Odd Perfect Number

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 (Ochem and Rao 2012). The following table summarizes the development of ever-higher bounds for the smallest possible odd perfect number.

author	bound
Kanold (1957)
Tuckerman (1973)
Hagis (1973)
Brent and Cohen (1989)
Brent et al. (1991)
Ochem and Rao (2012)

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 stated 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 the elliptic curve factorization method at mersenneforum.org and OddPerfect.org. Ochem and Rao (2012) subsequently showed that any odd perfect number has at least 101 not necessarily distinct prime factors.

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. Ochem and Rao (2012) subsequently showed that the largest component (i.e., divisor with prime) is greater than .

For the smallest prime factor of an odd perfect number with all even powers lower than six, 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)