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Perfect numbers and their properties

10 min read

From: veritasium

The question of whether odd perfect numbers exist is considered the oldest unsolved problem in mathematics, dating back 2000 years [00:00:00]. It was listed by Italian mathematician Piergiorgio Odifreddi in 2000 among four of the most pressing open problems [00:00:18]. Mathematicians have used computers to check numbers up to 10 to the power of 2,200, but no odd perfect numbers have been found [00:00:26].

What is a Perfect Number?

A perfect number is a positive integer that is equal to the sum of its proper divisors (divisors excluding the number itself) [00:00:55].

Examples of Perfect Numbers

  • 6: Its proper divisors are 1, 2, and 3. Adding them up (1 + 2 + 3) equals 6, making it a perfect number [00:00:58].
  • 28: This is also a perfect number [00:01:38].

Most numbers either overshoot or undershoot when their proper divisors are summed [00:01:32]. For example, the proper divisors of 10 (1, 2, 5) sum to 8, so 10 is not perfect [00:01:21].

Between 1 and 100, only 6 and 28 are perfect numbers [00:01:35]. Going up to 10,000, the next two perfect numbers are 496 and 8,128 [00:01:42]. These four numbers were the only perfect numbers known to the ancient Greeks and for over a thousand years thereafter [00:01:51].

Properties of Known Perfect Numbers

Early perfect numbers reveal several interesting patterns:

  • Each successive perfect number is one digit longer than the previous one [00:02:08].
  • The ending digit alternates between 6 and 8 [00:02:15].
  • All known perfect numbers are even [00:02:18].
  • They are triangular numbers, meaning they can be written as the sum of consecutive integers:
  • Every perfect number except 6 is the sum of consecutive odd cubes:
  • When written in binary, perfect numbers are a string of ones followed by a series of zeros, which corresponds to consecutive powers of two:

Euclid’s Formula for Even Perfect Numbers

Around 300 BC, Euclid discovered a pattern for generating even perfect numbers [00:04:03]. His method involves powers of two:

  1. Start with 1, double it (2, 4, 8, 16, etc.) [00:04:11].
  2. Sum these powers of two consecutively (1+2=3, 1+2+4=7, etc.) [00:04:20].
  3. If the sum is a prime number, multiply it by the last power of two in the sequence to get a perfect number [00:04:25].

This pattern can be formalized. A sum of consecutive powers of two, from 2⁰ to 2^(n-1), is equal to 2^n - 1 [00:05:30]. Thus, Euclid’s formula for a perfect number is: 2^(P-1) * (2^P - 1) [00:06:32]

This formula generates a perfect number whenever 2^P - 1 is a prime number [00:06:36]. Since the formula involves multiplication by an even number (2^(P-1)), it will always produce an even number [00:06:42]. Euclid found a way to generate even perfect numbers, but he did not prove that this was the only way to get them, leaving open the possibility of odd perfect numbers [00:06:50].

Nicomachus’s Conjectures

Four hundred years after Euclid, the Greek philosopher Nicomachus published Introductio Arithmetica, a standard arithmetic text for the next thousand years [00:07:05]. In it, he stated five conjectures about perfect numbers that he believed to be true but did not prove [00:07:13]:

  1. The nth perfect number has n digits [00:07:20].
  2. All perfect numbers are even [00:07:26].
  3. All perfect numbers end in 6 and 8 alternately [00:07:29].
  4. Euclid’s algorithm produces every even perfect number [00:07:33].
  5. There are infinitely many perfect numbers [00:07:37].

For a millennium, these conjectures were considered facts [00:07:42]. However, in the 13th century, Egyptian mathematician Ibn Fallus published a list of 10 perfect numbers [00:07:51].

  • The fifth perfect number was eight digits long, disproving Nicomachus’s first conjecture [00:08:05].
  • Both the fifth and sixth perfect numbers ended in 6, disproving Nicomachus’s third conjecture about alternating 6 and 8 endings [00:08:12].

Mersenne Primes and Descartes’ Contributions

The problem reached Renaissance Europe, leading to the rediscovery of the fifth, sixth, and seventh perfect numbers [00:08:29]. All found perfect numbers so far adhered to Euclid’s form [00:08:37]. The key to finding new ones became identifying values of P that make 2^P - 1 prime [00:08:41].

French polymath Marin Mersenne extensively studied numbers of this form, publishing a list in 1644 of 11 values of P for which he claimed they corresponded to primes [00:08:47]. Numbers of the form 2^P - 1 that are prime are now called Mersenne Primes [00:09:01]. The first seven exponents on his list did result in primes, corresponding to the first seven perfect numbers [00:09:06]. Mersenne acknowledged the difficulty of checking large numbers for primality [00:09:14].

René Descartes, corresponding with Mersenne, believed he could show that no even perfect numbers existed outside Euclid’s form [00:09:36]. He also conjectured that if an odd perfect number did exist, it must be the product of a prime and the square of a different number [00:09:44]. However, Descartes could not prove either statement [00:10:01].

Euler’s Breakthroughs

Around 100 years later, Leonhard Euler took up the problem [00:10:15].

  1. In 1732, he discovered the eighth perfect number by verifying that 2^31 - 1 is prime, as Mersenne had predicted [00:10:56].
  2. For his second breakthrough, Euler invented the sigma function, which sums all divisors of a number, including the number itself [00:11:09]. For a perfect number, the sigma function returns twice the number itself [00:11:28]. This function is powerful because it allows the [[mathematical_uniqueness_of_prime_numbers | sigma function]] of a composite number to be split into the [[mathematical_uniqueness_of_prime_numbers | sigma functions]] of its prime powers [00:12:30]. Using this, Euler proved the Euclid-Euler theorem: every even perfect number has Euclid’s form [00:13:01]. This solved a 1600-year-old problem and proved Nicomachus’s fourth conjecture [00:13:09].
  3. For his third breakthrough, Euler proved Descartes’ conjecture about the specific form an odd perfect number must take [00:13:29]. Using the sigma function, he showed that if an odd perfect number (n) exists, it must have exactly one prime factor raised to an odd power, while all other prime factors must be raised to an even power [00:15:08]. This can be expressed as n = p^k * m^2, where p is a prime number and k is an odd exponent, and m is an integer [00:09:50]. Despite this, Euler could not prove or disprove the existence of odd perfect numbers [00:15:45].

The Modern Search for Perfect Numbers

After Euler, progress in finding new perfect numbers stalled for 150 years [00:15:56]. The focus shifted to verifying Mersenne’s list of proposed primes [00:16:27].

  • In 1876, Edouard Lucas proved that 2^67 - 1 was not prime, disproving another of Mersenne’s claims [00:16:45].
  • In 1903, Frank Nelson Cole famously demonstrated the factors of 2^67 - 1 manually at a meeting of the American Mathematical Society [00:16:57].

The advent of computers revolutionized the search:

  • By 1952, only 12 Mersenne Primes (and thus 12 perfect numbers) had been discovered [00:17:53].
  • In 1952, American mathematician Raphael Robinson wrote a computer program for the SWAC (Standards Western Automatic Computer) that found the next five Mersenne Primes within 10 months [00:18:06].
  • Over the next 50 years, new Mersenne Primes were discovered rapidly using computers [00:18:24].

Great Internet Mersenne Prime Search (GIMPS)

In 1996, computer scientist George Woltman launched the Great Internet Mersenne Prime Search (GIMPS) [00:19:02]. This distributed computing project allows volunteers to contribute their computer’s processing power to search for Mersenne Primes [00:19:09].

  • GIMPS has discovered 17 new Mersenne Primes, 15 of which were the largest known primes at the time [00:19:17].
  • In 2017, John Pace discovered the 50th Mersenne Prime, 2^(77,232,917) - 1, which is more than 23 million digits long [00:19:45]. A Japanese publisher even printed this number in a 719-page book [00:20:06].
  • A year later, the 51st Mersenne Prime was discovered, 2^(82,589,933) - 1, which has over 24 million digits [00:20:31]. This remains the largest known prime as of today [00:21:11].

The Enduring Question: Odd Perfect Numbers

As of today, only 51 perfect numbers have been found, all of which are even [00:21:31].

Infinitely Many Perfect Numbers?

Nicomachus’s fifth conjecture, that there are infinitely many perfect numbers, remains unproven [00:07:37]. However, the Lenstra and Pomerance Wagstaff conjecture predicts an infinite number of Mersenne Primes, and thus infinitely many even perfect numbers [00:21:48]. While this is a strong prediction, it is not a mathematical proof [00:22:17].

The Search for Odd Perfect Numbers

The question of whether odd perfect numbers exist is still the oldest unsolved problem in mathematics [00:22:26].

Applications of Pure Mathematics

While the problem of perfect numbers may seem to lack real-world applications [00:27:32], the history of mathematics demonstrates the value of pursuing pure curiosity:

  • For over 2000 years, number theory had no real-world applications, but in the 20th century, it became the foundation for modern cryptography, protecting everything from text messages to government secrets [00:27:50].
  • Einstein’s general relativity was built on non-Euclidean geometries, which were developed as intellectual curiosities without foresight of their future impact [00:28:45].

Unsolved problems in mathematics, even those without immediate applications, drive the development of new ideas and build foundational knowledge that may prove useful in unexpected ways in the future [00:28:16].

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