Perfect Numbers

In this post I’ll move away from my usual topics of astronomy and space science and talk about something more abstract from the world of pure mathematics, perfect numbers.

 Definition of a Perfect Number

A perfect number is a positive whole number that is equal to the sum of its proper divisors, that is the divisors excluding the number itself.  For instance, 6 has proper divisors 1, 2, and 3 and 1 + 2 + 3 = 6. So, 6 is a perfect number. In fact, it is the first perfect number.

• The second perfect number is 28. It has proper divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28.
• After 28, the third perfect number is 496, which has proper divisors 1, 2, 4, 8, 16, 31, 62, 124 and 248.
• The fourth perfect number is 8128, which has proper divisors 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, and 4064.

As you get to higher numbers, perfect numbers get rarer and rarer. The next four are:

• 33 550 336
• 8 589 869 056
• 137 438 691 328
• 2 305 843 008 139 952 128

In recent years computer searches have identified more perfect numbers. But even today, only 52 are known. The largest, discovered in October 2024, has 82 048 640 digits! (see notes at the end of this post). All the known perfect numbers are even.

Some Other Properties of Perfect Numbers

All even perfect numbers are also triangular numbers.  These are numbers formed by summing up the number of dots in a triangle where the first row has 1 dot, the second 2 dots, the third three dots etc. However, the relationship doesn’t work the other way round; clearly not all triangular numbers are perfect.

Another property is that every even perfect number other than 6 is the sum of a sequence of consecutive odd numbers cubed. For example:

28 = 1³  + 3³   (sum of the first two odd numbers cubed)

496 = 1³  + 3³   + 5³ + 7³ (sum of the first four odd numbers cubed)

8128 = 1³  + 3³   + 5³ + 7³ + 9³ + 11³ + 13³ + 15³ (sum of the first eight odd numbers cubed)

Perfect numbers have held a fascination for mathematicians for thousands of years. René Descartes, the great 17th-century French mathematician and philosopher, who had a keen interest in perfect numbers, allegedly said that:

“Perfect numbers, like perfect men, are very rare.” 

Unsolved Problems relating to Perfect Numbers

Most unsolved problems in mathematics are difficult for anyone without an advanced mathematical background to comprehend. Interestingly, there are two unsolved problems in pure mathematics relating to perfect numbers which can be easily understood by the layperson.

• Are there infinitely many perfect numbers? Or is there a largest perfect number with no further ones above this value? Most mathematicians believe that there are infinitely many, but no one has been able to prove this (or alternatively provide a proof that there can only be a finite number of them).
• Are there any odd perfect numbers? The general belief among mathematicians is that all  perfect numbers are even. However, although no odd ones have been found, nobody has been able to prove that they don’t exist. This question has taxed the greatest mathematical minds for centuries.

The Importance of Perfect Numbers in Ancient Times

The ancient Greeks had many beliefs which to us seem strange and mystical. Numbers, in particular small numbers, were often assigned personalities and genders. For example, the number 5 represented marriage, because it is the sum of 2 which was assigned a female gender and 3 which was assigned a male gender.

Around the year 100 CE the Ancient Greek philosopher Nicomachus wrote a famous text Introductio Arithmetica which divided whole numbers into three classes: [1]

• Abundant numbers – the sum of their proper divisors is greater than the number, e.g. 12 has proper divisors 1, 2, 3, 4 and 6 which sum up to 16.
• Deficient numbers  – the sum of their proper divisors is less than the number, e.g. 8 has proper divisors 1, 2 and 4 which sum up to 7. Prime numbers, whose only proper divisors are 1,  are all deficient.
• Perfect numbers – the sum of their proper divisors is equal to the number.

Nicomachus had a strange philosophical way of looking at groups of numbers and went on to say the following.

[Referring to Abundant Numbers] In the case of the too much, is produced excess, superfluity, exaggerations and abuse;

[Referring to Deficient Numbers] In the case of too little, is produced wanting, defaults, privations and insufficiencies.

[Referring to Perfect Numbers] And in the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort – of which the most exemplary form is that type of number which is called perfect.

Perfect numbers had an important place in religion and because the first two perfect numbers are 6 and 28, early theologians used them to explain the Universe. For example:

• God created the Universe in six days because the number 6 is perfect. God was then able to rest on the seventh day.
• The Moon orbits the Earth in approximately 28 days.

Perfect Numbers in Recent Culture

Perfect numbers appear in modern culture, usually in quite subtle ways which can be only picked up by people with mathematical knowledge.

Perhaps the most interesting example is in a 2006 episode of The Simpsons  “Marge and Homer Turn a Couple Play”. This contained some  secret mathematics.  A question appeared on the large screen at Springfield Stadium asking fans to estimate the attendance at the game. [2]

Although these numbers might appear to be purely random, there is a hidden meaning for pure mathematics geeks. (Of which I am definitely not one 😊. I certainly would not have noticed this if I’d been watching the episode)

The number 8128

As I’ve mentioned earlier 8128 is a perfect number.

The number 8208

8208 has an interesting property. It has four digits and if you take each digit, raise it to the fourth power and then add all four numbers together you get 8208.

And adding the four numbers together gives

      8⁴ + 2⁴ + 0⁴ + 8⁴ = 8208.

Because 8208 can create itself from its own components, by raising each digit to the power of the number of digits in the number and then summing them, it is termed a  narcissistic number. An example of a narcissistic number with three digits is 153. If you take each digit, raise it to the third power and add the three numbers together you get 153.

Narcissistic numbers are very rare. For example, there are:

• No two digit narcissistic numbers
• Four three digit narcissistic numbers: 153, 370, 371 and 407
• Only three four digit narcissistic numbers: 1634, 8208 and  9474

 In fact among the infinity of numbers, it can be shown that only 88 narcissistic numbers exist. [4]

The number 8191

The final number on the scoreboard 8191 is a prime number, it has no divisors other than 1 and the number itself. In fact, it is a special type of prime number known as a Mersenne prime.

And finally…

I hope you have enjoyed this article, which is on a topic very different from my usual posts. I certainly enjoyed reseaching and writing on a subject which I don’t cover in my blog. Many areas of obscure mathematics, which were originally thought to be very abstract, later turn out to have a practical value in understanding the Universe. For example much of Einstein’s theory of general relativity is built on the complex mathematics of non-Euclidean geometry.

Even so , I just can’t see perfect numbers ever making such an important contribution. If a mathematical genius were able to PROVE that no odd perfect numbers exists, something that has eluded the best minds for centuries, it would be a amazing achievement but it is difficult see how this would be of any benefit to our scientic understanding of the world around us

I have blogging since 2014 and over the last 12 years I have written two other posts on more abstract mathematical topics which may be of interest.

• An unsolved problem in mathematics – Is the number 196 a Lychrel number?
• One of the oldest and most famous unsolved mathematical problems the Goldbach conjecture.  Every even number greater than 2 can be expressed as the sum of two prime numbers.

For those of you who want a little more detail, here’s some extra information on how perfect numbers can be calculated.

Calculating Perfect numbers – relationship with Mersenne primes.

A Mersenne number is a number that is one less than a power of two. That is a number of the form M_n = 2ⁿ − 1 where n is a positive whole number. Some examples are 1, 3, 7 and 15.

• M₁ = 1, which is 2¹ – 1
• M₂ = 3, which is 2² – 1
• M₃ = 7, which is 2³ – 1
• M₄ = 15, which is 2⁴ – 1

They are named after the French mathematician Marin Mersenne (1588 -1648).

If a Mersenne number is also a prime number, then it is known as a Mersenne prime. For this to be the case the power which 2 is raised to (the exponent) must also be a prime number. A few examples are below.

Not all Mersenne numbers with a prime number exponent are Mersenne primes. For example, in the case of M₁₁, 11 is prime but 2¹¹ – 1 = 2047 which is not prime. In fact in turns out that for the vast majority of prime numbers (2^p– 1 ) is not prime and so far only 52 Mersenne primes have been discovered, and it is an unsolved problem in mathematics whether or not there are infinitely many.

Around the year 300 BCE Euclid proved that

If (2^p – 1) is a prime number then the number 2^p−1(2^p − 1) is a perfect number.

However, it was 2000 years after Euclid before the Swiss mathematician Leonhard Euler (1707 – 1783) was able to prove the result also held the other way round.

Every even perfect number has the form  2^p−1(2^p – 1) where  (2^p – 1) is a Mersenne prime.

The combination of these two results is the Euclid–Euler theorem. This states:

An even number is perfect if and only if it has the form 2^p−1(2^p − 1), where (2^p – 1) is a Mersenne prime.

Using this rule the first eight even perfect numbers and their corresponding Mersenne primes (2^p – 1)  are listed below.

Interestingly, because any even perfect number has the form 2^p−1(2^p − 1), it means that when expressed in binary (base 2) it consists of a sequence of p ones followed by (p – 1) zeros. This is shown below for the first four perfect numbers.

How New Perfect Numbers are Calculated

There is a large distributed computing project known as the Great Internet Mersenne Prime Search which has been running since 1996 The largest Mersenne prime found so far is 2^136279841-1 which has 41 024 320 digits. [3] Because of the connection between Mersenne primes and perfect numbers, it means that the largest even perfect number known is 2^1362798410 x (2^136279841 – 1), which has 82 048 640 digits.

References

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[1] Maths History. (2020). Perfect numbers. [online] Available at: https://mathshistory.st-andrews.ac.uk/HistTopics/Perfect_numbers [Accessed 6 Jan. 2026].

[2] the Guardian. (2013). The Simpsons’ secret formula: it’s written by maths geeks. [online] Available at: https://www.theguardian.com/tv-and-radio/2013/sep/22/the-simpsons-secret-formula-maths-simon-singh [Accessed 6 Jan. 2026].

[3] Mersenne.org. (2026). Great Internet Mersenne Prime Search – PrimeNet. [online] Available at: https://www.mersenne.org/. [Accessed 6 Jan. 2026].

[4] Weisstein, E.W. (2026.). Narcissistic Number. [online] mathworld.wolfram.com. Available at: https://mathworld.wolfram.com/NarcissisticNumber.html. [Accessed 6 Jan. 2026].