Perfect Numbers

Numbers of the Gods: From the Pythagorean Cult to a Million-Digit Hunt

The **Pythagoreans** believed numbers were the language of the Universe. Among all numbers they singled out the special ones - **perfect**. The number 6 equals the sum of its divisors (1+2+3). That is also the number of days of creation! 28 is the lunar cycle, also perfect. Coincidence? The Pythagoreans didn't believe in coincidences. These numbers are the **signature of the Creator**.

God arithmetizes. - Carl Jacobi

**GIMPS** (Great Internet Mersenne Prime Search) is a project where ordinary people search for perfect numbers using home computers. In 2018, a volunteer found the 51st perfect number - with **nearly 50 million digits**. A $150,000 prize awaits whoever finds the first Mersenne prime exceeding 100 million digits. Maybe it will be you?

6 = 1 + 2 + 3. A number that equals the sum of its divisors! The Pythagoreans called such numbers divine. The next one is 28 (the lunar cycle). Then 496, 8128... and they become increasingly rare. Does an odd perfect number exist? Nobody knows - it's an open problem over a thousand years old.

• **Cryptography:** Mersenne primes in key generation
• **History of mathematics:** from Pythagoras to distributed computing
• **Number theory:** connection to divisor functions

Defining Perfect Numbers

A **perfect number** is a natural number equal to the sum of all its proper divisors (excluding itself). This is one of the oldest concepts in number theory.

**Definition:** n is perfect if: σ(n) - n = n or equivalently: σ(n) = 2n where σ(n) is the sum of ALL divisors of n.

The Pythagoreans considered perfect numbers mystical. 6 is the number of days of creation (without Sunday). 28 is the lunar cycle.

Examples of Perfect Numbers

Perfect numbers are extremely rare. Only 51 are known (as of 2024). All known perfect numbers are even.

**Open questions:** • Do odd perfect numbers exist? (None found) • Are there infinitely many perfect numbers? (Unknown) • If an odd perfect number exists, it is > 10^2000

The search for perfect numbers is tied to the search for Mersenne primes. Each new Mersenne prime yields a new perfect number.

Amicable Numbers

**Amicable numbers** are a pair of numbers where the sum of divisors of one equals the other, and vice versa. They generalize the concept of perfect numbers.

**History:** The Pythagoreans considered 220 and 284 a symbol of friendship. In the Middle Ages, amulets bearing these numbers were given as wedding gifts. Fermat and Descartes found new pairs in the 17th century. Euler found 59 pairs.

More than 1.2 billion amicable pairs are known. But whether there are infinitely many is unknown.

Connection to Mersenne Numbers

The Euclid-Euler theorem links perfect numbers to **Mersenne primes**. Every even perfect number is generated by a Mersenne prime.

**GIMPS - Great Internet Mersenne Prime Search:** A distributed project searching for Mersenne primes. Anyone can participate. $150,000 prize for the first Mersenne prime with over 100 million digits!

Searching for perfect numbers means searching for Mersenne primes. Both problems are open: it is unknown whether there are infinitely many Mersenne primes (and perfect numbers).

Key Ideas

• Perfect number = sum of its own divisors
• Euclid-Euler formula: 2^(p-1)(2^p - 1)
• Amicable numbers - a generalization: σ(a) = b, σ(b) = a
• No odd perfect numbers found

Related Topics

Perfect numbers are connected to divisibility theory:

• Factorization — Breaking into divisors
• Prime Numbers — Mersenne primes
• Special Primes — Mersenne primes in depth

Вопросы для размышления

• Why does the Euclid formula only work for Mersenne primes?
• How would you prove there are no odd perfect numbers (if that's true)?
• Why have amicable numbers fascinated people for thousands of years?

Связанные уроки

• dm-01