Key Moments
The Oldest Unsolved Problem in Math
VeritasiumKey Moments
The oldest unsolved math problem: do odd perfect numbers exist? Intense efforts but no proof yet.
Key Insights
Perfect numbers are numbers where the sum of their proper divisors equals the number itself (e.g., 6, 28).
Euclid discovered a formula to generate even perfect numbers using Mersenne primes (2^p - 1 where p is prime).
Mathematicians have extensively searched for odd perfect numbers, establishing many conditions they must satisfy.
Despite extensive computer searches and theoretical advancements, no odd perfect number has ever been found.
The search for perfect numbers, initially an abstract pursuit, has laid groundwork for modern cryptography.
The existence of infinitely many even perfect numbers is conjectured but not proven, while odd perfect numbers remain a complete mystery.
DEFINING PERFECT NUMBERS AND EARLY DISCOVERIES
The video introduces the concept of perfect numbers, which are positive integers divisible by the sum of their proper divisors. Examples like 6 (1+2+3=6) and 28 (1+2+4+7+14=28) are highlighted. Early mathematicians discovered only a few of these numbers: 6, 28, 496, and 8128. These numbers exhibit intriguing patterns, such as being sums of consecutive integers, cubes of odd numbers, and having specific binary representations. Their rarity and the patterns they exhibit have fascinated mathematicians for millennia, sparking the quest to understand their nature.
EUCLID'S FORMULA AND THE GENERATION OF EVEN PERFECT NUMBERS
Euclid devised a formula to generate even perfect numbers: (2^p - 1) * 2^(p-1), provided that (2^p - 1) is a prime number (a Mersenne prime). This formula successfully generates all known even perfect numbers found so far. The discovery meant that any even perfect number must be of this form, significantly narrowing the search space for even perfect numbers and proving one of Nicomachus's conjectures, a key step in understanding the problem.
NICOMACHUS'S CONJECTURES AND EARLY CHALLENGES
Nicomachus, an ancient Greek mathematician, proposed several conjectures about perfect numbers, including that all perfect numbers are even. For centuries, these conjectures were accepted as fact. However, later discoveries, like the fifth and sixth perfect numbers, disproved two of his conjectures by showing that perfect numbers do not necessarily have an alternating ending digit and that they don't always have n digits. This highlighted the incomplete understanding and the potential for exceptions.
MERsenNE PRIMES AND THE BREAKTHROUGHS OF EULER
Marin Mersenne studied numbers of the form 2^p - 1, which are now known as Mersenne numbers. If p is prime, the number 2^p - 1 might also be prime (a Mersenne prime). Euler made significant contributions, proving that all even perfect numbers must follow Euclid's formula. He also developed the sigma function, which sums all divisors of a number. Using this, he proved that if an odd perfect number exists, it must have a specific form, with only one prime factor raised to an odd power.
THE PERSISTENT SEARCH FOR ODD PERFECT NUMBERS
Despite extensive efforts and the proof for even perfect numbers, the existence of odd perfect numbers remains unproven. Mathematicians have established numerous conditions that an odd perfect number must satisfy, such as its size (currently greater than 10^1500) and its prime factorization. Yet, no odd perfect number has ever been found, leading some to believe they may not exist, supported by heuristic arguments suggesting their extreme rarity.
COMPUTERS, GIANT NUMBERS, AND THE FUTURE OF THE PROBLEM
Modern computational power has revolutionized the search for Mersenne primes and, consequently, even perfect numbers, leading to the discovery of many large ones through projects like GIMPS. While computers have drastically increased the number found and pushed the lower bound for odd perfect numbers, they haven't solved the existence question. The problem's longevity underscores the nature of mathematical inquiry: a pursuit driven by curiosity with the potential for unforeseen applications, such as in cryptography.
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Common Questions
The oldest unsolved problem in mathematics is the question of whether odd perfect numbers exist. This problem dates back about 2,000 years and has challenged generations of mathematicians.
Topics
Mentioned in this video
A number that is equal to the sum of its proper divisors. Examples include 6, 28, 496, and 8128.
The theorem proving that every even perfect number must be of the form 2^(p-1) * (2^p - 1), where 2^p - 1 is a Mersenne prime.
A prime number of the form 2^p - 1, where p is also a prime number. These are crucial for generating even perfect numbers.
A function that sums all the divisors of a number, used by Euler to prove the Euclid-Euler theorem.
Mathematician who formulated a heuristic argument predicting the rarity and likely non-existence of large odd perfect numbers.
French polymath who studied numbers of the form 2^p - 1 and published a list associated with primes, now known as Mersenne primes.
Church Deacon who discovered the 50th Mersenne prime in 2017 using GIMPS.
Mathematician who famously demonstrated that 2^67 - 1 was not prime by calculating both sides of the equation to show they were equal, taking him three years.
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