Project 2: Mathematical Toolkit (TAOCP Foundations)

Build a toolkit of number-theory and combinatorial primitives used throughout TAOCP.

Quick Reference

Attribute Value
Difficulty Intermediate
Time Estimate 1-2 weeks
Language C
Prerequisites Basic math, integer arithmetic
Key Topics gcd, modular arithmetic, combinatorics

1. Learning Objectives

By completing this project, you will:

  1. Implement core number-theory functions (gcd, lcm, modexp).
  2. Implement combinatorial functions (nCr, permutations).
  3. Build reusable utilities for later TAOCP projects.
  4. Validate results with known identities.

2. Theoretical Foundation

2.1 Core Concepts

  • Euclidean algorithm for gcd.
  • Modular arithmetic and modular exponentiation.
  • Combinatorial identities for validation.

2.2 Why This Matters

TAOCP relies on precise arithmetic primitives; correctness here affects every later project.

2.3 Historical Context / Background

These algorithms are centuries old and still underpin cryptography and algorithm analysis.

2.4 Common Misconceptions

  • “nCr fits in int”: It often does not.
  • “Modulo is just remainder”: Sign rules matter.

3. Project Specification

3.1 What You Will Build

A library with functions for gcd, lcm, modular exponentiation, factorials, binomial coefficients, and primes.

3.2 Functional Requirements

  1. gcd/lcm for 64-bit integers.
  2. modular exponentiation with fast exponentiation.
  3. factorial and nCr with overflow checks.
  4. prime test and simple sieve.

3.3 Non-Functional Requirements

  • Correctness: Verified by identities.
  • Usability: Clear API and docs.
  • Performance: Efficient for moderate inputs.

3.4 Example Usage / Output

gcd(48,18)=6
modexp(3,13,17)=12
nCr(10,3)=120

3.5 Real World Outcome

A header/library you can import into later projects for mathematical foundations.

4. Solution Architecture

4.1 High-Level Design

┌──────────────┐     ┌──────────────┐
│ math API     │────▶│ implementations│
└──────────────┘     └──────────────┘

4.2 Key Components

Component Responsibility Key Decisions
Number theory gcd, modexp, primes iterative algorithms
Combinatorics factorial, nCr overflow-safe methods

4.3 Data Structures

typedef struct {
    uint64_t n;
    uint64_t k;
} BinomArgs;

4.4 Algorithm Overview

Key Algorithm: fast exponentiation

  1. Square base, halve exponent.
  2. Multiply when exponent bit is 1.

Complexity Analysis:

  • Time: O(log n) for modexp.
  • Space: O(1).

5. Implementation Guide

5.1 Development Environment Setup

gcc --version

5.2 Project Structure

project-root/
├── include/mathkit.h
├── src/mathkit.c
└── tests/

5.3 The Core Question You’re Answering

“What minimal math primitives do all TAOCP algorithms depend on?”

5.4 Concepts You Must Understand First

  • Euclidean algorithm
  • Modular arithmetic rules
  • Binomial coefficient identities

5.5 Questions to Guide Your Design

  1. How do you avoid overflow in nCr?
  2. What types are sufficient for each function?
  3. How will you validate correctness?

5.6 Thinking Exercise

Show that gcd(a,b)=gcd(b,a mod b) using a simple proof.

5.7 The Interview Questions They’ll Ask

  1. Why is Euclid’s algorithm correct?
  2. How does fast exponentiation work?
  3. How do you compute nCr without factorial overflow?

5.8 Hints in Layers

Hint 1: Start with gcd and lcm. Hint 2: Add modexp and sieve. Hint 3: Add combinatorics with overflow checks.

5.9 Books That Will Help

Topic Book Chapter
Number theory TAOCP Vol 1 Ch. 1
Combinatorics Concrete Mathematics relevant sections

5.10 Implementation Phases

Phase 1: Foundation (3-4 days)

  • gcd, lcm, modexp

Phase 2: Core Functionality (3-4 days)

  • factorial, nCr

Phase 3: Polish & Edge Cases (2-3 days)

  • sieve, tests, docs

5.11 Key Implementation Decisions

Decision Options Recommendation Rationale
nCr method factorial vs multiplicative multiplicative reduces overflow

6. Testing Strategy

  • Identity tests: gcd(a,0)=a
  • Binomial symmetry: nCr(n,k)=nCr(n,n-k)
  • Known prime counts for small n

7. Common Pitfalls & Debugging

  • Overflow in factorial
  • Wrong modulo with negatives
  • Slow prime checks

8. Extensions & Challenges

  • Implement extended gcd
  • Add modular inverse
  • Add big integer fallback

9. Real-World Connections

  • Cryptography, hashing, probability, algorithm analysis.

10. Resources

  • TAOCP Vol 1 Ch. 1
  • Concrete Mathematics

11. Self-Assessment Checklist

  • I can compute gcd and modexp correctly.
  • I can verify combinatorial identities.

12. Submission / Completion Criteria

Minimum: Core functions implemented and tested. Full: Full toolkit with docs. Excellence: Extended gcd and modular inverse.

This guide was generated from LEARN_TAOCP_DEEP_DIVE.md. For the complete learning path, see the parent directory README.