Junior Villafana
Math 301
Dr. Meredith
Odd Perfect Numbers
Arguably the oldest unsolved problem in mathematics is still giving mathematicians a headache,
even with the aid of technology; the search for an odd perfect number is still going on today. There has
not been a lot of success with finding what an odd perfect number would be, but there have been
discoveries that explain what it could possibly look like. I will give some background information on
perfect numbers in general and those who are pioneers to this field. A perfect number is a number
whose divisors, excluding the number itself, sum up to itself. In less modern terms, the aliquot parts of
the digit would be summed up. Aliquot parts are the proper quotient of the number one is dealing with.
8
8
8
4
8
2
So the aliquot parts of 8 are 1, 2 and 4; this is because 1 = , 2 = and 4 = . 8 is not considered an
aliquot number since 8 =
8
1
is not a proper quotient. When dealing with perfect numbers we are only
looking at the positive integers, 𝒁≥0 ; there are no irrational, imaginary, or any other sort of numbers
that would be looked at as perfect from a mathematical point of view. Although it is not certain who
were the first to take interest in this idea of perfect numbers, but there is still a lot of history behind it.
In fact the first four perfect numbers were discovered in Ancient times, yet no one knows by whom. One
thing that is known is that these interesting numbers were studied by Pythagoras and his followers,
from a less theoretical standpoint than mathematicians that followed them. Now let’s look at an
example of the most basic case of a perfect number, this number would be the 6. The divisors of 6 are 1,
2, and 3. When taking the sum of these, 1 + 2 + 3 = 6, so 6 would indeed be considered a perfect
number. This is actually the first perfect number that exists. There are many perfect numbers that have
been found over the thousands of years that they have existed, but no one has been able to find one
that is odd or prove if one even exists.
Many of the greatest and most famous mathematicians through the ages have tried to come up
with an answer for odd perfect numbers, yet no one has been able to succeed. The first written
discussion of perfect numbers was found in Euclid’s Elements. In his book, Euclid explains how to find
perfect numbers, there was no distinction whether they would be even or odd. Although now we know
that Euclid’s formula can only yield even numbers. His explanation and proof were then turned into a
more modern proposition which reads:
𝐹𝑜𝑟 𝑘 > 1, 𝑖𝑓 2𝑘 − 1 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑡ℎ𝑒𝑛 2𝑘−1 2𝑘 − 1 𝑖𝑠 𝑎 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑛𝑢𝑚𝑏𝑒𝑟.
Nicomachus was another mathematician that too interest in Euclid’s discoveries. Nicomachus broke up
the integers into 3 different groups; superabundant, deficient and perfect. This next quote describes
how these groups were characterized, “Among simple even numbers, some are superabundant, others are
deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle
position between the two, they are said to be perfect. And those which are said to be opposite to each other, the
superabundant and the deficient, are divided in their condition, which is inequality, into the too much and the too
little”. Nicomachus was referring to the aliquot numbers of a digit. A number whose sum of aliquot
number is less than itself makes it deficient, greater than itself makes it superabundant and equal to
itself makes it perfect. Along with his 3 classes he also made a few bold properties that he believed
were true about perfect numbers: 1) the nth perfect number would have n digits 2) all perfect numbers
would be even 3) all perfect numbers would end in 6 or 8 alternating between the two 4) Euclid’s
method would give all possible perfect numbers and 5) there exist infinitely many perfect numbers. All
of these assertions were made without any type of proof; they were just believed to be true at the time.
As it turns out, not all of these would be valid assertions.
A familiar name that many of us know is Pierre de Fermat, he too was trying to contribute to the
theory of perfect numbers. Fermat had the intentions of publishing the work that he had done on this
topic but never had the time to get to it and he was not as successful on the subject as he had hoped.
While exploring perfect numbers Fermat was able to come up with his famous theorem, “ Fermat’s little
theorem”. With that being said, Fermat was writing to other mathematicians at the time and explaining
what he was able to find out about these perfect numbers. One person who really utilized Fermat’s
information was Mersenne, he would eventually go down in mathematical history for his discovery. He
claimed that for prime numbers, p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257 would produce perfect
numbers. Mersenne was a little off when stating this but he was very close to being correct; There was
no way of checking these primes at that time, but later it was shown that 67 did not produce a perfect
number and also that Mersenne had missed the numbers 61, 89, and 107. Mersenne would have a
particular prime named after him, namely the prime 2𝑘 − 1, which today we know as the Mersenne
primes. This special type of prime is found only when k itself is a prime number. Euclid and Mersenne
both knew this. Hence the reason that Mersenne had started off with a list of primes to come up with
this new number. These specific primes would go hand in hand when finding perfect numbers. As it
turns out, these Mersenne primes are extremely rare. Not every prime number, k, put into 2𝑘 − 1 will
result in a prime. It turns out to be really tedious work showing that a Mersenne prime is indeed a
prime, the latest Mersenne prime took 26 days to check to ensure that the number in fact could not be
factored; but finding a Mersenne prime would ensure another perfect number. Yet the problem persists
to be that none have been odd.
Euler would later come to show a way to describe the form of an odd perfect number by using
some information that Descartes had written in a letter for Mersenne. Euler made an observation that
every odd number could be written as 4n+1 or 4n+3. He showed that when an odd number, q, is
written as 4n+1, any exponent of q will be 1 (mod4). Euler then also proved that an odd perfect prime
would have to be in this form,
4𝑛 + 1
4𝑘+1
(𝑏2 ) where b is an odd number and 4n+1 is a prime.
Euler contributed towards the efforts on even perfect numbers as well. He discovered a perfect number
that was larger than any other before him. Euler in 1772 proved that 230(231 - 1) was a perfect number,
this was only the 8th perfect number that was found. It took 111 years for someone to find the next
perfect number, Pervushin discovered that 260(261 - 1) was indeed the 9th perfect number. Also today,
computers are used to compute Mersanne primes which is in direct correlation to finding a new perfect
number. The numbers are gigantic in size, some ranging well over 12 million digits. GIMPS, Great
Internet Mersenne Prime Search, was founded by Richard Crandall in 1996. This is an organization
dedicated to finding new Mersenne primes.
Descartes was the first person to hint at the existence of an odd perfect number. He had taken a
look at what Euclid had discovered and saw no reason there would not be one. Many in this day believe
that there does exist such a number and it is known that it would have to be a very large number at that.
James Joseph Sylvester was the next to make an important step towards finding this very rare integer.
He would go on to show that if an odd perfect number were to exist it would have to contain at least 5
distinct prime factors. Mathematicians and computers have helped out a lot in the last 120 years, in
1888 Catalan had proved that if an odd perfect number was not divisible by 3, 5, or 7 then it had at least
26 distinct primes. Then later Norton improved on Catalan’s work to show that it had to actually have 27
distinct primes instead of 26. Other discoveries of that sort have been made, Neilson in 2006 showed
that if an odd perfect number was not divisible by 3 then it had at least 12 distinct primes. Also in 2006,
Neilson had shown that the minimum number of distinct primes for a general odd perfect number had
to be 9. The most recent contribution has come from Kevin Hare who has shown that 75 prime factors
would have to be contained in any given odd perfect number. He was able to show this by using ECM
(Elliptical curve factorization method) factorization, this computes large multiple point on an elliptical
curve modulo the number that you desire to be factored. This method is the same method that other
organizations are using to find Mersenne primes. In 1953 Jaques Touchard had proved that any odd
perfect number would have to be in the form 12k + 1 or 36k + 9. We also know that an odd perfect
number would have to be greater than 300 digits and it would have a prime divisor greater than
1,000,000. There has been no overwhelming evidence showing that it is impossible for an odd perfect
number to exist, but no one will disagree that is will be a very difficult task to show that one in fact exist.
There is some hope though in finding such a number, when looking at how large the even
perfect numbers discovered are, there is no overwhelming evidence that states that an odd perfect
number might just be larger. Many of the Pioneers of Mathematics believed there existed such a
number; they were just not able to produce one. Today it is unknown if perfect numbers have any
practical use. On a more religious point of view, it took God six days to construct Earth and also God
chose to make the moon take 28 days to travel around the Earth. Both 6 and 28 are perfect numbers,
some people believe that there is a direct relationship between these numbers and religion. Saint
Augustine wrote in his famous text The City of God, “ Six is a number perfect in itself, and not because
God created all things in six days; rather, the converse is true. God created all things in six days because
the number is perfect... “. Although I do not believe that is the main reason for Mathematicians to strive
to find the first odd perfect prime, I believe that it is the theory of an odd perfect number is what seems
to get their attention. Looking ahead into the future technology is growing ever so fast; programs are
being created to help out in this cause. People are still discovering new perfect numbers that satisfy the
old formulas derived by Euclid, Fermat and Mersanne. Euler’s proof of the form of an odd perfect
number could serve as a template for someone that could very well lead them to find what
mathematicians throughout the ages could not. The question of does there really exist an odd perfect
number, is what seems to keep the interest alive in the topic. Famous Mathematicians throughout the
ages, Euclid, Descartes, Mersenne, and Euler are some of the more recognizable names but certainly not
the only ones that have contributed to finding an odd perfect number. There still seems to be an
interest in the topic since countless mathematicians have been finding new discoveries. Names like Karl
Norton, Peter Hagis, Scott Kurowski and George Woltman are just a few names that have been dealing
with perfect numbers over the last 60 years. The excitement of uncertainty and the long history behind
the theory of this problem must also be in the back of the minds of those working to solve this question.
I’m sure that there would be a nice amount of money given to the person that finds the first odd perfect
number, looking at the prize money given to those who found the 45th Mersenne prime, $100,000.00,
one would assume that finding the first odd perfect number would be worth more. One thing is for
certain, if anyone were to find an odd perfect number they would go down in Mathematical History.
Bibliography
Odd Perfect Number
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2006. Print.
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"Perfect Numbers." Welcome to the Turnbull Server. Web. 08 Mar. 2011. <http://www-groups.dcs.stand.ac.uk/~history/HistTopics/Perfect_numbers.html>.
"Odd Perfect Number -- from Wolfram MathWorld." Wolfram MathWorld: The Web's Most Extensive
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