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ODD PERFECT NUMBERS
The Existence of Odd Perfect Numbers:
The Conditions That Must Be Satisfied
Sara Valco
San Francisco State University
ODD PERFECT NUMBERS
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Abstract
A perfect number is a positive integer that is equal to the sum of its positive
divisors, and can be represented by the equation  (n)  2n . Even perfect numbers have
been discovered, and there is a search that continues for odd perfect number(s) A list of
conditions for odd perfect numbers to exist has been compiled, and there has never been a
proof against their existence.
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The Existence of Odd Perfect Numbers:
The Conditions That Must Be Satisfied
The great philosopher of his day, Saint Augustine, believed that the number 6 was
a truly perfect number, because God had created the Earth in 6 days (Voight, 2003). He
was in fact correct, the number 6 is a perfect number, but for a much more mathematical
reason. The factors of 6 include 1, 2, 3, and 6. When added together these factors are
equal to 12. When divided by 2, 12 becomes equal to 6. A general definition of a perfect
number is an integer where all of its possible divisors add up to the integer. Today the
present equation that denotes a perfect number x , is  (x)  2x , where  (x) is the sum
of all the positive divisors of x , as well as 1 and itself (Firoozbakht & Hasler, 2010).
Around 300 BCE, Euclid was the first to discover a formula for even perfect numbers,
2 p1 (2 p  1) , where (2 p  1) is prime number, called a Mersenne prime (Firoozbakht &
Hasler, 2010). There is no proven formula for odd perfect numbers and no odd perfect
number has ever been found, however there is no proof of their nonexistence. What is
known about odd perfect numbers is that they must satisfy a number of conditions to be
classified as an odd perfect number. In the paper titled, Odd Perfect Numbers Have at
Least Nine Distinct Prime Factors, by Nielsen (2007), he lists the necessary conditions
for the existence of an odd perfect number. Those conditions include Eulerian Form,
Lower Bound, Upper Bound, Large Factors, Small Factors, Number of Total Prime
Factors, Number of Distinct Prime Factors, and the Exponents. This paper will go into
detail of the history and some of the methods used by mathematicians to prove these
conditions.
If an odd perfect number, N, exists, then let it be in the formula N   ik1 piai ,
where each pi is a prime number and k is the number of distinct prime factors (Nielsen,
2007). The Swiss mathematician, Leonhard Euler, was able to show that the prime
factorization of an odd perfect number must be in the form of
n  p0e0 p12e1 ...pk2ek , p0  e0  1 (mod 4), with n beings the odd perfect number (Goto &
Ohno, 2008). This illustration of the factorization of n has become known as Eulerian
Form and is first on the list of the necessary conditions for an odd perfect number to pass.
Research has found lower bounds for odd perfect numbers, which is a condition
that was previously mentioned in Nielsen (2007). Brent, Cohen, and te Riele are able to
show in their paper, Improved Techniques for Lower Bounds for Odd Perfect Numbers,
that the lower bound for odd perfect numbers is N  10 300 (1991). In an earlier paper by
Brent and Cohen (1989), A New Lower Bound for Odd Perfect Numbers, they describe
the algorithm they use to calculate the lower bound for odd perfect numbers, which they
again use in their later work. In the earlier paper they discover the lower bound to then be
10160 . Their method uses Eulerian Form as the standard for N, and because  is
multiplicative and  (N)  2N , the sets {q : q prime, q  2,q  ( piai ) for some i=0, 1,…,
t} and { p0 ,..., pt } are equal (Brent & Cohen, 1989). The researchers use the factor chain
method for the problems with odd perfect numbers and from any component p a of N,
further prime factors of N may be generated, mainly the odd prime factors of  ( p a ) . A
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computer program would factorize until either a contradiction was reached or they found
an odd perfect number, which unfortunately has not yet been the case.
We revisit Nielsen in another one of his works titled, An Upper Bound for Odd
Perfect Numbers (2003). Throughout the paper, it is assumed that n,d Z . Theorem 1
in Nielsen’s paper states that if N is an odd perfect number with k distinct prime factors
k
then N  (d  1)4 (Nielsen, 2003). As a result of Theorem 1, Corollary 1 then says that if
k
N is an odd perfect number then the upper bound is N  2 4 (Nielsen, 2003). The bounds
are tightened even more when we know which small prime factors divide N, for example
k1
Corollary 2 states that if N is an odd perfect number and 5 N , then N  5  4 4 (Nielsen,
2003).
The Large Factors condition has been previously proven by Hagis and Cohen
(1995), Jenkins (2003), and finally Goto and Ohno (2008). Hagis and Cohen (1995)
wrote their paper to prove that if there is an odd perfect number N, then N has a prime
factor that exceeds 10 6 . The researchers prove their theory using contradiction. They
knew that they could have proved pk  10 7 , but had the resources to compute and prove
pk  10 6 . A computer, called the CYBER 860, at the Temple University Computing
Center contained in its memory a list of the 78498 primes up to 10 6 (Hagis & Cohen,
1995). They determined that to compute 10 7 , the computer’s memory would have to
contain a list of 664579 primes, which would take a significant amount of more time.
Jenkins (2003) followed the method used by Hagis and Cohen, to prove that pk  10 7 .
Goto and Ohno (2008) increased the number so that every perfect number is divisible by
a prime exceeding 10 8 . The purpose of their paper, Odd Perfect Numbers Have a Prime
Factor Exceeding 10 8 , is to show that this is true for odd perfect numbers as well. To test
this the researchers designed a computer program. They found that in the case of the
bound 10 8 , if r ( p) is acceptable, where r ( p) are cyclotomic numbers that are
acceptable values for the cyclotomic polynomial, denoted by d ( p) , then r  47 , where
r is a possibly even prime and p is an odd prime. The cyclotomic polynomial used for
the proof is  ( pe )  1  p  p2  ...  pe   d (e1),d 1 d ( p) (Goto & Ohno, 2008).
2
k  2 , where N is
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an odd perfect number (as cited in Voight, 2003). Later in his paper titled, On Odd
Perfect, Quasiperfect, and Odd Almost Perfect Numbers, Kishore proved that
Grun proved that the smallest prime factor p1 N satisfies p1 
M   i 1 piai , where M is an odd perfect number, pi  22 (r  i  1) for 2  i  6
r
i1
(Kishore, 1981). Therefore, the bounds for the prime factors of odd perfect numbers
currently stands as the upper bounds exceeding 10 8 and the lower bounds being less than
i1
22 (r  i  1) for 2  i  6 . Without being sure of the value of r in the lower bound, this
appears to be a large span of possible prime factors of an odd perfect number. However,
since no number has yet been proven to exist, this large difference between the bounds
can be reasonably expected.
Hare (2004) proved that the number of total prime factors for an odd perfect
number, N, that are not necessarily distinct, must be at least 47. This total number of
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prime factors is defined as (N ) :   2 kj 1  j . Previous to Hare’s findings, Sayers
proved (N)  29 and then Iannucci and Sorli extended the bounds to (N)  37 (Hare,
2004). To prove (N)  47 , Hare assumed (N ) :    2i  45 and obtained a
contradiction for every combination of  and i (Hare, 2004). The main contradictions
that are tested for are excess of a given prime, excess of the number of primes, partition
cannot be satisfied, and/or excess of  1 .
Chein and Hagis, independent of each other, proved that the number of distinct
prime factors for an odd perfect number is represented as  (N)  8 (Nielsen, 2007). Then
Nielsen was able to increase this number to  (N)  9 . To test this, Nielsen used a
computer to run his algorithm, which utilized a factor chain argument. Every possible
number of distinct prime divisors of N was tested for contradictions. Nielsen describes
the case where k, which is the number of distinct prime factors of N, is equal to 4 and p,
which is the number of prime divisors, is equal to 3 or 5. There are an infinite number of
cases possible for each value of p, so it is impossible to be able to analyze each case.
However, each case is inspected until a contradiction is found.
The Exponents condition is for pii N non-special, let  i  2i for each i
(Voight, 2003). Voight (2003) cites that Kanold (1942) proved that i  2 is impossible
and that if d  gcd( i  1)  1 , then 9, 15, 21, 33  d .
The question of whether or not odd perfect numbers are in fact real or just an
urban legend continues. There are two main arguments against the existence of odd
perfect numbers that are cited oddperfect.org, which include Sylvester’s Web and
Pomerance’s Heuristic. James Joseph Sylvester in 1888, referred to the conditions
described above as, “the complex web of conditions,” and thought that it would be just
short of a miracle for an actual odd perfect number to exist through the web
(oddperfect.org/against.html). The reason this problem is so difficult is that numbers
exceeding thousands and millions of digits must be factored to solve. However, Sylvester
made this claim before computers, when work by hand was the only considerable option.
Dr. Carl Pomerance wrote the Pomerance Heuristic, which implies that there should be
no large perfect numbers, which we know is untrue for even perfect numbers, so may
also be untrue for odd perfect numbers. Dr. Pomerance claims that Mersenne Primes are a
conspiracy and that there may be a conspiracy around producing odd perfect numbers.
So the search for odd perfect numbers continues. Mathematicians believe that
there possibly exists an infinite number of perfect numbers. Similar to the search for odd
perfect numbers is the search for more even perfect numbers. There is a relatively small
finite list of even perfect numbers and mathematicians are seeking to increase that list.
With no proof against odd perfect numbers’ existence, and the list of conditions for the
odd perfect numbers to meet, the search could be infinitely long.
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References
Brent, R. P. & Cohen, G. L. (1989). A new lower bound for odd perfect numbers.
Mathematics of Computation, 53(187), 431-437.
Brent, R. P., Cohen, G. L., & te Riele, J. J. (1991). Improved techniques for lower bounds
for odd perfect numbers. Mathematics of Computation, 57(196), 857-868.
Firoozbakht, F. & Hasler, M. F. (2010). Variations on Euclid’s formula for perfect
numbers. Journal of Integer Sequences, 13(10.3.1), 1-18.
Goto, T. & Ohno, Y. (2008). Odd perfect numbers have a prime factor exceeding 10 8 .
Mathematics of Computation, 77(263), 1859-1868.
Hagis, P. Jr. & Cohen, G.L. (1998). Every odd perfect number has a prime factor which
exceeds 10 6 . Mathematics of Computation, 67(223), 1323-1330.
Jenkins, P. M. (2003). Odd perfect numbers have a prime factor exceeding 10 7 .
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Kishore, M. (1981). On odd perfect, quasiperfect, and almost perfect numbers.
Mathematics of Computation, 36(154), 583-586.
Nielsen, P. P. (2003). An upper bound for odd perfect numbers. Electronic Journal of
Combinatorial Number Theory, 3(A14), 1-9.
Nielsen, P. P. (2007). Odd perfect numbers have at least nine distinct prime factors.
Mathematics of Computation, 76(260), 2109-2126.
OddPerfect.org. Retrieved from http://www.oddperfect.org/
Voight, J. (2003). On the nonexistence of odd perfect numbers. American Mathematical
Society, 293-300.
Weinsstein, E. (2011, March 7). Perfect number. Retrieved from
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http://mathworld.wolfram.com/PerfectNumber.html
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