The search for odd perfect numbers is one of the oldest unsolved problems in mathematics, captivating mathematicians for over two millennia. This question combines simplicity with depth, demonstrating the beauty and mystery within mathematical inquiry. Despite advances in technology and computation, this puzzle remains open today. Let’s break down this fascinating topic into parts.

A Brief History

Mathematicians have been intrigued by perfect numbers since ancient times. The concept of a perfect number can be traced back over 2,000 years to the Greek mathematician Euclid, who studied them as part of his explorations in number theory. Even with the advent of high-powered computers and sophisticated algorithms, no one has yet confirmed the existence of an odd perfect number. It’s this enduring mystery that keeps mathematicians interested, highlighting how some simple questions can have complex or even elusive answers.

Understanding Odd and Even Numbers

To approach this question, let’s go through some basics,

• Even Numbers: An even number is any integer that can be divided by 2 without leaving a remainder. Examples include 2, 4, 6, and so on. Mathematically, even numbers are represented as 2n, where n is an integer.
• Odd Numbers: To the contrary, an odd number, is an integer that, when divided by 2, leaves a remainder of 1. Examples include 1, 3, 5, etc. These numbers are represented as 2n+1, where n is an integer.

What Is a Perfect Number?

A perfect number is an integer that equals the sum of its proper divisors (excluding the number itself). To illustrate this, let’s look at a simple example.

Take 6. The divisors of 6 are 1, 2, 3, and 6. If we exclude 6 and sum the remaining divisors, we get: 1 + 2 + 3 = 6 Since the sum matches the original number, 6 is a perfect number.

Now, take 9. The divisors of 9 are 1, 3, and 9. If we exclude 9 and sum the remaining numbers, we get: 1 + 3 = 4 Since the sum does not match 9, it is not a perfect number.

As you can see within the first 10,000 numbers, there are only four perfect numbers. A pattern emerges—each next perfect number is one digit longer than the one before it, and the last digit alternates between 6 and 8.

In 300 BC, Euclid saw a pattern,

• 6 = (1+2) × 2¹
• 28 = (1+2+4) × 2²
• 496 = (1+2+4+8+16) × 2⁴

Observing this, he formulated,

• 6 = (2² – 1) × 2¹
• 28 = (2³ – 1) × 2²
• 496 = (2⁵ – 1) × 2⁴

This led to the general formula,

Perfect number = (2^p – 1) × 2^(p-1), where (2^p – 1) is a prime.

Since 2^(p-1) is always even, all perfect numbers generated this way are even. But is this the only way to find perfect numbers? Are there any odd perfect numbers?

Do Any Odd Perfect Numbers Exist?

After 400 years, Nicomachus proposed five conjectures,

1. The n^th perfect number has n digits.
2. All perfect numbers are even.
3. All perfect numbers end in 6 or 8 alternately.
4. Euclid’s algorithm produces every even perfect number.
5. There are infinitely many perfect numbers.

However, the first and third conjectures were later proven false.

In 1644, French polymath Marin Mersenne published a list of primes

P = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257

Numbers were called now Mersenne primes. From these, he predicted seven perfect numbers. However, checking whether these large numbers were prime was a major challenge.

In 1732, Leonhard Euler made significant progress,

1. He found the 8th perfect number (2³¹ – 1) × 2³⁰, as Mersenne had predicted.
2. He proved that every even perfect number follows Euclid’s formula, confirming Nicomachus’ fourth conjecture.
3. He proved that if odd perfect numbers exist, they must follow the form,

N = p^(4k+1) × M², where p = (4n+1)

Despite this, Euler could not prove whether odd perfect numbers exist.

In 1952, Raphael M. Robinson wrote a mathematical program to check Mersenne numbers for primality.

Over the next 50 years, thanks to computers, many new Mersenne primes were discovered. In 1996, George Woltman launched the Great Internet Mersenne Prime Search (GIMPS), allowing volunteers to contribute computing power to this search. As of 2018, 51 Mersenne primes have been discovered, the largest containing more than 24 million digits. The fun fact is if your computer find a new Mersenne prime you will list as it’s discoverer and adding yourself to a list that includes some of the best mathematicians. Worth the try?

Searching for Odd Perfect Numbers

By 1991, an algorithm called factor chain proved that if an odd perfect number exists, it must be larger than 10³⁰⁰. In 2012, Pascal Ochem and Michael Rao raised the lower bound to 10¹⁵⁰⁰, and today, it has been pushed further to 10²²⁰⁰.

Interestingly, mathematicians have found numbers very close to odd perfect numbers, known as spoofs. Spoofs share many properties with odd perfect numbers, except for a few key differences. In 2022, Pace Nielsen and his team identified 21 spoof numbers, but not a single odd perfect number.

At this point, one might ask,

Why continue the search if there are no practical applications?

The answer lies in the essence of mathematics itself. Mathematicians often pursue knowledge purely out of curiosity, without immediate practical applications in mind. History has shown that abstract mathematical discoveries, once considered purely theoretical, later became the foundation for groundbreaking advancements. Take number theory, for example—once seen as an intellectual curiosity, it now underpins cryptography, internet security, and modern computing.

So, while the search for an odd perfect number may seem like an unsolved riddle, it represents something much greater: the pursuit of knowledge for its own sake. Who knows? Perhaps one day, this ancient mystery will unlock insights we have yet to imagine

Written By: Mahinsa Rajawardana

References:

1. https://medium.com/starts-with-a-bang/happy-perfect-number-day-4590158cf8b4  
2. https://mathworld.wolfram.com/OddPerfectNumber.html 
3. https://www.quantamagazine.org/mathematicians-open-a-new-front-on-an-ancient-number-problem-20200910/

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