A new lower bound for odd perfect numbers
Authors:
Richard P. Brent and Graeme L. Cohen
Journal:
Math. Comp. 53 (1989), 431437, S7
MSC:
Primary 11A25; Secondary 11Y05, 11Y70
MathSciNet review:
968150
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: We describe an algorithm for proving that there is no odd perfect number less than a given bound K (or finding such a number if one exists). A program implementing the algorithm has been run successfully with , with an elliptic curve method used for the vast number of factorizations required.
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 W. Beck & R. Najar, "A lower bound for odd triperfects," Math. Comp., v. 38, 1982, pp. 249251. MR 637303 (83m:10006)
 [2]
 R. P. Brent, "Some integer factorization algorithms using elliptic curves," Australian Computer Science Communications, v. 8, 1986, pp. 149163.
 [3]
 R. P. Brent, G. L. Cohen & H. J. J. te Riele, An Improved Technique for Lower Bounds for Odd Perfect Numbers, Report TRCS8808, Computer Sciences Laboratory, Australian National University, August 1988.
 [4]
 J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman & S. S. Wagstaff, Jr., Factorizations of Up to High Powers, Contemp. Math., vol. 22, Amer. Math. Soc., Providence, R.I., 1983. MR 715603 (84k:10005)
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 M. Buxton & S. Elmore, "An extension of lower bounds for odd perfect numbers," Notices Amer. Math. Soc., v. 23, 1976, p. A55.
 [6]
 M. Buxton & B. Stubblefield, "On odd perfect numbers," Notices Amer. Math. Soc., v. 22, 1975, p. A543.
 [7]
 G. L. Cohen & P. Hagis, Jr., "Results concerning odd multiperfect numbers," Bull. Malaysian Math. Soc., v. 8, 1985, pp. 2326. MR 810051 (87a:11010)
 [8]
 R. K. Guy, Unsolved Problems in Number Theory, SpringerVerlag, New York, 1981. MR 656313 (83k:10002)
 [9]
 P. Hagis, Jr., "A lower bound for the set of odd perfect numbers," Math. Comp., v. 27, 1973, pp. 951953. MR 0325507 (48:3854)
 [10]
 H.J. Kanold, "Über mehrfach vollkommene Zahlen. II," J. Reine Angew. Math., v. 197, 1957, pp. 8296. MR 0084514 (18:873b)
 [11]
 T. Nagell, Introduction to Number Theory, Chelsea, New York, 1981.
 [12]
 B. M. Stewart, Math. Rev., 81m:10011.
 [13]
 B. Stubblefield, "Lower bounds for odd perfect numbers (beyond the googol)" in Black Mathematicians and Their Works, Dorrance, Ardmore, PA, 1980, pp. 211222. MR 573929 (81m:10011)
 [14]
 B. Tuckerman, "A search procedure and lower bound for odd perfect numbers," Math. Comp., v. 27, 1973, pp. 943949. MR 0325506 (48:3853)
 [15]
 S. Wagon, "Perfect numbers," Math. Intelligencer, v. 7, 1985, pp. 6668. MR 784945 (86f:11010)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909681502
PII:
S 00255718(1989)09681502
Article copyright:
© Copyright 1989 American Mathematical Society
