NZMATH Files

"""
The latest version of this file and the files in this directory are in
    https://sourceforge.net/p/nzmath/code/ci/default/tree/enttakagi/
however they are under construction so there are usually many bugs.
====================================
             enttakagi version 0.0.2
Part 1 (sec01-04) release 2024/03/28
Part 2 (sec05-07) release 2024/08/08
====================================
Lectures on Elementary Number Theory
      (2nd Edition) by TAKAGI, Teiji
           notebook in Python-NZMATH(*1)
====================================
The purpose of this notebook is an easy introduction to Algorithmic
Number Theory --- ANT.  You can study three topics
    'Number Theory'
    'Python Programming'
    'English for Mathematics'(*2)
at a time, which are necessary for ANT, only by running and reading the
programs sec01.py, ..., sec60.py in this directory.  For that, you need
three preparations:
    Get the book by Takagi in the title above.(*3)
    Let Python & package NZMATH usable on your machine.(*4)
    Download all files in this directory to your machine.(*5)
Then, on the command prompt or interactive shell, do
    $ python sec01.py
etc. and read the printed messages.(*6)  The programs themselves in the
files sec01.py, ... are easy English text to read.  That is all!(*7)
                        Tanaka, Satoru (TMCIT); NAKAMULA, Ken (TMU)(*8)
                                          2022/06/23 --- 2024/08/08
    (*1) Based on Python 3.9.16 and NZMATH 3.0.1.
    (*2) 'Japanese for Mathematics' to English readers.
    (*3) https://www.kyoritsu-pub.co.jp/book/b10011316.html
    (*4) https://www.python.org/ and https://nzmath.sourceforge.io/
    (*5) https://sourceforge.net/projects/nzmath/files/nzmath-enttakagi/
    (*6) When running programs, you are requested to 'Hit Return!'
         so that you can continue after reading the printed messages.
    (*7) Finding and fixing bugs of Python calculator NZMATH on ANT is
         another important purpose for us developers.
    (*8) Special thanks to MATSUI, Tetsushi; OGURA, Naoki; MIYAMOTO,
         Yasunori and others on ACKNOWLEDGEMENTS.txt in the directory
         https://sourceforge.net/p/nzmath/code/ci/default/tree/dist/
         Home Page (NAKAMULA) https://tnt-nakamu-lab.fpark.tmu.ac.jp/

<<<< CONTENTS (functions) >>>>
Chapter 1   Elementary Number Theory
====================================
Section 1   Divisibility of Integerse
------------------------------------------------------
    sec01.py    Thm1_01, Thm1_02 (divmod), Thm1_02_rem
Section 2   Greatest Common Divisors, Least Common Multiplese
------------------------------------------------------------------------------
    sec02.py    Thm1_03, Thm1_04, Thm1_05 (lcm), Thm1_06,
                Prob1 (gcd, Eucledean Algorithm == GCD),
                Prob1_rem (gcd_, general GCD by modl),
                Prob1_rem_eg, Prob2 (lcm_, by gcd_)
Section 3   Linear Indeterminate Equationse
------------------------------------------------------------
    sec03.py    Thm1_07 (gcd_of_list, general extended GCD),
                Thm1_07_eg (general extended GCD by extgcd_gen), Prob1,
                <<skip Prob2>>
Section 4   Prime Numbers
-----------------------------------------
    sec04.py    Thm1_08, Thm1_09 (prime factorization)
    sec04a.py   Prob1 (tau function), Prob2 (sigma function),
                Prob3 (multiplicative function), Prob4,
                PerfNumb (perfect number),
                Prob5 (Mersenne number, Lucas-Lehmer test)
    sec04b.py   Prob6, Prob7, gcdlcmFI, Prob8, Prob9, Prob10, Prob11
                    (FactoredInteger plays important role)
    sec04c.py   Prob12, Prob12_rem, Prob13, Prob14, Prob14_rem
                    (binomial coefficients, partial fraction decomposition)
    sec04d.py   Thm1_10, PrimeTable (eratosthenes sieve),
                Thm1_10_rem (primes of arithmetic progression),
                PrimeNumberTheorem (prime.generator), 
                Tschebyschef (Tschebyschef, nextPrime),
                twinPrime (twin prime), gapPrime, Goldbach (Goldbach conjecture)
Section 5   Congruences
-----------------------
    sec05.py    CongEquiv, SysRes_eg, Thm1_11, Thm1_12
                    (fundamental arithmetic of congruence)
                Prob1 (congruence arithmetic by digital criterion)
Section 6   Congruences of Degree One
-------------------------------------
    sec06.py    Thm1_13 (e1_ZnZ, extended GCD), Thm1_13_rem, Thm1_13_eg
    sec06a.py   Thm1_14 (CRT_, Chinese Remainder Theorem),
                Thm1_14_eg (CRT_Gauss by moduli symmetric),
                Thm1_14_rem (partial fraction decomposition)
    sec06b.py   Prob1 (CRT of moduli m, n, GCD(m, n) > 1),
                Prob2 (CRT of general non-coprime moduli),
                <<skip Remark on Ring of Fraction>>
Section 7   Introduction to Solving Congruences
-----------------------------------------------
    sec07.py    Thm1_15 (allroots_Fp, cyclicity of  Fp*),
                Thm1_15_rem (digital method, Horner),
                Thm1_16 (lift up prime power modulus, Hensel lift), Thm1_16_eg
    sec07a.py   Prob1, Thm1_17 (allroots_ZnZ, complete solutions modulo n)
                    (allroots_Fp, liftup_ZpnZ, CRT_, allroots_ZnZ)

Bugs of imported NZMATH functions & classes / newly defined ones
================================================================
    See fix_nzmath.py about everything on this matter!

List of utility functions of utils.py
=====================================
    HitRet, again, strInt, randFactLists, randmodPolyList, printPoly
    lcm_def, allDivisors_def, gcd_def, countDivisors_def, sumDivisors_def
    allroots_ZnZ_def
=========
Copyright
=========
The package is a part of NZMATH, and is distributed under the BSD
license.  See https://nzmath.sourceforge.io/LICENSE.txt for detail.
"""