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)

REFERENCES:

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, 1987.

Brent, 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, 1975.

Cohen, G. L. "On the Largest Component of an Odd Perfect Number." J. Austral. Math. Soc. Ser. A 42, 280-286, 1987.

Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, pp. 3-33, 2005.

Goto, T. and Ohno, Y. "Odd Perfect Numbers Have a Prime Factor Exceeding " Preprint, Mar. 2006. https://www.ma.noda.tus.ac.jp/u/tg/perfect.html.

Guy, R. K. "Perfect Numbers." §B1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 44-45, 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, K. G. "Odd Perfect Number - Update." 5 Oct 2004. https://listserv.nodak.edu/scripts/wa.exe?A2=ind0409&L=nmbrthry&F=&S=&P=1064.

Hare, 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." https://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. Monthly 109, 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." https://mersenneforum.org/showthread.php?t=3101.

Nielsen, P. P. "Odd Perfect Numbers Have at Least Nine Distinct Prime Factors." 22 Feb 2006. https://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, 2012.

OddPerfect.org. "Odd Perfect Number Search." https://www.oddperfect.org/.

Pegg, E. Jr. and Weisstein, E. W. "Seven Mathematical Tidbits." MathWorld Headline News. Nov. 8, 2004. https://mathworld.wolfram.com/news/2004-11-08/seventidbits/#2.

Steuerwald, R. "Verscharfung einen notwendigen Bedingung fur die Existenz einen ungeraden vollkommenen Zahl." Sitzungsber. Bayer. Akad. Wiss., 69-72, 1937.

Subbarao, M. V. "Odd Perfect Numbers: Some New Issues." Period. Math. Hungar. 38, 103-109, 1999.

Touchard, J. "On Prime Numbers and Perfect Numbers." Scripta Math. 19, 35-39, 1953.

Tuckerman, B. "Odd Perfect Numbers: A Search Procedure, and a New Lower Bound of ." Not. Amer. Math. Soc. 15, 226, 1968.

Tuckerman, B. "A Search Procedure and Lower Bound for Odd Perfect Numbers." Math. Comput. 27, 943-949, 1973.

Voight, J. "On the Nonexistence of Odd Perfect Numbers." MASS Selecta. Providence, RI: Amer. Math. Soc., pp. 293-300, 2003.

Yamada, T. "On the Divisibility of Odd Perfect Numbers by a High Power of a Prime." 16 Nov 2005. https://arxiv.org/abs/math.NT/0511410.