Statement

Number Theory & Algorithms Collection: Statement of Work (SOW) ?????
The following paper presents a summary of the scope, objectives, algorithms, and technical specifications of a complete software repository under the general title "Number Theory & Algorithms Collection" to deal with properties of integers and computational number theory.
1. Overview and Project Objectives

The collection below consists of 34 self-contained Python modules and demonstrates advanced algorithmic skills with a focus on mathematical rigor, efficiency, and clarity of the code.

Primary objectives are to:

To implement a complete catalog of algorithms ranging from basic number manipulation and prime number theory through cryptography to discrete mathematics.

Ensure all solutions operate using only native Python Standard Library-math, time, sys, and random Guaranteed to have zero external dependencies.

Include performance profiling out-of-the-box by using time and sys/tracemalloc to measure execution speed and memory overhead for most of the scripts.

2. Scope of Algorithms- Complete Catalog

Core concepts and algorithms to be used in the implementation are as follows:

A. Divisibility and Digit Properties

Concept	Algorithm Examples

Combinatorics

factorial(n)

Digit Manipulation



is_palindrome(n), mean_of_digits(n), digital_root(n)

is_abundant(n), is_deficient(n), aliquot_sum(n)

Special Numbers

is_harshad(n) (Niven), is_pronic(n) (Oblong), is_highly_composite (HCN)

Persistencia

multiplicative_persistence

B. Prime numbers and factorization

Concept	Algorithm Examples

Factorization

prime_factors(n) (Trial Division), count_distinct_factors, count_divisors(n)

Primality Tests

is_mersenne_prime(p), is_fibonacci_prime(n)

Advanced Primes

twin_primes(limit) (Twin Prime Conjecture), is_carmichael(n) (Pseudoprimes)

Probabilistic Algorithms

miller_rabin(n) (Randomized Algo), pollards_rho(n) (Sub-exponential Factorization)

C. Modular Arithmetic and Cryptography

Concept	Algorithm Examples

Cryptography

modular_exponentiation Returns (b

e

)modm)

Euclidean Algorithm
mod_inverse(a, m) (Using Extended GCD)
System of Congruences
crt(remainders, moduli) (Chinese Remainder Theorem)
Group Theory


order_mod(a, n) (Multiplicative Order)

Quadratic Reciprocity

is_quadratic_residue via Euler's Criterion


3. Implementation Highlights

Chinese Remainder Theorem (CRT)

Purpose: solve a system of congruences x=r


i

​

(modm


i In principle, there can never be a truly perfect crystal; all real crystals contain imperfections that play an important role in many practical applications. ) 
Methodology: Constructs solution x using modular inverses of partial products. This is necessary for any high-performing implementation of RSA decryption. Collatz Conjecture (The 3n+1 Problem) Algorithm If the number is even then divide by 2, if odd then multiply by 3 and add 1. 
output: traces the path length-halting time for n to converge to 1. Carmichael Numbers Checks if a number 'n' is composite and for which there exists an 'a' satisfying: n−1 ≡1(modn) for all a coprime to n. Pollard's Rho Algorithm finds a non-trivial factor of a composite integer n, more rapidly than with the general trial division technique. 
Approach: It involves a randomizing function as F(x)= x2. +c (mod n), and incorporates Floyd's cycle-finding logic to conclude about factors in a relatively more efficient manner.