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
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R. P. and Cohen, G. L. "A New Bound for Odd Perfect Numbers."
Math. Comput.53, 431-437 and S7-S24, 1989.Brent, R. P.;
Cohen, G. L.; te Riele, H. J. J. "Improved Techniques for Lower
Bounds for Odd Perfect Numbers." Math. Comput.57, 857-868, 1991.Burton,
D. M. Elementary
Number Theory, 4th ed. Boston, MA: Allyn and Bacon, 1989.Buxton,
M. and Elmore, S. "An Extension of Lower Bounds for Odd Perfect Numbers."
Not. Amer. Math. Soc.22, A-55, 1976.Buxton, M. and Stubblefield,
B. "On Odd Perfect Numbers." Not. Amer. Math. Soc.22, A-543,
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Have a Prime Factor Exceeding " Preprint, Mar. 2006. http://www.ma.noda.tus.ac.jp/u/tg/perfect.html.Guy,
R. K. "Perfect Numbers." §B1 in Unsolved
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1994.Hagis, P. Jr. "A Lower Bound for the Set of Odd Perfect Numbers."
Math. Comput.27, 951-953, 1973.Hagis, P. Jr. "An
Outline of a Proof that Every Odd Perfect Number has at Least Eight Prime Factors."
Math. Comput.34, 1027-1032, 1980.Hagis, P. Jr.; and Cohen,
G. L. "Every Odd Perfect Number Has a Prime Factor Which Exceeds." Math. Comput.67, 1323-1330, 1998.Hare,
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K. "New Techniques for Bounds on the Total Number of Prime Factors of an Odd
Perfect Number." Math. Comput.74, 1003-1008, 2005.Hare,
K. G. "Some Factorizations that I Want." http://www.math.uwaterloo.ca/~kghare/ODDPERFECT/MissingValues.html.Heath-Brown,
D. R. "Odd Perfect Numbers." Math. Proc. Cambridge Philos. Soc.115,
191-196, 1994.Holdener, J. A. "A Theorem of Touchard and the
Form of Odd Perfect Numbers." Amer. Math. Monthly109, 661-663,
2002.Iannucci, D. E. "The Second Largest Prime Divisor of
an Odd Perfect Number Exceeds Ten Thousand." Math. Comput.68,
1749-1760, 1999.Iannucci, D. E. "The Third Largest Prime Divisor
of an Odd Perfect Number Exceeds One Hundred." Math. Comput.69,
867-879, 2000.Jenkins, P. M. "Odd Perfect Numbers Have a Prime
Factor Exceeding ."
Math. Comput.72, 1549-1554, 2003.Kanold, H.-J. "Über
mehrfach vollkommene Zahlen. II." J. reine angew. Math.197, 82-96,
1957.Kishore, M. "On Odd Perfect, Quasiperfect, and Odd Almost
Perfect Numbers." Math. Comput.36, 583-586, 1981.mersenneforum.org.
"Odd Perfect Numbers--A Factoring Challenge." http://mersenneforum.org/showthread.php?t=3101.Nielsen,
P. P. "Odd Perfect Numbers Have at Least Nine Distinct Prime Factors."
22 Feb 2006. http://arxiv.org/abs/math.NT/0602485.Norton,
K. K. "Remarks on the Number of Factors of an Odd Perfect Number."
Acta Arith.6, 365-374, 1960.Ochem, P. and Rao, M. "Odd
Perfect Numbers Are Greater than ." Math. Comput.81, 1869-1877,
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E. Jr. and Weisstein, E. W. "Seven Mathematical Tidbits." MathWorld
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R. "Verscharfung einen notwendigen Bedingung fur die Existenz einen ungeraden
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M. V. "Odd Perfect Numbers: Some New Issues." Period. Math. Hungar.38,
103-109, 1999.Touchard, J. "On Prime Numbers and Perfect Numbers."
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