ODD PERFECT NUMBER
SUKI YAN YE
Abstract. Perfect numbers are not only special but also rare. Since the first
perfect number was found, there were only a limited number of perfect numbers are known today, and all of them are even numbers. Is there an odd
perfect number? Since the number can go indefinitely large, can we sure there
is no odd perfect number exist somewhere larger than the largest perfect number we do know today?
1. Introduction
It takes God six days to bulit the world; 28 days for Moon revolve the earth.
Why did God use six days, not five days or even eight days? How come the Moon
revolves around the earth in twenty-eight days, not thirty days? Is there any relationship between 6 and 28? The ancients of Greek named these two number as
”Perfect number”, not because these two numbers were generated by God nor the
Moon, was because they believed 6 and 28 were the best two numbers. The great
ancients of Greek found out that both six and eight had the same kind of pattern,
the sum of the proper divisors of these numbers are both equal to the number itself. For instance: the proper divisor of 6 : 1, 2, 3 and the sum is 1 + 2 + 3 = 6;
the proper divisor of 28 : 1, 2, 3, 4, 7, 14 and the sum 1 + 2 + 3 + 4 + 7 + 14 = 28.
Therefore, perfect number is any number that is equal to the sum of its proper
divisors. Euclider first discovered perfect number as early as 300 B.C, that ”if a
number of the form 2n − 1 is prime, then the perfect number would have the form
of ( Gimbel and Jaroma):
2n−1 (2n − 1)
Ever since the fist found of perfect number, only a limit amount of perfect numbers were found during throusand years without computer. Still, some mathematicans wrer firmly believe that the existence of perfect number was true. According
to Gimbel and Jaroma, ”Euler proved that all even perfect number are necessarily
of Euclid’s form” eventhough he did not know how many even perfect number were
existed at that time, not to metion odd ones.
2. History of finding odd perfect number
According to Dickson, on November 15, 1638, Rene Descartes first considered
the exitence of odd perfect number. He wrote a letter to Mersenne stated that he
Date: May 2, 2011.
Term Paper final draf.
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could prove the exitence of even perfect number that fitted the form of Euclid’s and
odd perfect number ”must have the form ps 2 , where p is a prime.” (Dickson, p12)
He saw no reason why an odd perfect number may not exist. For
p = 22021, s = 3 · 7 · 11 · 13, ps 2 would be perfect if p were prime
[but p = 61 · 192 ]
In 1657, Frenicle proved Descartes’ hypothesis and observed that ”p must be of
the form 4n + 1” (Dickson, p322).
In the early 19th century, the idea of odd perfect number and its existence
seemed to be more opened minded. In 1832, American mathematician Benjamin
Peirce proved that the first odd perfect number must have at least four factors.
(Gimbel and Jaroma, p2)
In 1888, British mathematician James Joseph Sylvester stated about the exixtence of odd perfect number in his eloquence that (oddperfect.org):
”...a prolonged meditation on the subject has satisfied me that the
existence of any once such – its escape, so to say, from the complex
web of conditions which hem it in on all sides – would be little short
of a miracle.”
In late 1888, Sylvester improved his odd perfect number finding result that a odd
perfect number can be divisible by at least five different prime divisors, and kept
working on the research. Coincidentally, Peirce and Sylvester both demonstrated
the similar result independently (Jennifer and Jaroma, p 49). Later that year,
Sylvester showed that there was no such case that ”a odd perfect number can be
divisible by 105, as well as put into place a lower bound of eight distinct prime
factors for an odd perfect number not divisible by 3.”(Gimbel and Jaroma, p3)
Futhermore, Hagis said that in 1913, L.E. Dickson had proved the distinct prime
factors of odd perfect numbers was finite; in 1972, Robbins from Brooklyn Polytechnical Institute and Pomerance from Harvard had proved that a odd perfect
number can be divisible by at least seven different prime divisors (Hagis, p1027).
Until today, no mathematician is able to find any odd perfect numbers.
3. The approach
Althought Euclid was not the first one who discovered the existence of perfect
number, yet he be the first one recorded in the history. The first recorded mathematical book that had metioned about perfect numbers was Elements written by
Euclid in around 300 B.C. Thought, Elements was a book about number theory,
people thought it as a geometry book due to the numbers were arranged by line
segments which cause it looked like a geometry book. According to O’Connor and
Robertson, the Proposition 36 had stated in Euclid’s book IX in the Elements:
ODD PERFECT NUMBER
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If as many numbers as we please beginning from a unit be set out
continuously in double propotion, until the sum of all becomes a
prime, and if the sum multipied into the last make some number,
the product will be perfect.
What Euclid mean of ”double propotion” was that, the sequence of the second
number is the twice of the previous number. For instance, 1 + 2 + 4 = 7 since 7
is prime number, so we stop and use 7 as the sum. Because the last number is
4, therefore, 7 ∗ 4 = 28 as the ”sum multiplied into the last number” which is the
second found perfect number. Try the next series,1 + 2 + 4 + 8 + 16 = 31 and
31 ∗ 16 = 496 also a perfect number. As result, Euclid gave a rigorous proof of his
propositiion and we could use the fact he had proof to establishe the form of ” the
sum” as :
∞
X
(1 + 2 + 4 + 8 + 16 + ... + 2n−1 ) = 2n − 1
n=2
and the form of perfect number when ”the sum” is prime:
(2n − 1)(2n−1 )
After Euclid’s first approach to perfect number, many mathematicians had worked
hard in the perfect number field and hoped to discover more perfect numbers or
the pattern of it. Although some found new perfect numbers, no one had successed
in finding odd perfect numbers nor to provf its existence.
As metinoed in the begining, Descartes was the first to come up with odd perfect
number. However, he could not able to get any result of odd perfect number, except
the ideal of its existence and the amount of work that required to go through. After Descartes, couple other famous mathematicians like Fermat, Frenicle de Bessy
and Cataldi, were also contribute to the perfect number, but only theorems and
propositions, none of them had solid result in odd perfect number, till Euler.
According to Sandifer’s research on How Euler Did It, Euler stated his approach to the perfect number including even and odd in an unpublished manuscript, Tractatusit de numerorum doctrina capita sedecim quae wupersunt, ”Tract
on the doctrine of numbers, consisting of sixteen chapters.” He got the similar even
perfect number formular with Euclid’s, but different for odd one. (Sandifer, p4):
Suppose that N is and odd perfect number and that it factors to
be N = ABCD etc. Euler implicitly assumes that these factors,
A, B , C , D, etc. arepowers of distinct primes, though he doesn not
say so explicitly. Since N is odd, all of these factors must be odd,
and since the prime factors are distinct, it must be that:
Z
2N =
Z
N=
Z
ABCDetc =
Z
A
Z
B
Z
C
D
etc.
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Euler and Euclid called 2N as ”oddly even” which is the double of odd number N ,
and is divisible by 2 but not 4 for the number itself and its
R factors.
R
R In the same
study by Sandifer, he addressed that Euler supposed that A, B, C.... would
had one oddly even and the rest of them must be odd to make the statement be
ture.
Refer to the long work that had done by Euler, Sandifer concluded that all of the
factors in odds, must be even power of a prime, so they must be perfect squares.
For the oddly even, Sandifer quote from Euler that, suppose A is the oddly even,
then (Sandifer, p4):
R
Euler tells us that if we write A = q m , then for A to be oddly
even, we must have q a prime number of the form 4n + 1 and also
m must be an odd number of the form 4λ + 1.
Therefore, odd perfect numbers must satisfy the form of (4n + 1)4λ+1 , where the
4n + 1 is a prime number.
For over hunder years, the progress of finding odd perfect number had been
frozen on what Euler had found. In 1888, another genius mathematician after
Euler, Sylvester, had finally made another step on finding odd perfect number.
According to Wagon, Sylvester proved that any odd perfect number must have at
least 5 distinct prime factors in 1888. Since then, no one has found any odd perfect,
but increasing the lower bound limit of odd perfect number larger and larger with
the help of computer. (O’Connor and Robertson)
...over the years, this has been steadily improved until today we
know that and odd perfect number would have to have at least
eight distinct prime factors, and at least twenty-nine prime factors
which are not necessarily distinct. It is also known that such a
number would have more than 300 digits and a prime divisor greater
than 106 . The problem of whether an odd perfect number exists,
however, remains unsolved.
In addition to Sylvester and other mathematician who fail to discover odd perfect
number, Dr. Carl Pomerance announced a heuristc that the odd perfect numbers
are unlikely to be existed. In 2003, Dr Pomerance wrote to Joshua Zelinsky about
his heuristc (oddperfect.org):
...There are amost log m possibilities for p, so the ”probability”
that at least one of these works is (log m)/m2 . Then sum this
expression over m. Since the sum converges, it ”follows” that there
are only finitely many odd perfect numbers. In Fact, since we’ve
searched up to 10300 with out finding any, and since m2 > 101 50
for an odd perfect number n > 10300 , it may be more appropriate
to sum (log m)/m2 for m > 1075 . This < 10−70 , which is so tiny,
that it is reasonable to conjecture that there are no odd perfect
numbers.
ODD PERFECT NUMBER
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4. Really? No odd Perfect number?
All the result relating to odd perfect number is the bounding limit being extended from twenty digits that found by Kanold in 1957 to thirty-six digits by
Tuckerman in 1973. Then, from Hagis’ result fifty digits in late 1973, to more that
200 digits by Buxton, Elmore and Stubblefield in 1976, sourced form Guardian.
Althought no one is capatable to find a single odd perfect number or to prove
the existence of odd perfect number for more than two thousand years, after generations of mathematician’s research, and the help ofcomputer, such number must
be huge, but how huge? As mathematician loves to solve problems that seems unable to solve, finding one odd perfect number or to proving such number ever exist
become something is electrifying. Before the first perfect number had discorved,
no one ever consider the exitence of perfect number. When the first couple perfect
number had been found out, no one dare to say all the perfect numbers were found.
Similary, no one can be sure that no such odd perfect number is existed, but what
is the use of odd perfect number if it is existed? We don’t know yet, since we
couldn’t find the whold series of perfect number, and there is not able to apply on
any reality situation. Searching for perfect number, like children chasing for birds,
no actual use but fun ( for some mathematicians). As the computer nowaday can
excute more and more digits, the prosibility of finding one would increase among
with the digits of number we are able to work with, hopefully someone would invent
the use of perfect number.
5. References
(1) Gimbel, Steven and Jaroma, John H. ”Sylvester: ushering in the mordern
era of research on odd perfect numbers”. Odd Perfect Number Search.
http:// oddperfect.org/against.html. October 23, 2003.
(2) Dickson, Leonard Eugene. ”History of the theory of numbers”. Carnegie
Institution of Washington. Print. Year 1919.
(3) O’Connor, J.J. and Robertson, E. F. ”Perfect Numbers”. http://wwwhistory.mcs.st-andrews.ac.uk/HistTopics/Perfect-numbers.html. May 2009.
(4) Euler, Leonhard. ”Tractatus de numerorum doctrina capita sedecim quae
supersunt, Commentationesarithmeticae 2, 1849, pp.503-575, reprinted in
Opera Omnia Series I vol 5 p. 182-283. www.eulerarchive.org.
(5) Sandifer, Ed. ”How Euler Did It” Odd PerfectNumbers.
http://www.maa.org/editorial/euler/HowEulerDidIt37oddperfectnumbers[1].pdf.
November, 2006.
(6) Betcher, Jennifer T., and John H. Jaroma.”An extension of the reults of
Servais and Cramer on odd perfect and odd multiply perfect numbers.”
TheAmericanMathematical Monthly. 110.1 (Jan 2003): 49-52. Print.
(7) Hagis, Peter Jr. ”Every Odd Perfect Number has at least Eight Prime
Factors”.Mathematicof Computation. Vol. 35. No.151( Jul., 1980), pp
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1027-1032. Print
(8) ”Computer Guardian: Odds on a perfectly odd number - Micromaths.”
Guardian [London, England] Apr 26, 1990:33. Print.