## A new lower bound for odd perfect numbers

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- by Richard P. Brent and Graeme L. Cohen PDF
- Math. Comp.
**53**(1989), 431-437 Request permission

## 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

- Walter E. Beck and Rudolph M. Najar,
*A lower bound for odd triperfects*, Math. Comp.**38**(1982), no. 157, 249–251. MR**637303**, DOI 10.1090/S0025-5718-1982-0637303-9
R. P. Brent, "Some integer factorization algorithms using elliptic curves," - John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman, and S. S. Wagstaff Jr.,
*Factorizations of $b^{n}\pm 1$*, Contemporary Mathematics, vol. 22, American Mathematical Society, Providence, R.I., 1983. $b=2,\,3,\,5,\,6,\,7,\,10,\,11,\,12$ up to high powers. MR**715603**
M. Buxton & S. Elmore, "An extension of lower bounds for odd perfect numbers," - Graeme L. Cohen and Peter Hagis Jr.,
*Results concerning odd multiperfect numbers*, Bull. Malaysian Math. Soc. (2)**8**(1985), no. 1, 23–26. MR**810051** - Richard K. Guy,
*Unsolved problems in number theory*, Problem Books in Mathematics, Springer-Verlag, New York-Berlin, 1981. MR**656313** - Peter Hagis Jr.,
*A lower bound for the set of odd perfect numbers*, Math. Comp.**27**(1973), 951–953. MR**325507**, DOI 10.1090/S0025-5718-1973-0325507-9 - Hans-Joachim Kanold,
*Über mehrfach vollkommene Zahlen. II*, J. Reine Angew. Math.**197**(1957), 82–96 (German). MR**84514**, DOI 10.1515/crll.1957.197.82
T. Nagell, - Beauregard Stubblefield,
*Lower bounds for odd perfect numbers (beyond the googol)*, Black mathematicians and their works, Dorrance & Co., Ardmore, Pa., 1980, pp. 211–222 (1 plate). MR**573929** - Bryant Tuckerman,
*A search procedure and lower bound for odd perfect numbers*, Math. Comp.**27**(1973), 943–949. MR**325506**, DOI 10.1090/S0025-5718-1973-0325506-7 - Stan Wagon,
*The evidence: perfect numbers*, Math. Intelligencer**7**(1985), no. 2, 66–68. MR**784945**, DOI 10.1007/BF03024179

*Australian Computer Science Communications*, v. 8, 1986, pp. 149-163. 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.

*Notices Amer. Math. Soc.*, v. 23, 1976, p. A-55. M. Buxton & B. Stubblefield, "On odd perfect numbers,"

*Notices Amer. Math. Soc.*, v. 22, 1975, p. A-543.

*Introduction to Number Theory*, Chelsea, New York, 1981. B. M. Stewart,

*Math. Rev.*,

**81m**:10011.

## Additional Information

- © Copyright 1989 American Mathematical Society
- Journal: Math. Comp.
**53**(1989), 431-437 - MSC: Primary 11A25; Secondary 11Y05, 11Y70
- DOI: https://doi.org/10.1090/S0025-5718-1989-0968150-2
- MathSciNet review: 968150