commit state before changing what noether considers primitive roots

README

@@ -0,0 +1,19 @@
+Noether is a project I plan to complete over a LONG period of time (years).
+A collection of useful commands for solving various math problems.
+
+A personal little MATLAB alternative I suppose :)
+
+
+In future I'd love to start using a developed TUI library like [textual](https://github.com/Textualize/textual?tab=readme-ov-file),
+but for now everything is just a custom ANSI wrapper thing.
+
+
+
+NOTES:
+1. could I define something like "primality classes"? 
+	ie   class one: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
+	     class two: 3, 5, 11, 17, 31, ... (class 1 with a prime index, ie 11 is the 5th class 1 prime)
+	   class three: 5, 11, 31, ... (class 2 with a prime index, ie 11 is the 3rd class 2 prime)
+
+2. figure out all possible patterns that can appear amongst primitive roots
+	can we then categorise primes by their primitive root behaviour?

prbraid/math.py

@@ -1,24 +0,0 @@
-# Euler's Totient (Phi) Function
-def totient(n):
-    phi = int(n > 1 and n)
-    for p in range(2, int(n ** .5) + 1):
-        if not n % p:
-            phi -= phi // p
-            while not n % p:
-                n //= p
-    #if n is > 1 it means it is prime
-    if n > 1: phi -= phi // n 
-    return phi
-
-
-def orbit(b, m):
-    generated = []
-    x = b
-    for i in range(m):
-        x = (x * b) % m
-        if x not in generated:
-            generated.append(x)
-    return generated
-
-def order(b, m):
-    return len(orbit(b, m))

py/NOTES

@@ -0,0 +1,8 @@
+TODO:
+NOTE: currently my "primitive roots" are actually just the numbers that generate every integer below the modulus, that is `a` such that `orbit(a) = Z_n`
+1. How I'm calculating something as a primtive root, and its order is not correct.
+	For instance, consider a modulus n=4, a=3 is a primitive root, and a=2 SHOULD have order 0 (it is aperiodic) 
+
+
+This site is useful as a reference:
+	https://owlsmath.neocities.org/Primitive%20Root%20Calculator/calculator

py/mquiet.py

@@ -0,0 +1,76 @@
+# Modulo Test
+from prbraid.math import *
+from prbraid.color import *
+
+import sys
+
+PROMPT = '[n]: '
+
+'''
+Pairwise orbit cumulative summation (cum orb) 
+'''
+def orb_cum(cum, orb):
+    for i in range(len(cum)):
+        cum[i] += orb[i]
+
+def main():
+    while True:
+        n = None
+        strlen_n = None
+        try:
+            uprint(PROMPT, color=Color.Blue, end='')
+            n = input()
+            strlen_n = len(n)
+            n = int(n)
+        except ValueError:
+            continue
+
+        # calculate phi of n
+        phi = totient(n)
+        # determine if n is prime
+        prime = n - 1 == phi
+        if prime:
+            uprint('', moveup=1, end='', flush=False)
+            column = len(PROMPT) + strlen_n + 1
+            sys.stdout.write(f'\033[{column}C')
+            uprint('[PRIME]', color=Color.Magenta, style=Color.Bold, flush=True)
+        # primitive root values
+        proot_v = []
+        # primitive root count
+        proot_c = 0
+        # cumulative sum of primitive root orbits
+        prorb_cum = []       
+  
+        # find all invertible elements (skipped for now)
+        for a in range(n):
+            orb, ord = orbit(a, n)
+            # check if `a` is a primitive root
+            proot = (ord + 1 == n)
+            proot_c += proot 
+
+            if proot:
+                proot_v.append(a)
+                if not prorb_cum:
+                    prorb_cum = orb
+                else:
+                    orb_cum(prorb_cum, orb)
+                print(a)            
+        uprint('Cum Orb:   ', end='', color=Color.Cyan)
+        uprint(f'{prorb_cum}', flush=True)
+        prorb_cum_mod = [x % n for x in prorb_cum]
+        uprint(f'           {prorb_cum_mod}', flush=True)
+
+        uprint('Roots:     ', end='', color=Color.Cyan)
+        uprint(proot_v, flush=True)
+        root_delta = [proot_v[i+1] - proot_v[i] for i in range(proot_c - 1)]
+        uprint('Delta:     ', end='', color=Color.Cyan)
+        uprint(root_delta, flush=True)
+        
+        uprint('Roots/Phi: ', end='', color=Color.Cyan)
+        uprint(f'{proot_c}/{phi}\n', flush=True)
+
+if __name__ == '__main__':
+    try:
+        main()
+    except (KeyboardInterrupt, EOFError):
+        pass     

py/noether.py

@@ -1,11 +1,20 @@
-# Modulo Test
+#!/usr/bin/env python3
-from prbraid.math import *
-from prbraid.color import *
-
 import sys
+import readline
+
+from noether.math import *
+from noether.cli import *
+
 
 PROMPT = '[n]: '
 
+'''
+Pairwise orbit cumulative summation (cum orb) 
+'''
+def orb_cum(cum, orb):
+    for i in range(len(cum)):
+        cum[i] += orb[i]
+
 def main():
     while True:
         n = None
@@ -22,13 +31,17 @@ def main():
         phi = totient(n)
         # determine if n is prime
         prime = n - 1 == phi
-        # sys.stdout.write('\x1b[1A')
-        # sys.stdout.flush()
         if prime:
             uprint('', moveup=1, end='', flush=False)
             column = len(PROMPT) + strlen_n + 1
             sys.stdout.write(f'\033[{column}C')
             uprint('[PRIME]', color=Color.Magenta, style=Color.Bold, flush=True)
+        # primitive root values
+        proot_v = []
+        # primitive root count
+        proot_c = 0
+        # cumulative sum of primitive root orbits
+        prorb_cum = []
         
         # calculate left padding to align i values
         # lpadded = strlen_n
@@ -37,11 +50,11 @@ def main():
     
         # find all invertible elements (skipped for now)
         for a in range(n):
-            orb = orbit(a, n)
+            orb, ord = orbit(a, n)
-            ord = len(orb) 
 
             # check if `a` is a primitive root
             proot = (ord + 1 == n)
+            proot_c += proot 
 
             # calculate padding
             lpad = ' ' * (strlen_n - len(str(a)))
@@ -51,20 +64,39 @@ def main():
             color_ord = None
             color_orb = None
             style_ord = None
-            style_orb = None
+            style_orb = None                
             if proot:
                 color_a = Color.Green
                 color_ord = Color.Yellow
                 color_orb = Color.Green
                 style_ord = Color.Bold
                 style_orb = Color.Bold
+                proot_v.append(a)
+                if not prorb_cum:
+                    prorb_cum = orb
+                else:
+                    orb_cum(prorb_cum, orb)
+            elif gcd(a, n) == 1:
+                color_a = Color.Yellow
             
             uprint(f'{lpad}{a}', color=color_a, style=Color.Bold, end=' ', flush=False)
-            uprint(f'->', end=' ', flush=False)
+            uprint('->', end=' ', flush=False)
             uprint(f'{ord}{rpad}', color=color_ord, style=style_ord, end=' ', flush=False)
-            uprint(f'|', end=' ', flush=False)
+            uprint('|', end=' ', flush=False)
             uprint(f'{orb}', color=color_orb, style=style_orb, flush=True)
-        print() # empty new line
+        uprint('Cum Orb:   ', end='', color=Color.Cyan)
+        uprint(f'{prorb_cum}', flush=True)
+        prorb_cum_mod = [x % n for x in prorb_cum]
+        uprint(f'           {prorb_cum_mod}', flush=True)
+
+        uprint('Roots:     ', end='', color=Color.Cyan)
+        uprint(proot_v, flush=True)
+        root_delta = [proot_v[i+1] - proot_v[i] for i in range(proot_c - 1)]
+        uprint('Delta:     ', end='', color=Color.Cyan)
+        uprint(root_delta, flush=True)
+        
+        uprint('Roots/Phi: ', end='', color=Color.Cyan)
+        uprint(f'{proot_c}/{phi}\n', flush=True)
 
 if __name__ == '__main__':
     try:

py/noether/ansi.py

@@ -0,0 +1,172 @@
+'''
+=== ANSI Escape Sequences ===
+This file exists to organise and name
+all important ANSI escape sequences
+for management of the CLI.
+'''
+
+from sys import stdout
+from enum import StrEnum
+
+RESET = '\x1b[0m'
+
+class Color(StrEnum):
+    BLACK   = '\x1b[30m'
+    RED     = '\x1b[31m'
+    GREEN   = '\x1b[32m'
+    YELLOW  = '\x1b[33m'
+    BLUE    = '\x1b[34m'
+    MAGENTA = '\x1b[35m'
+    CYAN    = '\x1b[36m'
+    WHITE   = '\x1b[37m'
+
+class Style(StrEnum):
+    BOLD      = '\x1b[1m'
+    DIM       = '\x1b[2m'
+    ITALICS   = '\x1b[3m'
+    UNDERLINE = '\x1b[4m'
+    BLINK     = '\x1b[5m'
+    REVERSE   = '\x1b[7m'
+    HIDE      = '\x1b[8m'
+
+    _DISABLE = [
+        '\x1b[21m', # BOLD
+        '\x1b[22m', # DIM
+        '\x1b[24m', # UNDERLINE
+        '\x1b[25m', # BLINK
+        '\x1b[27m', # REVERSE
+        '\x1b[28m'  # HIDE
+    ]
+
+'''
+Implements cursor movement functionality.
+NOTE: 
+    The Cursor class currently has no ability
+    to handle EXACT line (row) numbers. Currently
+    I'm assuming all functionality of noether can
+    be implemented solely via relative movements.
+'''
+class Cursor(StrEnum):
+    # SAVE current / RESTORE last saved cursor position
+    SAVE = f'\x1b[7'
+    RESTORE = f'\x1b[8'
+    
+    _MV_UP     = 'A'
+    _MV_DOWN   = 'B'
+    _MV_LEFT   = 'C'
+    _MV_RIGHT  = 'D'
+    # NEXT/PREV are the same as DOWN/UP (respectively)
+    # except that the cursor will reset to column 0
+    _MV_NEXT   = 'E' 
+    _MV_PREV   = 'F'
+    # move cursor to the start of the current line
+    _MV_START  = '\r' # there is no ESC CSI for carriage return
+    
+    _MV_COLUMN = 'G'
+    
+    '''
+    Generates an ESC code sequence corresponding
+    to a horizontal movement relative to the 
+    cursor's current vertical position.
+        +ive = right
+        -ive = left   
+    '''
+    @staticmethod
+    def _XMOVE_REL(n: int) -> str:
+        if n > 0:
+            return f'\x1b[{n}{Cursor._MV_RIGHT}'
+        if n < 0:
+            return f'\x1b[{n}{Cursor._MV_LEFT}'
+        return ''
+
+    '''
+    Generates an ESC code sequence corresponding
+    to a horizontal movement to an EXACT column.  
+    '''
+    @staticmethod
+    def _XMOVE(n: int) -> str:
+        return f'\x1b[{n}{Cursor._MV_COLUMN}'
+
+    '''
+    Generates an ESC code sequence corresponding
+    to a vertical movement relative to the 
+    cursor's current vertical position.
+        +ive = down
+        -ive = up      
+    '''
+    @staticmethod
+    def _YMOVE_REL(n: int, reset: bool = False) -> str:
+        if n > 0:
+            if reset:
+                return f'\x1b[{n}{Cursor._MV_NEXT}'
+            return f'\x1b[{n}{Cursor._MV_DOWN}'
+        if n < 0:
+            if reset:
+                return f'\x1b[{n}{Cursor._MV_PREV}'
+            return f'\x1b[{n}{Cursor._MV_UP}'
+        return ''
+
+    '''
+    Sets the cursor column (horizontal) position to an exact value.
+    NOTE: does NOT flush stdout buffer
+    '''
+    @staticmethod
+    def set_x(n: int) -> None:
+        stdout.write(Cursor.XMOVE(n))
+
+    '''
+    Moves the cursor left/right n columns (relative).
+    NOTE: does NOT flush stdout buffer
+    '''
+    @staticmethod
+    def move_x(n: int) -> None:
+        stdout.write(Cursor._XMOVE_REL(n))
+    
+    '''
+    Moves the cursor up/down n rows and resets the
+    cursor to be at the start of the line
+    NOTE: does NOT flush stdout buffer
+    '''
+    @staticmethod
+    def move_y(n: int, reset: bool = True) -> None:
+        stdout.write(Cursor._YMOVE_REL(n, reset=reset))
+
+    '''
+    Saves the current cursor position.
+    NOTE: does NOT flush stdout buffer
+    '''
+    @staticmethod
+    def save() -> None:
+        stdout.write(Cursor.SAVE)
+
+    '''
+    Restores the cursor position to a saved position.
+    NOTE: does NOT flush stdout buffer
+    '''
+    @staticmethod
+    def restore() -> None:
+        stdout.write(Cursor.RESTORE)
+
+'''
+Handles erasing content displayed on the screen.
+NOTE that the cursor position is NOT updated
+via these sequences. The \r code should be given after.
+'''
+class Erase(StrEnum):
+    # erase everything from the current cursor
+    # position to the START/END of the screen
+    ER_SCR_END    = '\x1b[0J'
+    ER_SCR_START  = '\x1b[1J'
+    # ER_SCR_ALL    = '\x1b[2J'
+    ER_SCR_ALL    = '\x1b[3J' # erase screen and delete all saved cursors
+    
+    # ER_SAVED      = '\x1b[3J' # erase saved lines
+
+    # erase everything from the current cursor
+    # position to the START/END of the current line
+    ER_LINE_END   = '\x1b[0K'
+    ER_LINE_START = '\x1b[1K'
+    ER_LINE_ALL   = '\x1b[2K'
+
+    @staticmethod
+    # TODO COME BACK HERE

py/noether/math.py

@@ -0,0 +1,42 @@
+from math import gcd
+
+# Euler's Totient (Phi) Function
+def totient(n):
+    phi = int(n > 1 and n)
+    for p in range(2, int(n ** .5) + 1):
+        if not n % p:
+            phi -= phi // p
+            while not n % p:
+                n //= p
+    #if n is > 1 it means it is prime
+    if n > 1: phi -= phi // n 
+    return phi
+
+def is_prime(n):
+    return n - 1 == totient(n)
+
+def orbit(b, m):
+    orb = []
+    ord = 0
+    x = 1 # start at identity
+    for i in range(m):
+        x = (x * b) % m
+        if x in orb:
+            break        
+        orb.append(x)
+        ord += 1
+    return orb, ord
+
+def order(b, m):
+    return len(orbit(b, m))
+
+
+'''
+Functions for Testing my Conjectures 
+
+The conjecture is specifically that, for all positive integers n
+where n + 1 is prime, then conj_phigcd(n) is also prime.
+'''
+def conj_phigcd(n):
+    phi = totient(n)
+    return phi / gcd(n, phi) - 1

py/primes.py

@@ -0,0 +1,80 @@
+# Modulo Test
+from prbraid.math import *
+from prbraid.color import *
+
+import sys
+from math import gcd
+from time import sleep
+
+PROMPT = '[n]: '
+
+'''
+Modify the body of this function, it will be run
+against every prime number ascending.
+'''
+def test_function(p: int, phi: int):
+    p_color = Color.Green
+    result_color = Color.Yellow
+    
+    result = phi / gcd(p, phi) - 1
+    if result < 0 or not is_prime(result):
+        p_color = Color.Red
+        result_color = Color.Red
+    uprint(p, color=p_color, style=Color.Bold, end=' -> ', flush=False)
+    uprint(result, color=result_color, flush=True)
+    sleep(0.1)
+    
+
+def main():
+    n = -1
+    while True:
+        n += 1
+
+        # calculate phi of n
+        phi = totient(n)
+        # determine if n is prime
+        prime = n - 1 == phi
+        if not prime:
+            continue
+        
+        # # primitive root values
+        # proot_v = []
+        # # primitive root count
+        # proot_c = 0
+        # # cumulative sum of primitive root orbits
+        # prorb_cum = []       
+  
+        # # find all invertible elements (skipped for now)
+        # for a in range(n):
+        #     orb, ord = orbit(a, n)
+        #     # check if `a` is a primitive root
+        #     proot = (ord + 1 == n)
+        #     proot_c += proot 
+
+        #     if proot:
+        #         proot_v.append(a)
+        #         if not prorb_cum:
+        #             prorb_cum = orb
+        #         else:
+        #             orb_cum(prorb_cum, orb)
+        #         print(a)            
+        # uprint('Cum Orb:   ', end='', color=Color.Cyan)
+        # uprint(f'{prorb_cum}', flush=True)
+        # prorb_cum_mod = [x % n for x in prorb_cum]
+        # uprint(f'           {prorb_cum_mod}', flush=True)
+
+        # uprint('Roots:     ', end='', color=Color.Cyan)
+        # uprint(proot_v, flush=True)
+        # root_delta = [proot_v[i+1] - proot_v[i] for i in range(proot_c - 1)]
+        # uprint('Delta:     ', end='', color=Color.Cyan)
+        # uprint(root_delta, flush=True)
+        
+        # uprint('Roots/Phi: ', end='', color=Color.Cyan)
+        # uprint(f'{proot_c}/{phi}\n', flush=True)
+        test_function(n, phi)
+
+if __name__ == '__main__':
+    try:
+        main()
+    except (KeyboardInterrupt, EOFError):
+        pass