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In Book IX of The Elements, Euclid gave a method for constructing perfect numbers (Dickson 2005, p. 3), although this method
applies only to even perfect numbers. In a 1638 letter to Mersenne, Descartes
proposed that every even perfect number is of Euclid's form, and stated that he saw
no reason why an odd perfect number could not exist (Dickson 2005, p. 12). Descartes
was therefore among the first to consider the existence of odd perfect numbers; prior
to Descartes, many authors had implicitly assumed (without proof) that the perfect
numbers generated by Euclid's construction comprised all possible perfect numbers
(Dickson 2005, pp. 6-12). In 1657, Frenicle repeated Descartes' belief that
every even perfect number is of Euclid's form and that there was no reason odd perfect
number could not exist. Like Frenicle, Euler also considered odd perfect numbers.
To this day, it is not known if any odd perfect numbers exist, although numbers up to  have been checked without success,
making the existence of odd perfect numbers appear unlikely (Brent et al. 1991;
Guy 1994, p. 44). The following table summarizes the development of ever-higher
bounds for the smallest possible odd perfect number. There is a project underway
at http://www.oddperfect.org/
seeking to extend the limit beyond  .
| author | bound | | Kanold (1957) |  | | Tuckerman (1973) |  | | Hagis (1973) |  | | Brent and Cohen (1989) |  | | Brent et al. (1991) |  |
Euler showed that an odd perfect number, if it exists, must be of the form
 |
(1)
|
where  is a prime of the form  (Fermat's 4n+1 theorem; Burton 1989), a result
similar to that derived by Frenicle in 1657 (Dickson 2005, pp. 14 and 19). In
other words, an odd perfect number must be of the form
 |
(2)
|
for distinct odd primes  ,  , ...,  with  (mod 4).
Steuerwald (1937) subsequently proved that the  s cannot all
be 1 (Yamada 2005).
Touchard (1953) proved that an odd perfect number, if it exists, must be of the form  or  (Holdener
2002).
In 1896, Stuyvaert proved that an odd perfect number must be a sum of two squares (Dickson 2005, p. 28). In 1887, Sylvester conjectured and in 1925, Gradshtein
proved that any odd perfect number must have at least six distinct prime factors (Ball and Coxeter 1987). Hagis (1980)
showed that odd perfect numbers must have at least eight distinct prime factors, in which case, the number is divisible
by 15 (Voight 2003).
In 1888, Catalan proved that if an odd perfect number is not divisible by 3, 5, or 7, it has at least 26 distinct prime
aliquot factors, and this was extended to 27 by Norton (1960). Norton (1960) showed
that odd perfect numbers not divisible by 3 or 5, it must have at least 15 distinct
prime factors. Neilsen (2006), improving the bound of Hagis (1980), showed that if
an odd perfect number is not divisible by 3, it must have at least 12 distinct prime factors. Nielsen (2006) also showed that a general
odd perfect number, if it exists, must have at least 9 distinct prime factors.
More recently, Hare (2005) has shown that any odd perfect number must have 75 or more prime factors. Improving this bound requires the factorization of several large
numbers (Hare), and attempts are currently underway to perform these factorizations
using ECM factorization at mersenneforum.org
and OddPerfect.org.
For the largest prime factor of an odd perfect number, Iannucci (1999, 2000) and Jenkins (2003) have worked to find lower bounds. The largest three factors must
be at least 100000007, 10007, and 101. Goto and Ohno (2006) verified that the largest
factor must be at least 100000007 using an extension to the methods of Jenkins.
For the smallest prime factor of an odd perfect number, Yamada (2005) determined an upper bound of 
For any odd perfect number with  prime factors and
 , Kishore (1981) has established
upper bounds for small factors of odd perfect numbers by showing that
 |
(3)
|
Portions of this entry contributed by Charles
Greathouse
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