TLDR;
This video explores the oldest unsolved problem in mathematics: the existence of odd perfect numbers. It defines perfect numbers, discusses the history of their discovery, and examines the properties and patterns of even perfect numbers as described by Euclid. The video also covers attempts to find odd perfect numbers, the work of mathematicians like Euler and Mersenne, and the use of computers to discover large prime numbers. It touches on the concept of "spoofs" and heuristic arguments suggesting that odd perfect numbers may not exist, while emphasising the value of pursuing mathematical problems even without immediate practical applications.
- Perfect numbers are those whose proper divisors add up to the number itself.
- Euclid discovered a formula for generating even perfect numbers.
- The existence of odd perfect numbers remains an open question.
Introduction to Perfect Numbers [0:00]
The video introduces the concept of perfect numbers, defining them as numbers whose proper divisors (excluding the number itself) sum up to the number itself. For example, the number 6 is perfect because its divisors 1, 2, and 3 add up to 6. Similarly, 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28. The video contrasts this with non-perfect numbers like 10, where the sum of its proper divisors (1, 2, and 5) equals 8, which is not 10. It notes that perfect numbers are rare; among numbers between 1 and 100, only 6 and 28 are perfect.
Patterns and Properties of Perfect Numbers [1:51]
The video explores patterns observed in perfect numbers. Early mathematicians only knew of four perfect numbers: 6, 28, 496, and 8,128. These numbers increase in length and alternate in their ending digit between 6 and 8, implying they are all even. Intriguingly, perfect numbers can be expressed as the sum of consecutive numbers (triangular numbers) and, except for 6, as the sum of consecutive odd cubes (e.g., 28 = 1³ + 3³). In binary, these numbers are strings of ones followed by zeros, corresponding to consecutive powers of two.
Euclid's Discovery and Nicomachus's Conjectures [4:03]
Around 300 BC, Euclid discovered a method for generating even perfect numbers. By doubling the number one repeatedly (1, 2, 4, 8, 16, etc.) and summing the sequence from one, if the sum is prime, multiplying it by the last number in the sequence yields a perfect number. This is expressed by the formula 2^(p-1) * (2^p - 1), where (2^p - 1) is prime. Euclid's formula always results in even numbers. 400 years later, Nicomachus made five conjectures about perfect numbers, including that the nth perfect number has n digits, all perfect numbers are even, they end in 6 or 8 alternately, Euclid's algorithm produces every even perfect number, and there are infinitely many perfect numbers.
Challenges to Nicomachus's Conjectures and the Work of Mersenne [7:42]
In the 13th century, Ibn Fallus disproved two of Nicomachus's conjectures by identifying a perfect number with a different number of digits and noting that consecutive perfect numbers could end in the same digit. Later, in Renaissance Europe, mathematicians rediscovered additional perfect numbers, all fitting Euclid's form. Marin Mersenne studied numbers of the form 2^p - 1, now known as Mersenne primes, as a means to find new perfect numbers. Despite his contributions, Mersenne admitted to not verifying the primality of some larger numbers on his list.
Euler's Breakthroughs and the Sigma Function [10:15]
Leonhard Euler made significant contributions to the study of perfect numbers. He discovered the eighth perfect number and proved that every even perfect number follows Euclid's formula, a result known as the Euclid-Euler theorem. Euler also used the sigma function, which sums all divisors of a number (including the number itself), to analyse perfect numbers. He demonstrated that the sigma function of a perfect number always equals twice the number itself. Euler further refined the understanding of odd perfect numbers, building on Descartes' work, but could not prove their existence.
Further Progress and the Advent of Computers [15:56]
For 150 years after Euler, progress on perfect numbers stalled. Despite this, mathematicians continued to explore Mersenne primes. Edouard Lucas proved that 2^67 - 1 was not prime, and Frank Nelson Cole demonstrated its factorisation in a silent presentation. The advent of computers in the mid-20th century revolutionised the search for Mersenne primes and perfect numbers. Raphael Robinson used a computer to discover five new Mersenne primes in 1952.
The Great Internet Mersenne Prime Search (GIMPS) [18:53]
In 1996, George Woltman launched the Great Internet Mersenne Prime Search (GIMPS), a distributed computing project that allows volunteers to use their computers to search for Mersenne primes. GIMPS has been highly successful, discovering numerous new Mersenne primes. In 2017, John Pace discovered the 50th Mersenne prime through GIMPS, a number with over 23 million digits. As of today, the largest known prime number is a Mersenne prime.
The Ongoing Search for Odd Perfect Numbers [22:17]
Despite the success in finding even perfect numbers, the question of whether odd perfect numbers exist remains unanswered. Researchers have established increasingly large lower bounds for potential odd perfect numbers, making them unlikely to be found by brute-force computation. Mathematicians are exploring conditions that odd perfect numbers must satisfy and studying "spoofs"—numbers that are nearly perfect—to identify properties that might preclude the existence of odd perfect numbers. Heuristic arguments suggest that odd perfect numbers are unlikely to exist, but these are not definitive proofs.
The Value of Unsolved Problems [27:29]
The video concludes by reflecting on the value of studying unsolved mathematical problems, even those without immediate practical applications. Number theory, once considered purely theoretical, now underpins modern cryptography. The pursuit of mathematical problems fosters new ideas and can lead to unexpected breakthroughs. The video encourages aspiring mathematicians to engage with unsolved problems like the search for odd perfect numbers, emphasising that progress is possible and that the effort itself is valuable. The video is sponsored by Brilliant, a platform for learning math, science, and technology through interactive problem-solving.