grscheller.boring-math 0.4.5

### Mathematical Libraries

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.

• Repositories
  • grscheller.boring-math project on PyPI
  • Source code on GitHub
• Detailed documentation
  • Detailed API documentation on GH-Pages

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

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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

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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, ...

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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

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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

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