From: Gerry Myerson Subject: Re: Perfect numbers Date: Tue, 13 Jun 2000 11:13:45 +1000 Newsgroups: sci.math Summary: [missing] In article , malai@winner.fr (Gianfranco Oldani) wrote: > Please, help me to find references to recent results about odd perfect > numbers and triperfect numbers (sigma(n)=3*n). Here are some recent papers on odd perfect numbers, from Math Reviews: [1] 1 651 762 (Review) Iannucci, Douglas E. The third largest prime divisor of an odd perfect number exceeds one hundred. Math. Comp. 69 (2000), no. 230, 867--879. (Reviewer: Duncan A. Buell) 11Y70 (11A2) [2] 1 651 761 (Review) Iannucci, Douglas E. The second largest prime divisor of an odd perfect number exceeds ten thousand. Math. Comp. 68 (1999), no. 228, 1749--1760. (Reviewer: Duncan A. Buell) 11Y70 (11A25) [3] 1 721 482 (Review) Subbarao, M. V. Odd perfect numbers: some new issues. Period. Math. Hungar. 38 (1999), no. 1-2, 103--109. (Reviewer: Aleksander Grytczuk) 11A25 [4] 1 728 236 (Review) Slowak, Jan Odd perfect numbers. Math. Slovaca 49 (1999), no. 3, 253--254. (Reviewer: Aleksander Grytczuk) 11A25 [5] 2000f:11007 Grytczuk, Aleksander; Wójtowicz, Marek There are no small odd perfect numbers. C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 12, 1101--1105. (Reviewer: G. Greaves) 11A25 (11N25) [6] 2000d:11010 Cook, R. J. Bounds for odd perfect numbers. Number theory (Ottawa, ON, 1996), 67--71, CRM Proc. Lecture Notes, 19, Amer. Math. Soc., Providence, RI, 1999. (Reviewer: D. R. Heath-Brown) 11A25 [7] 2000c:11009 Ewell, John A. On necessary conditions for the existence of odd perfect numbers. Rocky Mountain J. Math. 29 (1999), no. 1, 165--175. (Reviewer: Aleksander Grytczuk) 11A25 [8] 99k:11005 Number theory. Proceedings of the 5th Conference of the Canadian Number Theory Association held at Carleton University, Ottawa, ON, August 17--22, 1996. Edited by Rajiv Gupta and Kenneth S. Williams. CRM Proceedings & Lecture Notes, 19. American Mathematical Society, Providence, RI, 1999. xxxii+392 pp. ISBN: 0-8218-0964-4 11-06 [9] 98k:11002 Hagis, Peter, Jr.; Cohen, Graeme L. Every odd perfect number has a prime factor which exceeds $10\sp 6$. Math. Comp. 67 (1998), no. 223, 1323--1330. (Reviewer: Aleksander Grytczuk) 11A25 (11Y70) [10] 96f:11009 Cook, Roger Factors of odd perfect numbers. Number theory (Halifax, NS, 1994), 123--131, CMS Conf. Proc., 15, Amer. Math. Soc., Providence, RI, 1995. (Reviewer: Neville Robbins) 11A25 (11Y70) [11] 96e:11007 Chen, Yi Ze; Chen, Xiao Song A condition for an odd perfect number to have at least $6$ prime factors $\equiv 1\bmod 3$. (Chinese) Hunan Jiaoyu Xueyuan Xuebao (Ziran Kexue) 12 (1994), no. 2, 1--6. (Reviewer: Mao Hua Le) 11A25 [12] 96c:11003 Number theory. Proceedings of the Fourth Conference of the Canadian Number Theory Association held at Dalhousie University, Halifax, Nova Scotia, July 2--8, 1994. Edited by Karl Dilcher. CMS Conference Proceedings, 15. Published by the American Mathematical Society, Providence, RI; for the Canadian Mathematical Society, Ottawa, ON, 1995. xxiv+431 pp. ISBN: 0-8218-0312-3 11-06 [13] 96b:11130 Heath-Brown, D. R. Odd perfect numbers. Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 2, 191--196. (Reviewer: P. Haukkanen) 11N64 (11A25) [14] 94m:11003 Schinzel, Andrzej Progress in number theory during the years 1989--1992. Discuss. Math. 13 (1993), 75--80. (Reviewer: Adolf Hildebrand) 11-02 [15] 94c:11003 Starni, Paolo Odd perfect numbers: a divisor related to the Euler's factor. J. Number Theory 44 (1993), no. 1, 58--59. (Reviewer: Armel Mercier) 11A25 [16] 92k:00014 Klee, Victor; Wagon, Stan Old and new unsolved problems in plane geometry and number theory. The Dolciani Mathematical Expositions, 11. Mathematical Association of America, Washington, DC, 1991. xvi+333 pp. ISBN: 0-88385-315-9 (Reviewer: F. J. Papp) 00A07 (01A05 11-02 51-02) [17] 92c:11004 Brent, R. P.; Cohen, G. L.; te Riele, H. J. J. Improved techniques for lower bounds for odd perfect numbers. Math. Comp. 57 (1991), no. 196, 857--868. (Reviewer: H. G. Diamond) 11A25 (11Y70) [18] 92a:11010 Starni, Paolo On the Euler's factor of an odd perfect number. J. Number Theory 37 (1991), no. 3, 366--369. (Reviewer: Katalin Kovács) 11A25 [19] 91d:01002 Rashed, Roshdi Ibn al-Haytham et les nombres parfaits. (French) [Ibn al-Haytham and perfect numbers] Historia Math. 16 (1989), no. 4, 343--352. (Reviewer: Jan P. Hogendijk) 01A30 (01A20 11-03) [20] 90k:11006 Hagis, Peter, Jr. Odd nonunitary perfect numbers. Fibonacci Quart. 28 (1990), no. 1, 11--15. (Reviewer: Neville Robbins) 11A25 [21] 90h:11111 te Riele, Herman; Lioen, Walter; Winter, Dik Factoring with the quadratic sieve on large vector computers. J. Comput. Appl. Math. 27 (1989), no. 1-2, 267--278. 11Y05 (11A25 65W05) [22] 90d:11140 Churchhouse, R. F. Presidential address: mathematics and computers. Bull. Inst. Math. Appl. 25 (1989), no. 3-4, 40--49. (Reviewer: Samuel S. Wagstaff, Jr.) 11Y50 (00A69) [23] 89m:11012 Zarrouk, Abdelwaheb Sur les nombres parfaits impairs. (French) [Odd perfect numbers] Mathematica (Cluj) 29(52) (1987), no. 2, 193--197. (Reviewer: Neville Robbins) 11A25 [24] 89m:11008 Brent, Richard P.; Cohen, Graeme L. A new lower bound for odd perfect numbers. Math. Comp. 53 (1989), no. 187, 431--437, S7--S24. (Reviewer: Samuel S. Wagstaff, Jr.) 11A25 (11Y05 11Y70) [25] 89f:11166 Churchhouse, R. F. Computers in number theory. 1987 CERN School of Computing (Troia, 1987), 357--377, CERN, 88-03, CERN, Geneva, 1988. (Reviewer: W. W. Adams) 11Yxx (11-02 11-04) [26] 87m:11005 Cohen, Graeme L. On the largest component of an odd perfect number. J. Austral. Math. Soc. Ser. A 42 (1987), no. 2, 280--286. (Reviewer: P. Erdös) 11A25 [27] 86f:11010 Wagon, Stan The evidence: perfect numbers. Math. Intelligencer 7 (1985), no. 2, 66--68. 11A25 [28] 86f:11009 Cohen, G. L.; Williams, R. J. Extensions of some results concerning odd perfect numbers. Fibonacci Quart. 23 (1985), no. 1, 70--76. (Reviewer: V. C. Harris) 11A25 [29] 85b:11004 Hagis, Peter, 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. (Reviewer: Masao Kishore) 11A25 (11-04) [30] 84m:10006 te Riele, H. J. J. Perfect numbers and aliquot sequences. Computational methods in number theory, Part I, 141--157, Math. Centre Tracts, 154, Math. Centrum, Amsterdam, 1982. (Reviewer: Masao Kishore) 10A20 [31] 84g:10011 Heyworth, Malcolm R. Continued fractions in a search for odd perfect numbers. New Zealand Math. Mag. 19 (1982/83), no. 2, 63--69. (Reviewer: Gerald Myerson) 10A20 (10A32 10B05) Here are a few about triperfects: [1] 94e:11003 Hagis, Peter, Jr. A new proof that every odd triperfect number has at least twelve prime factors. A tribute to Emil Grosswald: number theory and related analysis, 445--450, Contemp. Math., 143, Amer. Math. Soc., Providence, RI, 1993. (Reviewer: Anthony A. Gioia) 11A25 (11A51) [2] 93j:11002 A tribute to Emil Grosswald: number theory and related analysis. Edited by Marvin Knopp and Mark Sheingorn. Contemporary Mathematics, 143. American Mathematical Society, Providence, RI, 1993. viii+612 pp. ISBN: 0-8218-5155-1 11-06 [3] 88c:11009 Kishore, Masao Odd triperfect numbers are divisible by twelve distinct prime factors. J. Austral. Math. Soc. Ser. A 42 (1987), no. 2, 173--182. (Reviewer: G. L. Cohen) 11A25 [4] 86k:11007 Kishore, Masao Odd triperfect numbers are divisible by eleven distinct prime factors. Math. Comp. 44 (1985), no. 169, 261--263. 11A25 [5] 85d:11009 Kishore, Masao Odd triperfect numbers. Math. Comp. 42 (1984), no. 165, 231--233. (Reviewer: Neville Robbins) 11A25 [6] 83m:10006 Beck, Walter E.; Najar, Rudolph M. A lower bound for odd triperfects. Math. Comp. 38 (1982), no. 157, 249--251. (Reviewer: Carl Pomerance) 10A20 [7] 81m:10009 Cohen, Graeme L. On odd perfect numbers. II. Multiperfect numbers and quasiperfect numbers. J. Austral. Math. Soc. Ser. A 29 (1980), no. 3, 369--384. (Reviewer: Masao Kishore) 10A25 That should keep you busy.... Gerry Myerson (gerry@mpce.mq.edu.au) [slightly reformatted --djr]