On the total number of prime factors of an odd perfect number
Authors:
D. E. Iannucci and R. M. Sorli
Journal:
Math. Comp. 72 (2003), 20772084
MSC (2000):
Primary 11A25, 11Y70
Published electronically:
May 8, 2003
MathSciNet review:
1986824
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: We say is perfect if , where denotes the sum of the positive divisors of . No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization of the form , where , , ..., are distinct primes and . We prove that if or for all , , then . We also prove as our main result that , where . This improves a result of Sayers given in 1986.
 [1]
N.
C. Ankeny, One more proof of the fundamental theorem of
algebra, Amer. Math. Monthly 54 (1947), 464. MR 0021624
(9,90b)
 [2]
E. Z. Chein, Ph.D. Thesis, Pennsylvania State University (1979).
 [3]
G. L. Cohen, On the total number of prime factors of an odd perfect number Appendix A.3, Ph.D. Thesis, University of New South Wales (1982).
 [4]
Graeme
L. Cohen, On the largest component of an odd perfect number,
J. Austral. Math. Soc. Ser. A 42 (1987), no. 2,
280–286. MR
869751 (87m:11005)
 [5]
G.
L. Cohen and R.
J. Williams, Extensions of some results concerning odd perfect
numbers, Fibonacci Quart. 23 (1985), no. 1,
70–76. MR
786364 (86f:11009)
 [6]
Peter
Hagis Jr., Outline of a proof that every odd
perfect number has at least eight prime factors, Math. Comp. 35 (1980), no. 151, 1027–1032. MR 572873
(81k:10004), http://dx.doi.org/10.1090/S00255718198005728739
 [7]
Peter
Hagis Jr., Sketch of a proof that an odd perfect
number relatively prime to 3 has at least eleven prime factors, Math. Comp. 40 (1983), no. 161, 399–404. MR 679455
(85b:11004), http://dx.doi.org/10.1090/S00255718198306794551
 [8]
HansJoachim
Kanold, Sätze über Kreisteilungspolynome und ihre
Anwendungen auf einige zahlentheoretische Probleme. II, J. Reine
Angew. Math. 188 (1950), 129–146 (German). MR 0044579
(13,443b)
 [9]
Masao
Kishore, Odd perfect numbers not divisible by
3. II, Math. Comp. 40
(1983), no. 161, 405–411. MR 679456
(84d:10009), http://dx.doi.org/10.1090/S00255718198306794563
 [10]
Wayne
L. McDaniel, The nonexistence of odd perfect numbers of a certain
form, Arch. Math. (Basel) 21 (1970), 52–53. MR 0258723
(41 #3369)
 [11]
Wayne
L. McDaniel, On the divisibility of an odd perfect
number by the sixth power of a prime, Math.
Comp. 25 (1971),
383–385. MR 0296013
(45 #5074), http://dx.doi.org/10.1090/S00255718197102960133
 [12]
M. Sayers, M.App.Sc. Thesis, New South Wales Institute of Technology (1986).
 [13]
R. Steuerwald, Verschärfung einer notwendigen Bedingung für die Existenz einer ungeraden vollkommenen Zahl S.B. Math.Nat. Abt. Bayer. Akad. Wiss. (1937), 6873.
 [1]
 A. Brauer, On the nonexistence of odd perfect numbers of the form Bull. Amer. Math. Soc. 49 (1943), 712718. MR 9:90b
 [2]
 E. Z. Chein, Ph.D. Thesis, Pennsylvania State University (1979).
 [3]
 G. L. Cohen, On the total number of prime factors of an odd perfect number Appendix A.3, Ph.D. Thesis, University of New South Wales (1982).
 [4]
 , On the largest component of an odd perfect number J. Austral. Math. Soc. Ser. A 42 (1987), 280286. MR 87m:11005
 [5]
 G. L. Cohen and R. J. Williams, Extensions of some results concerning odd perfect numbers J. Fibonacci Quart. 23 (1985), 7076. MR 86f:11009
 [6]
 P. Hagis, Outline of a proof that every odd perfect number has at least eight prime factors Math. Comp. 35 (1980), 10271032. MR 81k:10004
 [7]
 P. Hagis, Sketch of a proof that every odd perfect number relatively prime to 3 has at least eleven prime factors Math. Comp. 40 (1983), 399404. MR 85b:11004
 [8]
 H.J. Kanold, Sätze über kreisteilungspolynome und ihre andwendungen auf einiger zahlentheoretische problem II J. Reine Angew. Math. 188 (1950), 129146. MR 13:443b
 [9]
 M. Kishore, Odd perfect numbers not divisible by three are divisible by at least eleven distinct primes Math. Comp. 40 (1983), 405411. MR 84d:10009
 [10]
 W. McDaniel, The nonexistence of odd perfect numbers of a certain form Arch. Math. 21 (1970), 5253. MR 41:3369
 [11]
 W. McDaniel, On the divisibility of an odd perfect number by the sixth power of a prime Math. Comp. 25 (1971), 383385. MR 45:5074
 [12]
 M. Sayers, M.App.Sc. Thesis, New South Wales Institute of Technology (1986).
 [13]
 R. Steuerwald, Verschärfung einer notwendigen Bedingung für die Existenz einer ungeraden vollkommenen Zahl S.B. Math.Nat. Abt. Bayer. Akad. Wiss. (1937), 6873.
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Additional Information
D. E. Iannucci
Affiliation:
University of the Virgin Islands, St. Thomas, Virgin Islands 00802
Email:
diannuc@uvi.edu
R. M. Sorli
Affiliation:
Department of Mathematical Sciences, University of Technology, Sydney, Broadway, 2007, Australia
Email:
rons@maths.uts.edu.au
DOI:
http://dx.doi.org/10.1090/S0025571803015229
PII:
S 00255718(03)015229
Keywords:
Odd perfect numbers,
factorization
Received by editor(s):
November 7, 2001
Published electronically:
May 8, 2003
Additional Notes:
The authors are grateful for the advice and assistance given by Graeme Cohen
Article copyright:
© Copyright 2003
American Mathematical Society
