grscheller.boring-math

### Mathematical Libraries


Keywords
math, mathematics, lcm, gcd, primes, comb, combinations, pythagorean, triples, ackermann, fibonacci
License
Apache-2.0
Install
pip install grscheller.boring-math==0.1.2

Documentation

Daddy's boring math library

Python package of modules of a mathematical nature. The project name was suggested by my then 13 year old daughter Mary.

Overview

Here are the modules and executables which make up the grscheller.boring-math PyPI project.

Library Modules

Integer Math Module

  • Number Theory
    • Function gcd(int, int) -> int
      • greatest common divisor of two integers
      • always returns a non-negative number greater than 0
    • Function lcm(int, int) -> int
      • least common multiple of two integers
      • always returns a non-negative number greater than 0
    • Function coprime(int, int) -> tuple(int, int)
      • make 2 integers coprime by dividing out gcd
      • preserves signs of original numbers
    • Function iSqrt(int) -> int
      • integer square root
      • same as math.isqrt
    • Function isSqr(int) -> bool
      • returns true if integer argument is a perfect square
    • Function primes(start: int, end: int) -> Iterator[int]
      • now using Wilson's Theorem
    • Function legendre_symbol(a: int, p: int) -> int
      • where p > 2 is a prime number
    • Function jacobi_symbol(a: int, n: int) -> int
      • where n > 0
  • Combinatorics
    • Function comb(n: int, m: int) -> int
      • returns number of combinations of n items taken m at a time
      • pure integer implementation of math.comb

Pythagorean Triple Module

  • Pythagorean Triple Class
    • Method Pythag3.triples(a_start: int, a_max: int, max: Optional[int]) -> Iterator[int]
      • Returns an iterator of tuples of primitive Pythagorean triples
    • A Pythagorean triple is a tuple in positive integers (a, b, c)
      • such that a² + b² = c²
      • a, b, c represent integer sides of a right triangle
      • a Pythagorean triple is primitive if gcd of a, b, c is 1
    • Iterator finds all primitive Pythagorean Triples
      • where 0 < a_start <= a < b < c <= max where a <= a_max
      • if max = 0 find all theoretically possible triples with a <= a_max

Recursive Function Module

Ackermann's Function

  • Function ackermann_list(m: int, n: int) -> int
    • an example of a total computable function that is not primitive recursive
    • becomes numerically intractable after m=4
    • see CLI section below for mathematical definition

Fibonacci Sequences

  • Function fibonacci(f0: int=0, f1: int=1) -> Iterator[int]

    • returns a Fibonacci sequence iterator where
      • f(0) = f0 and f(1) = f1
      • f(n) = f(n-1) + f(n-2)
    • yield defaults to 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
  • Function rev_fibonacci(f0: int=0, f1: int=1) -> Iterator[int]

    • returns a Reverse Fibonacci sequence iterator where
      • rf(0) = f0 and rf(1) = f1
      • rf(n) = rf(n-1) - rf(n-2)
        • rf(0) = fib(-1) = 1
        • rf(1) = fib(-2) = -1
        • rf(2) = fib(-3) = 2
        • rf(3) = fib(-4) = -3
        • rf(4) = fib(-5) = 5
    • yield defaults to 1, -1, 2, -3, 5, -8, 13, -21, ...

CLI Applications

Implemented in an OS and package build tool independent way via the project.scripts section of pyproject.toml.

Ackermann's function CLI scripts

Ackermann, a student of Hilbert, discovered early examples of totally computable functions that are not primitively recursive.

A fairly standard definition of the Ackermann function is recursively defined for m,n >= 0 by

   ackermann(0,n) = n+1
   ackermann(m,0) = ackermann(m-1,1)
   ackermann(m,n) = ackermann(m-1, ackermann(m, n-1))

CLI program ackermann_list

  • Given two non-negative integers, evaluates Ackermann's function
  • Implements the recursion via a Python array
  • Usage: ackerman_list m n

Pythagorean triple CLI script

Geometrically, a Pythagorean triangle is a right triangle with positive integer sides.

CLI program pythag3

  • Generates primitive Pythagorean triples
    • A primitive Pythagorean triple is a 3-tuple of integers (a, b, c) such that
      • a³ + b³ = c³ where a,b,c > 0 and gcd(a,b,c) = 1
    • The integers a, b, c represent the sides of a right triangle
  • Usage: pythag3 [m [n [max]]
    • 3 args print all triples with m <= a <= n and a < b < c <= max
    • 2 args print all triples with m <= a <= n
    • 1 arg prints all triples with a <= m
    • 0 args print all triples with 3 <= a <= 100