The Oldest Unsolved Problem in Math

The video examines whether odd perfect numbers exist, the longest-standing open question in mathematics.

Core Definition

• A perfect number equals the sum of its proper divisors; 6 and 28 are the smallest examples (0:00-10:02).
• Euclid gave the formula 2^(p-1)(2^p-1) that generates all known even perfect numbers when 2^p-1 is prime (0:00-10:02).

Euler's Contributions

• Euler proved every even perfect number matches Euclid's form and showed any odd perfect number must have exactly one prime to an odd power (10:04-20:05).
• He invented the sigma function to sum divisors and confirm these properties (10:04-20:05).

Modern Search

• GIMPS distributed computing has found 51 Mersenne primes, yielding 51 even perfect numbers (10:04-20:05).
• Lower bounds now require any odd perfect number to exceed 10^2200 (20:06-30:09).

Heuristics and Spoofs

• Pomerance's heuristic and the web of conditions argue odd perfect numbers are unlikely (20:06-30:09).
• Spoof numbers nearly satisfy the conditions but fail, offering clues toward a possible non-existence proof (20:06-30:09).

00:00:00Major Topic

Subtopic

• detailed notes / examples / explanations

00:00:00Major TopicIntroduction to the oldest unsolved problem in math

• The video introduces the oldest unsolved problem in mathematics, a problem dating back roughly 2000 years. It explains that in the year 2000, Piergiorgio Odifreddi listed it among the four most pressing open problems of his time. The central question: do odd perfect numbers exist?
• A perfect number is defined as a positive integer n such that the sum of its proper divisors (divisors excluding n itself) equals n. Example: 6 has divisors 1, 2, 3; 1+2+3 = 6. In contrast, 10 has proper divisors 1, 2, 5; 1+2+5 = 8 ≠ 10, so 10 is not perfect.
• Historical progression: ancient Greeks knew the first few perfect numbers (6, 28, 496, 8128). Up to 10,000, those are the only ones known to them. These numbers share patterns: each next perfect number is one digit longer than the previous, the final digit alternates 6 and 8, and they can be expressed as sums of consecutive integers (triangular numbers). They also can be written as sums of consecutive odd cubes: 6 = 1^3 + 2^3? (note: 6 is not a sum of consecutive odd cubes; the video notes a pattern where 28 = 1^3 + 3^3, 496 = 1^3 + 3^3 + 5^3 + 7^3, etc.). The speaker emphasizes the surprising connections between these numbers and simple sequences.
• The video also highlights how representations in binary look: 6 = 110₂, 28 = 11100₂, 496 = 111110000₂, 8128 also a string of ones followed by zeros, hinting at a link to powers of two.
• Euclid’s insight (pre-Euler) is introduced: starting from doubling 1, summing powers of two, then if the partial sum is prime, multiply by the last power of two to get a perfect number. This leads to the Euclid form for even perfect numbers: if 2^P − 1 is prime (a Mersenne prime), then 2^(P−1) * (2^P − 1 is a perfect number). This is the basis for even perfect numbers.
• The narrator notes Euclid didn’t prove that this is the only way to get perfect numbers; odd perfect numbers could exist in principle, but are not yet found.

> "This is a video about the oldest unsolved problem in math that dates back 2000 years." > "It's old, it's simple, it's beautiful. What else could you want?"

00:10:04Major TopicHistorical development up to Euler and the sigma function

• Nicomachus of Gerasa, in Introducción Arithmetica, proposed five conjectures about perfect numbers (e.g., nth perfect number has n digits; all even; end in 6 and 8 alternately; Euclid algorithm yields every even perfect number; infinitely many perfect numbers). These conjectures guided thinking for centuries until evidence contradicted several of them.
• Ibn Fallus (Ibn al-Farûs) later listed perfect numbers that caused some conjectures to be disproved (e.g., the fifth perfect number was eight digits long, disproving Nicomachus’s first conjecture; the pattern of ending digits also varied).
• Marin Mersenne studied numbers of the form 2^P − 1. If this is prime (now called a Mersenne prime), then 2^(P−1)*(2^P−1) is an even perfect number. Mersenne’s list of exponents P produced the first several known perfect numbers; however, he sometimes did not verify primality for very large exponents.
• Descartes engaged with Mersenne’s work and speculated about the form of potential odd perfect numbers (if any exist). He considered that odd perfect numbers, if they exist, must have special structure (e.g., prime times a square), but he did not prove it.
• Euler’s breakthroughs: 1) Euclid-Euler theorem: every even perfect number has the Euclid form 2^(P−1)*(2^P−1) with (2^P−1) prime. 2) He invented the sigma function, σ(n), which sums all divisors of n (including n itself). This function is multiplicative over co-prime factors and allows decomposition into products of sigma functions of prime powers. 3) He showed that if any odd perfect number exists, it must have a single prime to an odd power with all other primes to even powers, using sigma and parity arguments. He concluded that the question of existence of odd perfect numbers is intensely difficult, but he could not settle it.
• Euler’s sigma function and its properties are explained with examples: σ(7) = 8, σ(7^k) = 1 + 7 + 7^2 + ... + 7^k, and σ is multiplicative across coprime factors (σ(ab) = σ(a)σ(b) when gcd(a,b) = 1). This leads to the conclusion that for perfect numbers, σ(n) = 2n.
• Euler established that all even perfect numbers must be of Euclid form; the question of odd perfect numbers remained unresolved for centuries.

> "The sigma function. All this function does is it takes all the divisors of a number, including the number itself and adds them up." > "If you have a number that is made up of other numbers that don't share factors with each other, then you can split up the sigma function into the sigma functions of the prime powers that make it up."

00:20:06Major TopicThe rise of GIMPS, large primes, and the status of the problem

• The narrative explains how the search for large Mersenne primes accelerated with computing power. For 500 BCE to 1952, only 12 Mersenne primes were known (and thus 12 even perfect numbers via Euclid-Euler). The breakthrough came with Raphael Robinson (1952), who wrote a computer program and began to discover several new Mersenne primes using fast machines (SWAC). This started a cascade: many large Mersenne primes were found in the ensuing decades as computers advanced.
• By 1994, 2^859,433 − 1 was the largest known Mersenne prime (258,716 digits). The search grew too resource-intensive for single machines, so the Great Internet Mersenne Prime Search (GIMPS) was launched in 1996, distributing computation across volunteers’ computers. GIMPS has discovered many of the largest known primes and thus the corresponding even perfect numbers. The program offers large recognition and a prize for the discovery of the first