Largest prime factor of an odd perfect number
[Japanese/English]
Professor Yasuo Ohno
(at Kinki University) and I have shown that every
odd perfect number must be divisible by a prime greater than 108.
The record of the bound were renewed as follows:
| bound |
year |
author(s) |
|---|
| 60 |
1944 |
Kanold [8] |
| 104 |
1973 |
Hagis and McDaniel [5] |
| 105 |
1975 |
Hagis and McDaniel [6] |
| 3 x 105 |
1978 |
Condict [2] |
| 5 x 105 |
1982 |
Brandstein [1] |
| 106 |
1996 |
Hagis and Cohen [4] |
| 107 |
2003 |
Jenkins [7] |
Our method is based on the paper by Hagis and Cohen for the bound 106,
however, using some property of cyclotomic polynomials,
we could shorten the computing time very much.
We also refer to the programs published by Jenkins, and improve on them.
This work was supported by Computing and Communications Center,
Kyushu University, and the computations needed about
26,000 hours for CPU time. We used ten CPU's simultaneously,
so it takes about four months. For the bound 107,
the computation needs only 42 hours by our programs and a PC.
For details, please see the text and the programs below.
text (PDF file)
programs:
ZIP file (recommended for Windows users)
GZIP file (recommended for UNIX users)
Note that UBASIC is available on only Windows (or MS-DOS),
GP2C on only UNIX, and PARI/GP on both Windows and UNIX.
References
[1] M. Brandstein, New lower bound for a factor of an odd perfect
number, Abstracts Amer. Math. Soc., 3 (1982), 257, 82T-10-240.
[2] J. Condict, On an odd perfect number's largest prime divisor,
Senior Thesis, Middlebury College, 1978.
[3] T. Goto and Y. Ohno, Odd perfect numbers have a prime factor exceeding
108, Math. Comp., to appear.
[4] P. Hagis, Jr. and G. L. Cohen, Every odd perfect number has a
prime factor which exceeds 106, Math. Comp., 67
(1998), 1323-1330.
[5] P. Hagis, Jr. and W. L. McDaniel, On the largest prime divisor of an
odd perfect number, Math. Comp., 27 (1973), 955-957.
[6] P. Hagis, Jr. and W. L. McDaniel, On the largest prime divisor of an
odd perfect number II, Math. Comp., 29 (1975), 922-924.
[7] P. M. Jenkins, Odd perfect numbers have a prime factor exceeding
107, Math. Comp., 72 (2003), 1549-1554.
[8] H. J. Kanold, Folgerungen aus dem Vorkommen einer Gauss'schen
Primzahl in der Primfaktorenzerlegung einer ungeraden vollkommenen Zahl,
J. Reine Angew. Math., 186 (1944), 25-29.
By Takeshi Goto, at Tokyo University of Science.
Since Mar. 2, 2006.
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