完全数
出典: フリー百科事典『ウィキペディア(Wikipedia)』 (2024/03/15 05:10 UTC 版)
外部リンク
- 足立恒雄『完全数』 - コトバンク
- 『完全数の一覧と性質』 - 高校数学の美しい物語
- Weisstein, Eric W. "Perfect Number". mathworld.wolfram.com (英語).
- Greathouse, Charles; Weisstein, Eric W. "Odd Perfect Number". mathworld.wolfram.com (英語).
注釈
出典
- ^ a b c d e f 「高数・数学者列伝」吉永良正『高校への数学』vol.20、1995年8月号
- ^ 淡中忠郎「メルセンヌ数物語」『数学セミナー』、1973年9月号。数学セミナー編集部(1982)、65-67頁に再録されている。
- ^ Nicomachus of Gerasa (1926). Introduction to Arithmetic. Martin Luther D'Oge (trans). The Macmillan Company. pp. 207–212
- ^ a b ハーディ & ライト 2001, p. 317
- ^ a b 和田 1981, pp. 59–61
- ^ Dickson (2005, p. 19)
- ^ "GIMPS Discovers Largest Known Prime Number: 282,589,933-1" (Press release) (英語). GIMPS. 21 December 2018. 2022年2月5日時点のオリジナルよりアーカイブ。2022年2月22日閲覧。
- ^ ハーディ & ライト 2001, p. 316
- ^ Dickson (2005, p. 98)
- ^ Nielsen, Pace P. (2003). “An upper bound for odd perfect numbers”. Integers 3: A14 .
- ^ Grün, Otto (1952). “Über ungerade vollkommene Zahlen”. Mathematische Zeitschrift 55 (3): 353--354. doi:10.1007/BF01181133.
- ^ M. Kishore, "On odd perfect, quasiperfect, and odd almost perfect numbers", Math. Comp. 36 (1981), 583-586.
- ^ W. L. McDaniel, "The non-existence of odd perfect numbers of a certain form", Arch. Math. (Basel) 21 (1970), 52-53.
- ^ Fletcher, S. Adam; Nielsen, Pace P.; Ochem, Pascal (2012). “Sieve methods for odd perfect numbers”. Mathematics of Computation 81 (279): 1753--1776. doi:10.1090/S0025-5718-2011-02576-7. ISSN 0025-5718. MR2904601 .
- ^ W. L. McDaniel and P. Hagis Jr., "Some results concerning the non-existence of odd perfect numbers of the form paM2β", Fibonacci Quart. 13 (1975), 25-28.
- ^ G. L. Cohen, R. J. Williams, "Extensions of some results concerning odd perfect numbers", Fibonacci Quart. 23 (1985), 70-76.
- ^ Yamada, Tomohiro (2019). “A new upper bound for odd perfect numbers of a special form”. Colloquium Mathematicum 156 (1): 15--21. doi:10.4064/cm7339-3-2018. ISSN 1730-6302.
- ^ J. Touchard, "On prime numbers and perfect numbers", Scripta Math. 19 (1953), 53-59.
- ^ M. Satyanarayana, "Odd perfect numbers", Math. Student 27 (1959), 17-18.
- ^ J. A. Holdener, "A theorem of Touchard on the form of odd perfect numbers". Amer. Math. Monthly, 109 (2002), 661-663.
- ^ T. Roberts, "On the Form of an Odd Perfect Number", Australian Mathematical Gazette, 35:4 (2008), 244
- ^ a b c Ochem, Pascal; Rao, Michaël (2012). “Odd perfect numbers are greater than 101500”. Mathematics of Computation 81 (279): 1869--1877. doi:10.1090/S0025-5718-2012-02563-4. ISSN 0025-5718. MR2904606. Zbl 1263.11005 .
- ^ R. P. Brent, Graeme L. Cohen, H. J. J. te Riele, "Improved techniques for lower bounds for odd perfect numbers", Math. Comp. 57 (1991), 857-868
- ^ Nielsen, Pace P. (2015). “Odd perfect numbers, Diophantine equations, and upper bounds”. Mathematics of Computation 84 (295): 2549--2567. doi:10.1090/S0025-5718-2015-02941-X. ISSN 0025-5718. MR3356038 .
- ^ a b Nielsen, Pace P. (2007). “Odd perfect numbers have at least nine distinct prime factors”. Mathematics of Computation 76 (260): 2109--2126. arXiv:math/0602485. doi:10.1090/S0025-5718-07-01990-4. ISSN 0025-5718. MR2336286 .
- ^ J. E. Z. Chein, "An odd perfect number has at least 8 prime factors", Doctoral Thesis, Pennsylvania State University, 1979.
- ^ P. Hagis Jr., "Outline of a proof that every odd perfect number has at least eight prime factors", Math. Comp. 35 (1980) 1027-1032.
- ^ G. L. Cohen, R. M. Sorli, "On the number of distinct prime factors of an odd perfect number", J. Discrete Algorithms 1 (2003), 21-35.
- ^ K. K. Norton, "Remarks on the number of factors of an odd perfect number", Acta Arith., 6 (1960/1961), 365-374.
- ^ 75個以上であることを示した、以前の結果は K. G. Hare, "New techniques for bounds on the total number of prime factors of an odd perfect number", Math. Comp. 76. (2007), 2241-2248. preprint
- ^ T. Goto and Y. Ohno, "Odd perfect numbers have a prime factor exceeding 108", Math. Comp. 77 (2008), 1859-1868. "奇数の完全数の最大素因子について" - preprint を入手可能。
- ^ P. M. Jenkins, "Odd perfect numbers have a prime factor exceeding 107", Math. Comp. 72 (2003), 1549-1554.
- ^ P. Hagis, Jr. and G. L. Cohen, "Every odd perfect number has a prime factor which exceeds 106", Math. Comp. 67 (1998), 1323-1330.
- ^ D. E. Iannucci, "The second largest prime divisor of an odd perfect number exceeds ten thousand", Math. Comp. 68 (1999), 1749-1760.
- ^ D. E. Iannucci, "The third largest prime divisor of an odd perfect number exceeds one hundred", Math. Comp. 69 (2000), 867-879.
- ^ Weisstein, Eric W. "Multiperfect Number". mathworld.wolfram.com (英語).
- ^ Weisstein, Eric W. "Deficient Number". mathworld.wolfram.com (英語).
- ^ Weisstein, Eric W. "Abundant Number". mathworld.wolfram.com (英語).
- ^ Weisstein, Eric W. "Amicable Pair". mathworld.wolfram.com (英語).
- ^ Weisstein, Eric W. "Sociable Numbers". mathworld.wolfram.com (英語).
- ^ Weisstein, Eric W. "Quasiperfect Number". mathworld.wolfram.com (英語).
- ^ Weisstein, Eric W. "Almost Perfect Number". mathworld.wolfram.com (英語).
- ^ Weisstein, Eric W. "Multiplicative Perfect Number". mathworld.wolfram.com (英語).
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