Perfect Number Network

A complete distributed computing system for discovering perfect numbers via the Euclid-Euler theorem, inspired by the Great Internet Mersenne Prime Search (GIMPS).

What Are Perfect Numbers?

A perfect number is a positive integer that equals the sum of its proper divisors (all divisors except the number itself).

Examples:

• 6 = 1 + 2 + 3 (divisors: 1, 2, 3)
• 28 = 1 + 2 + 4 + 7 + 14 (divisors: 1, 2, 4, 7, 14)
• 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
• 8128 = 1 + 2 + 4 + 8 + ... + 4064

The Euclid-Euler Theorem

Every even perfect number can be expressed as:

P = 2^(p-1) × (2^p - 1)

where 2^p - 1 is a Mersenne prime.

This remarkable connection between perfect numbers and Mersenne primes (discovered by Euclid ~300 BC, proven by Euler in 1772) means:

• Finding a Mersenne prime → Finding a perfect number
• We test Mersenne numbers using the Lucas-Lehmer primality test
• If M(p) = 2^p - 1 is prime, we've found a perfect number!

Known perfect numbers: Only 51 are known (as of 2024). The largest has 49,724,095 digits!

Overview

This distributed system coordinates searches for perfect numbers by testing Mersenne primes. When we verify that 2^p - 1 is prime via the Lucas-Lehmer test, we've discovered a new perfect number P = 2^(p-1) × (2^p - 1).

System Components

🖥️ server.py - Coordination Server

The main server that distributes work and tracks discoveries.

Features:

• Work assignment with time-based reservations (24-72 hours)
• Progress tracking and reporting
• Automatic reassignment of expired work
• User credit and contribution tracking
• SQLite database for persistence
• Priority-based work queue
• Result verification and storage

Usage:

python server.py [port]
python server.py 5555

💻 client.py - Computational Worker

The worker client that performs Lucas-Lehmer tests to verify perfect numbers.

Features:

• Lucas-Lehmer primality testing for Mersenne numbers
• Progress reporting every 5 minutes
• Checkpoint system (saves every 10,000 iterations)
• Resume capability for interrupted tests
• Real-time progress display with ETA
• Automatic work request and submission
• Perfect number calculation and display

Usage:

python client.py [server_host] [server_port]
python client.py                    # localhost:5555
python client.py 192.168.1.100     # remote server
python client.py localhost 8080    # custom port

🌐 dashboard.py - Web Dashboard

Real-time web-based monitoring dashboard.

Features:

• Live statistics and progress
• Active search tracking
• User leaderboard
• Perfect number discovery announcements
• Recent results feed with digit counts
• Auto-refresh every 10 seconds
• Responsive design
• Euclid-Euler theorem display

Usage:

python dashboard.py [http_port] [db_file]
python dashboard.py 8080 perfectnet.db

Then open: http://localhost:8080

📊 monitor.py - Status Monitor

Command-line tool to query server and database status.

Features:

• Server statistics via network
• Detailed database analysis
• Top contributors
• Recent activity feed
• Current searches
• Discovered perfect numbers list
• Work queue preview

Usage:

python monitor.py [server_host] [server_port]
python monitor.py localhost 5555

🔧 admin.py - Administration Tool

Database management and administrative functions.

Features:

• Add work to queue
• Add ranges of exponents
• Reset expired assignments
• Clear user assignments
• Export results to CSV
• Database optimization
• Detailed statistics

Usage:

python admin.py [command] [options]

# Add specific exponents
python admin.py add-work 521 607 1279

# Add range of prime exponents
python admin.py add-range 10000 20000

# Reset expired assignments
python admin.py reset

# Clear user's work
python admin.py clear-user alice

# Export results
python admin.py export results.csv

# Show statistics
python admin.py stats

# Optimize database
python admin.py vacuum

⚡ benchmark.py - Performance Benchmark

Test system performance and estimate completion times.

Features:

• Quick, medium, and large benchmark suites
• Test specific exponents
• Performance estimates
• Speed calculations
• Recommendations for optimal work size
• Perfect number verification display

Usage:

python benchmark.py [test_level]

# Quick benchmark (recommended first run)
python benchmark.py quick

# Medium benchmark (~1 minute)
python benchmark.py medium

# Large benchmark (~several minutes)
python benchmark.py large

# Test specific exponent
python benchmark.py 2281

Quick Start Guide

1. Start the Server

python server.py

Output:

╔════════════════════════════════════════════════════════════════════╗
║                 Perfect Number Network Server                      ║
╚════════════════════════════════════════════════════════════════════╝
Server: 0.0.0.0:5555
Database: perfectnet.db

💡 Searching for perfect numbers via the Euclid-Euler theorem:
   P = 2^(p-1) × (2^p - 1) where 2^p - 1 is prime

Status: Waiting for clients...

2. (Optional) Start Web Dashboard

In another terminal:

python dashboard.py

Open browser to: http://localhost:8080

3. Run Benchmark (Recommended)

In another terminal:

python benchmark.py quick

This shows your system's performance and estimates completion times.

4. Start Client(s)

In one or more terminals:

python client.py

Enter a username when prompted. The client will:

• Connect to the server
• Receive perfect number candidates
• Perform Lucas-Lehmer tests
• Report progress
• Submit results (including perfect number if found!)

5. Monitor Progress

python monitor.py

or visit the web dashboard at http://localhost:8080

System Architecture

┌──────────────────┐
│  Web Dashboard  │ ← HTTP :8080
│  (dashboard.py) │
└────────┬─────────┘
         │
    ┌────▼────────────────────────────────┐
    │                                     │
    │         Server (server.py)          │
    │         ├─ Work Queue               │
    │         ├─ Assignment Manager       │
    │         ├─ Result Collector         │
    │         └─ SQLite Database          │
    │                                     │
    └────────┬────────────────────────────┘
             │ TCP :5555
      ┌──────┴──────┬───────────┬─────────────┐
      │             │           │             │
┌─────▼──────┐ ┌────▼─────┐ ┌───▼──────┐ ┌────▼──────┐
│  Client   │ │ Client  │ │ Client  │ │  Monitor │
│  (alice)  │ │  (bob)  │ │ (carol) │ │   Tool   │
└───────────┘ └─────────┘ └─────────┘ └──────────┘

How It Works

Perfect Numbers via Mersenne Primes

Mersenne number: M(p) = 2^p - 1

A Mersenne number is prime only for prime values of p.

The Connection:

• Test if M(p) = 2^p - 1 is prime (using Lucas-Lehmer)
• If M(p) is prime → P = 2^(p-1) × M(p) is a perfect number!

Known Perfect Numbers (via Mersenne primes):

• P₁ = 6 (p=2)
• P₂ = 28 (p=3)
• P₃ = 496 (p=5)
• P₄ = 8,128 (p=7)
• P₅ = 33,550,336 (p=13)
• ... (51 known as of 2024)

Lucas-Lehmer Test

The most efficient primality test for Mersenne numbers:

1. Let s = 4
2. For i = 1 to p-2:
      s = (s² - 2) mod M(p)
3. If s = 0, then M(p) is prime
   → Therefore P = 2^(p-1) × M(p) is a perfect number!

Work Distribution Flow

1. Server starts and initializes work queue with exponents to test
2. Client connects and registers with username
3. Server assigns smallest untested exponent with time limit
4. Client performs Lucas-Lehmer test on M(p) = 2^p - 1
5. Client reports progress every 5 minutes
6. Client checkpoints every 10,000 iterations
7. Client submits result:
   • If M(p) is prime → Perfect number P = 2^(p-1) × M(p) discovered!
   • If M(p) is composite → Candidate rejected
8. Server records result and credits user
9. Server celebrates if perfect number found!
10. Server assigns next work item
11. If work expires, server automatically reassigns to another client

Database Schema

users

• username - User identifier
• total_ghz_days - Total computational contribution
• exponents_tested - Number of completed tests
• perfect_numbers_found - Perfect numbers discovered
• last_active - Last activity timestamp

assignments

• exponent - Exponent being tested (for candidate P = 2^(p-1) × M(p))
• username - Assigned user
• assigned_at - Assignment timestamp
• expires_at - Expiration timestamp
• status - 'assigned' or 'completed'
• progress - Percentage complete (0-100)
• last_update - Last progress update

results

• exponent - Tested exponent
• username - User who completed test
• is_perfect - Boolean: is this a perfect number?
• perfect_number - The perfect number value (if verified)
• digit_count - Number of digits in the perfect number
• discovered_at - Completion timestamp
• residue - Final residue (for verification)
• time_seconds - Test duration

work_queue

• exponent - Exponent to test
• priority - Priority value (higher = sooner)
• added_at - Queue entry timestamp

Performance Estimates

Based on modern hardware (approximate):

Exponent	Perfect Number Digits	Time Estimate
127	77	< 1 second
521	314	< 1 second
1,279	770	~1 second
2,281	1,373	~5 seconds
9,941	5,985	~30 seconds
21,701	13,066	~3 minutes
44,497	26,790	~15 minutes
100,000	60,206	~2 hours
1,000,000	602,060	~7 days

Note: Times vary greatly by CPU. Run benchmark.py for your system.

Running a Network

Local Testing (Single Machine)

# Terminal 1: Server
python server.py

# Terminal 2: Dashboard
python dashboard.py

# Terminal 3: Client 1
python client.py
# Username: alice

# Terminal 4: Client 2
python client.py
# Username: bob

Multi-Machine Network

# Server machine (192.168.1.100)
python server.py 5555

# Client machines
python client.py 192.168.1.100 5555

Adding Work

# Add known perfect number exponents (for testing)
python admin.py add-work 2 3 5 7 13 17 19 31 61 89 107 127

# Add range for real searching
python admin.py add-range 10000 20000

# Add large exponents (serious searching!)
python admin.py add-range 100000 200000

Management Tasks

Monitor Status

python monitor.py
python admin.py stats

Reset Stuck Work

python admin.py reset

Clear User's Assignments

python admin.py clear-user username

Export Results

python admin.py export results.csv

Optimize Database

python admin.py vacuum

Example Session Output

Server:

Client registered: alice from 127.0.0.1:54321
Assigned P(p=127) to alice (77 digits)
Progress: alice - candidate P(p=127) - 100.0% complete

══════════════════════════════════════════════════════════════════
🎉 PERFECT NUMBER DISCOVERED! 🎉
Perfect Number: 2^126 × (2^127 - 1)
Digits: 77
Discovered by: alice
Value: 14474011154664524427946373126085988...
(Verified via Mersenne prime M(127) = 2^127 - 1)
══════════════════════════════════════════════════════════════════

Client:

╔════════════════════════════════════════════════════════════════╗
║  New Perfect Number Candidate Assignment                       ║
╚════════════════════════════════════════════════════════════════╝
Candidate: P = 2^126 × (2^127 - 1)
Digits: 77
Time allowed: 24 hours

Verifying via Lucas-Lehmer test of M(127)...

Lucas-Lehmer test for M(127) = 2^127 - 1
Iterations needed: 125
If prime → P = 2^126 × M(127) is a perfect number

Iteration 125/125 (100.00%) - ETA: 0s

✓ Test completed in 0.02 seconds
✓ Result submitted successfully

══════════════════════════════════════════════════════════════════
🎉 PERFECT NUMBER DISCOVERED! 🎉
P = 2^126 × (2^127 - 1)
Digits: 77
Value: 14474011154664524427946373126085988...

✓ This equals the sum of all its proper divisors!
(Verified via Mersenne prime M(127) = 2^127 - 1)
══════════════════════════════════════════════════════════════════

Fun Perfect Number Facts

1. All known perfect numbers are even. Whether odd perfect numbers exist is one of the oldest unsolved problems in mathematics!

2. All even perfect numbers end in 6 or 28 (in base 10)

3. The sum of reciprocals of divisors of a perfect number equals 2:

   • For 6: 1/1 + 1/2 + 1/3 + 1/6 = 2

4. Perfect numbers are rare: Only 51 known in all of mathematics

5. The largest known perfect number (discovered 2018) has 49,724,095 digits!

6. Ancient Greeks knew the first 4 perfect numbers (6, 28, 496, 8128)

7. Euclid (300 BC) discovered the formula connecting them to Mersenne primes

8. Euler (1772) proved all even perfect numbers follow this formula

Comparison to GIMPS

This project is inspired by GIMPS (Great Internet Mersenne Prime Search) but focuses on the beauty of perfect numbers:

Feature	GIMPS/mprime	This Implementation
Primary Goal	Mersenne primes	Perfect numbers
Display	"M(p) is prime!"	"P is a perfect number!"
Communication	PrimeNet API (HTTP)	JSON over TCP
Database	MySQL/PostgreSQL	SQLite
Test algorithm	LL + FFT + optimizations	Lucas-Lehmer (basic)
Checkpointing	Every 10-30 min	Every 10,000 iterations
Work reservation	Yes (days/weeks)	Yes (24-72 hours)
User credit	Yes	Yes
Web interface	Yes	Yes (included)

Future Enhancements

• Trial factoring: Pre-screen candidates with small factors
• P-1 factoring: Pollard's p-1 method
• FFT multiplication: Speed up large exponent tests dramatically
• Double-checking: Verify results independently
• GPU acceleration: OpenCL/CUDA support
• Email notifications: Alert on perfect number discoveries
• Perfect number properties: Calculate and display divisor sums
• Historical context: Display when each perfect number was discovered

System Requirements

• Python 3.7+ (no external dependencies!)
• RAM: 1GB+ (more for large exponents)
• CPU: Any (faster = better)
• Disk: 100MB+ for database
• Network: For distributed operation

Educational Value

This project demonstrates:

• Number theory: Perfect numbers, Mersenne primes, Euclid-Euler theorem
• Distributed computing: Coordinating work across multiple machines
• Algorithms: Lucas-Lehmer primality test
• Network programming: TCP sockets, JSON protocols
• Database design: SQLite for persistence
• Web development: Real-time dashboards
• Mathematics history: Ancient Greeks to modern discoveries

Contributing

This is a system focused on the beauty of perfect numbers. Feel free to contribute by creating a pull request or raising an issue.

Credits

• Inspired by GIMPS (Great Internet Mersenne Prime Search)
• Lucas-Lehmer test algorithm
• Euclid (~300 BC) - Discovery of perfect number formula
• Euler (1772) - Proof of Euclid-Euler theorem
• Perfect number research community

License

MIT License

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Happy Perfect Number Hunting! ✨🔢

"Perfect numbers, like perfect men, are very rare." - René Descartes

"The study of perfect numbers... has, from the earliest times, engaged the attention of mathematicians." - Leonard Eugene Dickson