The Oldest Unsolved Problem in Math

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Perfect Numbers

A perfect number is a positive integer that is equal to the sum of its proper divisors.
The oldest unsolved problem in mathematics is to determine if any odd perfect numbers exist.
The only even perfect numbers known to the ancient Greeks were 6, 28, 496, and 8,128.
Euclid discovered a pattern that generates even perfect numbers: (2^{p-1} \times (2^p - 1)), where (p) is a prime number.
Nicomachus published five conjectures about perfect numbers, including that all perfect numbers are even and end in 6 or 8 alternately.
Ibn Fallus published a list of 10 perfect numbers, but three of them were incorrect.
Marin Mersenne studied numbers of the form (2^p - 1) and identified several Mersenne primes, which correspond to perfect numbers.
Pierre de Fermat and Rene Descartes believed that if an odd perfect number exists, it must have a special form.
Descartes proposed the existence of perfect numbers but couldn't prove it.
Leonhard Euler made three breakthroughs in studying perfect numbers.
Euler proved that every even perfect number has Euclid's form, solving a 1600-year-old problem.
Euler conjectured that every odd perfect number must have a specific form but couldn't prove its existence.
Progress in finding new perfect numbers was slow until the advent of computers.
Raphael Robinson used a computer program to find five new Mersenne primes and corresponding perfect numbers in 1952.
The Great Internet Mersenne Prime Search (GIMPS) was launched in 1996 to distribute the search for Mersenne primes over many computers, leading to the discovery of 17 new Mersenne primes.
The largest known prime number is 2 to the power of 82,589,933 minus 1, discovered in 2018.
Mersenne primes are a special type of prime numbers that are almost always the largest known prime numbers.
The Lenstra and Pomerance Wagstaff conjecture predicts that there are infinitely many Mersenne primes and even perfect numbers.

Odd Perfect Numbers

Odd perfect numbers are numbers that are the sum of their proper divisors and are odd.
No odd perfect numbers have been found yet, and the largest known lower bound for an odd perfect number is 10 to the power of 2,200.
Spoofs are numbers that are very close to being odd perfect numbers, but are not perfect because they have a composite prime factor.
A heuristic argument suggests that there are no odd perfect numbers between 10 to the power of 2,200 and infinity.

Applications and Significance

Number theory had no real-world applications for over 2000 years, but it later became the foundation of cryptography.
The problem of perfect numbers has no known applications, but it has led to the development of new mathematical ideas and techniques.
Around 10 to 15 mathematicians are currently working on the problem of perfect numbers.

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