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^ a b ハーディ & ライト 2001, p. 317
^ a b 和田 1981, pp. 59?61
^ Dickson (2005, p. 19)
^ .mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation.cs-ja1 q,.mw-parser-output .citation.cs-ja2 q{quotes:"「""」""『""』"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}"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. ⇒http://math.colgate.edu/~integers/vol3.html.
^ Grun, Otto (1952). “Uber 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. ⇒http://www.lirmm.fr/~ochem/opn/OPNS_Adam_Pace.pdf.
^ 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, Michael (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. ⇒http://www.lirmm.fr/~ochem/opn/opn.pdf.
^ 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. https://math.byu.edu/~pace/BestBound_web.pdf.
^ 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. https://math.byu.edu/~pace/NotEight_web.pdf.
^ 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).
参考文献
数学セミナー編集部 編『数の世界』日本評論社、東京〈数学セミナー増刊 数学セミナー・リーディングス〉、1982年9月30日。
ハーディ, G.H.、ライト, E.M. 著、示野信一・矢神毅 訳『数論入門 I』丸善出版〈シュプリンガー数学クラシックス8〉、2001年7月。ISBN 978-4-621-06226-5。
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