number from one numeric system into another

• PrimeFactor - page that can factor integer number, or check first N digits from 1/P finging its prime factors

Requirements

PySide2 is not possible to deliver for ARM MacOs, so I switched into using PySide6.

pip install PySide6
pip install factordb-pycli
pip install sympy
pip install gmpy2

FactorDB - is a library for prime numbers factorisation, it uses connection to internet. Sympy is optional and used for prime factorisation of small numbers, but usage of full reptend prime page is much slower without it. Also you may epxerience issue of installing gmpy2 on windows, in this case you may use unofficial wheel: https://www.lfd.uci.edu/~gohlke/pythonlibs/

You may also experience issues with PySide2 using 64 bit Python interpriter on Windows, so suggestion is to use 32bit version.

For using midi features you will also need to instal midiutil

pip install midiutil

Midi options are availible on IntervalScales component, and can be accessed through the "Save midi" button.

It accessible on IntervalScales component.

Also generation availible on page MusicFromFractions.

Another option is to generate midi from fractions in certain numric system.

python ath.py -midi num denom numericSystem [all/else] [local/sum/diff]

if argument "all" passed - then there would be done generation from all rationals: 1/denom .. denom-1/denom. In other case generation will be done only for num/denom in certain numeric system.

Arguemnts "local/sum/diff" make different ways to produce midi notes values;

There is a component that allows you to rotate periodic fraction on digit circe, and generate animation from all of possible rotations.

There is a component that helps you to generate animation for all of 1/N .. N-1/N periodic fractions on digit circle. And same component helps to generate 'oroboric' and 'non-oroboric' connections animation.

Project was running at Linux/MacOs/Windows - but the very last version tested only on Windows.

Excuses

This project is far from being nice coded, and right now it mostly abadoned.

In this repository may appear later some new work related to full reptend prime numbers and geometric progressions, but the rest of the code is not going to be supported in Python.

This code is published for education purpose, as a support for the paper 'Cyclic prime numbers'.

Originaly project was planned as part of interactive engine that was developed for chat messaging, this project was failed to support using Python on mobile devices. For now it is abadoned - but if it would be reincarneded link would appear here. I tried my best to make it nice, but with time limits its still far from I wanted.

Anyway, I hope you will enjoy the mathematics behind this code :)

Notice

This repo is a clone of https://github.com/eversearch/cyclicprime

Both repositories would be present for historical resons, but further development would appear in current one.

Cyclic prime numbers propbably wouldn't be involed in this code, but few other themes for following papers would be done here.

There is also repo for OEIS sequence generation related to cyclic prime numbers: https://github.com/constcut/cyclicprime-cli/blob/main/cyclicprime.py

Yet a little abadoned :)