Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



A new lower bound for odd perfect numbers

Authors: Richard P. Brent and Graeme L. Cohen
Journal: Math. Comp. 53 (1989), 431-437, S7
MSC: Primary 11A25; Secondary 11Y05, 11Y70
MathSciNet review: 968150
Full-text PDF

Abstract | References | Similar Articles | 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 $ K = {10^{160}}$, with an elliptic curve method used for the vast number of factorizations required.

References [Enhancements On Off] (What's this?)

  • [1] W. Beck & R. Najar, "A lower bound for odd triperfects," Math. Comp., v. 38, 1982, pp. 249-251. MR 637303 (83m:10006)
  • [2] R. P. Brent, "Some integer factorization algorithms using elliptic curves," Australian Computer Science Communications, v. 8, 1986, pp. 149-163.
  • [3] R. P. Brent, G. L. Cohen & H. J. J. te Riele, An Improved Technique for Lower Bounds for Odd Perfect Numbers, Report TR-CS-88-08, 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 $ {b^n} \pm 1, b = 2,3,5,6,7,10,11,12$ Up to High Powers, Contemp. Math., vol. 22, Amer. Math. Soc., Providence, R.I., 1983. MR 715603 (84k:10005)
  • [5] M. Buxton & S. Elmore, "An extension of lower bounds for odd perfect numbers," Notices Amer. Math. Soc., v. 23, 1976, p. A-55.
  • [6] M. Buxton & B. Stubblefield, "On odd perfect numbers," Notices Amer. Math. Soc., v. 22, 1975, p. A-543.
  • [7] G. L. Cohen & P. Hagis, Jr., "Results concerning odd multiperfect numbers," Bull. Malaysian Math. Soc., v. 8, 1985, pp. 23-26. MR 810051 (87a:11010)
  • [8] R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 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. 951-953. MR 0325507 (48:3854)
  • [10] H.-J. Kanold, "Über mehrfach vollkommene Zahlen. II," J. Reine Angew. Math., v. 197, 1957, pp. 82-96. 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. 211-222. MR 573929 (81m:10011)
  • [14] B. Tuckerman, "A search procedure and lower bound for odd perfect numbers," Math. Comp., v. 27, 1973, pp. 943-949. MR 0325506 (48:3853)
  • [15] S. Wagon, "Perfect numbers," Math. Intelligencer, v. 7, 1985, pp. 66-68. MR 784945 (86f:11010)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11A25, 11Y05, 11Y70

Retrieve articles in all journals with MSC: 11A25, 11Y05, 11Y70

Additional Information

Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society