The Oldest Unsolved Problem in Math

An ancient mathematical puzzle dating back 2000 years that some of the brightest mathematicians have failed to solve. The question: Do any odd perfect numbers exist?

• Listed among four most pressing open problems in mathematics in 2000 • Could be solved by finding a single number • Has captured imaginations of countless mathematicians

Mathematicians have used computers to check numbers up to 10 to the power of 2,200 with no success so far.

• It's old • It's simple • It's beautiful

A perfect number equals the sum of its proper divisors (all divisors except the number itself).

• 6 has proper divisors 1, 2, 3 which sum to 6 • 10 has proper divisors 1, 2, 5 which sum to 8 (not perfect) • 28 is the second perfect number • 496 is the third perfect number

Between 1 and 100, only 6 and 28 are perfect. Up to 10,000, only 496 and 8,128 are perfect.

The ancient Greeks knew only four perfect numbers: 6, 28, 496, and 8,128. These remained the only known perfect numbers for over 1,000 years.

• Each next perfect number is one digit longer than the previous • Ending digits alternate between 6 and 8 • All known perfect numbers are even

Perfect numbers equal the sum of consecutive integers: 6 = 1+2+3, 28 = 1+2+3+4+5+6+7, and so on.

• 28 = 1³ + 3³ • 496 = 1³ + 3³ + 5³ + 7³ • 8,128 = 1³ + 3³ + 5³ + 7³ + 9³ + ... + 15³

In binary, perfect numbers consist of consecutive 1s followed by consecutive 0s: 6 = 110, 28 = 11100, 496 = 111110000, 8,128 = 1111111000000

Around 300 BC, Euclid discovered that perfect numbers have the form 2^(P-1) × (2^P - 1) when 2^P - 1 is prime.

• Start with 1 and repeatedly double: 1, 2, 4, 8, 16, 32... • Add consecutive terms: 1+2 = 3 (prime), so 2 × 3 = 6 • 1+2+4 = 7 (prime), so 4 × 7 = 28 • 1+2+4+8+16 = 31 (prime), so 16 × 31 = 496

Using algebraic manipulation, the sum of consecutive powers of 2 up to 2^(n-1) equals 2^n - 1, making the pattern clear.

Euclid proved this generates perfect numbers but didn't prove it was the only way, leaving open the possibility of odd perfect numbers.

• The nth perfect number has n digits • All perfect numbers are even • All perfect numbers end in 6 and 8 alternately • Euclid's algorithm produces every even perfect number • There are infinitely many perfect numbers

Published in Introdutio Arithmetica, these conjectures were treated as facts for the next 1,000 years without proof.

In the 13th century, Ibn Fallus published a list of 10 perfect numbers; three were incorrect but the others were correct.

• The fifth perfect number has 8 digits, disproving Conjecture 1 • Both the fifth and sixth perfect numbers end in 6, disproving Conjecture 3

French polymath Marin Mersenne extensively studied numbers of the form 2^P - 1 in the 17th century.

In 1644, Mersenne published a list claiming 11 values of P correspond to primes. The first seven were correct and matched the first seven perfect numbers.

For larger numbers like 2^67 - 1, Mersenne admitted he hadn't checked whether they were prime, saying it would take more time than he had.

Mersenne discussed perfect numbers with contemporaries Pierre de Fermat and René Descartes, advancing understanding of the problem.

• Believed no even perfect numbers exist except those of Euclid • Conjectured odd perfect numbers must be the product of a prime and the square of a different number

These would have been the biggest breakthroughs since Euclid 2,000 years earlier, but Descartes couldn't prove either statement.

Descartes believed odd perfect numbers likely exist but admitted the search takes a very long time.

His insights set the stage for future mathematicians to build upon his conjectures about odd perfect numbers.

In 1729, Christian Goldbach introduced the 20-year-old prodigy Leonhard Euler to the work of Fermat, sparking his passion for number theory.

In 1732, Euler discovered the eighth perfect number by verifying that 2^31 - 1 is prime, confirming Mersenne's prediction.

Euler invented a powerful tool that sums all divisors of a number including the number itself. For perfect numbers, sigma(n) = 2n.

If a number is the product of coprime factors, sigma can be split into the product of sigmas of each factor, making complex calculations tractable.

Euler proved that every even perfect number must have Euclid's form 2^(P-1) × (2^P - 1), solving a 1,600-year-old problem.

This result proved Nicomachus's fourth conjecture correct, that Euclid's algorithm produces every even perfect number.

Mathematician William Dunham called it the greatest mathematical collaboration in history, bridging Euclid and Euler across millennia.

The theorem established that all known perfect numbers follow one pattern, making the search for odd perfect numbers the remaining central question.

For an odd perfect number, sigma(n) = 2n, where n is odd. Euler used his sigma function to analyze what this means.

• A prime to an odd power gives even sigma • A prime to an even power gives odd sigma • Since 2n is even and n is odd, exactly one sigma factor must be even

Any odd perfect number must have exactly one prime factor to an odd power and all others to even powers, confirming Descartes' prediction.

Despite this insight, Euler couldn't prove whether odd perfect numbers exist, writing that it is a 'most difficult question'.

For 150 years after Euler, very little progress was made on perfect numbers and no new ones were discovered.

English mathematician Peter Barlow claimed Euler's eighth perfect number would be the last ever discovered, as they are merely curious without usefulness.

Barlow was wrong. Mathematicians continued to pursue perfect numbers despite the century-long hiatus in discoveries.

Researchers began systematically testing Mersenne's list of proposed primes to find new Mersenne primes and their corresponding perfect numbers.

In 1876, Édouard Lucas proved that 2^67 - 1 was not prime but couldn't find its factors.

27 years later, Frank Nelson Cole silently wrote 2^67 - 1 = 147,573,952,589,676,412,927 on one side of a blackboard.

Cole then multiplied 193,707,721 × 761,838,257,287 on the other side, getting the same answer. The audience erupted in applause.

Cole later admitted it took three years of work on Sundays to solve this. Modern computers solve it in less than a second.

From 500 BC until 1952, only 12 Mersenne primes and corresponding perfect numbers were discovered over 2,500 years.

In 1952, Raphael Robinson wrote a program to test Mersenne numbers on the SWAC, the fastest computer at the time.

Within 10 months, Robinson found five new Mersenne primes and corresponding perfect numbers, dramatically accelerating the pace of discovery.

• By end of 1952: largest was 2^2,281 - 1 with 687 digits • By end of 1994: largest was 2^859,433 - 1 with 258,716 digits

In 1996, computer scientist George Woltman launched GIMPS (Great Internet Mersenne Prime Search) to distribute the computational burden across many computers.

GIMPS allows anyone to volunteer their computer power to help search for Mersenne primes through distributed computing.

The project has discovered 17 new Mersenne primes, 15 of which were the largest known primes at their time of discovery.

• Discoverers are credited as the prime's discoverer • Added to a list including history's greatest mathematicians • A $250,000 prize offered for the first billion-digit prime

In 2017, church deacon John Pace using GIMPS discovered 2^77,232,917 - 1, over 23 million digits long, the largest known prime at the time.

Japanese publisher Nanairosha published the entire number in a 719-page book titled 'The Largest Prime Number of 2017'.

The book became a bestseller on Amazon and sold out in four days, demonstrating public fascination with mathematical achievement.

A year later, the 51st Mersenne Prime was found: 2^82,589,933 - 1 with 24,862,048 digits, still the largest known prime.

With only 51 Mersenne primes found, some suspect there might be only finitely many, contradicting Nicomachus's fifth conjecture.

This conjecture predicts how often Mersenne primes should appear based on the size of P and has performed remarkably well with actual data.

The conjecture predicts infinitely many Mersenne primes and therefore infinitely many even perfect numbers, though this remains unproven.

Mersenne primes are so large and rare that finding them requires enormous time and computer resources, making further progress slow.

Do any odd perfect numbers exist? This remains the oldest unsolved problem in mathematics.

Researchers in 1991 proved that if odd perfect numbers exist, they must be larger than 10^300, making brute-force search impractical.

• By 2012, Pascal Ochem and Michael Rao raised the lower bound to 10^1,500 • Recent progress pushed it to 10^2,200 • Unlikely that computers will find one soon

Rather than checking every large odd number, mathematicians seek to prove odd perfect numbers cannot exist through logical constraints.

Mathematicians have been finding more and more conditions that any odd perfect number must satisfy, creating a 'web of conditions'.

• Must have at least 10 prime factors • Must have thousands of non-distinct prime factors • Must be larger than 10^3,000 • Must satisfy many other properties

The hope is that eventually there will be so many constraints that no number can satisfy all of them simultaneously, proving they don't exist.

Since Euler, mathematicians have continued adding new conditions, but so far this hasn't been enough to prove non-existence.

Spoofs are numbers very close to being odd perfect numbers, sharing most properties but failing by one factor. Example: Descartes found 198,585,576,189.

The number 3² × 7² × 11² × 13² × 22021 would be perfect if 22021 were prime, but it equals 19² × 61, so it fails.

By finding properties that all spoofs must have but odd perfect numbers cannot, mathematicians hope to prove odd perfect numbers don't exist.

In 2022, Pace Nielsen and a team at BYU found 21 spoofs and discovered new spoof properties, but haven't yet ruled out odd perfect numbers.

Some experts believe odd perfect numbers simply don't exist, based on mathematical intuition rather than proof.

Carl Pomerance's heuristic argument predicts between 10^2,200 and infinity, there are at most 10^(-540) odd perfect numbers of the form pm².

The same heuristic unexpectedly predicts no large even perfect numbers either, yet infinitely many are expected to exist, undermining confidence in the method.

Experts acknowledge the heuristic can be strengthened with additional considerations, but remain uncertain whether to trust it for odd perfect numbers.

The problem of odd perfect numbers has no known direct applications to real-world problems, unlike applied mathematics.

Number theory had no applications for over 2,000 years while mathematicians built foundational knowledge purely from curiosity.

In the 20th century, number theory became the foundation of modern cryptography, protecting everything from text messages to government secrets.

Einstein's general relativity was built on non-Euclidean geometries developed as intellectual curiosities without foreseeable application, showing curiosity-driven math creates breakthroughs.

About 10-15 people currently have research papers in the area of perfect numbers and related problems.

High school students passionate about mathematics can work on this problem and make genuine progress discovering new things.

• Hundreds of people have studied this problem for thousands of years • That doesn't prevent newcomers from contributing • Each person can discover something new

Doing mathematics is the only way to know what the outcome will be. The problem might be a dead end or remarkably helpful, but only trying reveals which.