Perfect number (English Wikipedia)

Dickson, L. E. (1919). History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington. p. 4.
In Introduction to Arithmetic, Chapter 16, he says of perfect numbers, "There is a method of producing them, neat and unfailing, which neither passes by any of the perfect numbers nor fails to differentiate any of those that are not such, which is carried out in the following way." He then goes on to explain a procedure which is equivalent to finding a triangular number based on a Mersenne prime.
Bayerische Staatsbibliothek, Clm 14908. See David Eugene Smith (1925). History of Mathematics: Volume II. New York: Dover. p. 21. ISBN 0-486-20430-8. {{cite book}}: ISBN / Date incompatibility (help)
Dickson, L. E. (1919). History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington. p. 10.
Dickson, L. E. (1919). History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington. p. 6.
Dickson, L. E. (1919). History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington. p. 25.

arxiv.org (Global: 49^th place; English: 31^st place)

Zelinsky, Joshua (July 2019). "Upper bounds on the second largest prime factor of an odd perfect number". International Journal of Number Theory. 15 (6): 1183–1189. arXiv:1810.11734. doi:10.1142/S1793042119500659. S2CID 62885986..
Nielsen, Pace P. (2007). "Odd perfect numbers have at least nine distinct prime factors" (PDF). Mathematics of Computation. 76 (260): 2109–2126. arXiv:math/0602485. Bibcode:2007MaCom..76.2109N. doi:10.1090/S0025-5718-07-01990-4. S2CID 2767519. Archived from the original (PDF) on 2021-11-03. Retrieved 30 March 2011.
Graeme Clayton, Cody Hansen (2023). "On inequalities involving counts of the prime factors of an odd perfect number" (PDF). Integers. 23. arXiv:2303.11974. Retrieved 29 November 2023.
Yamada, Tomohiro (2019). "A new upper bound for odd perfect numbers of a special form". Colloquium Mathematicum. 156 (1): 15–21. arXiv:1706.09341. doi:10.4064/cm7339-3-2018. ISSN 1730-6302. S2CID 119175632.

austms.org.au (Global: low place; English: low place)

Roberts, T (2008). "On the Form of an Odd Perfect Number" (PDF). Australian Mathematical Gazette. 35 (4): 244. Archived from the original (PDF) on 2013-05-14. Retrieved 2009-12-18.

books.google.com (Global: 3^rd place; English: 3^rd place)

Pickover, C (2001). Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning. Oxford: Oxford University Press. p. 360. ISBN 0-19-515799-0.
Peterson, I (2002). Mathematical Treks: From Surreal Numbers to Magic Circles. Washington: Mathematical Association of America. p. 132. ISBN 88-8358-537-2.
Yan, Song Y. (2012), Computational Number Theory and Modern Cryptography, John Wiley & Sons, Section 2.3, Exercise 2(6), ISBN 9781118188613.
Redmond, Don (1996). Number Theory: An Introduction to Pure and Applied Mathematics. Chapman & Hall/CRC Pure and Applied Mathematics. Vol. 201. CRC Press. Problem 7.4.11, p. 428. ISBN 9780824796969..

byu.edu (Global: 1,502^nd place; English: 829^th place)

math.byu.edu

Nielsen, Pace P. (2015). "Odd perfect numbers, Diophantine equations, and upper bounds" (PDF). Mathematics of Computation. 84 (295): 2549–2567. doi:10.1090/S0025-5718-2015-02941-X. Archived from the original (PDF) on 8 July 2015. Retrieved 13 August 2015.
Nielsen, Pace P. (2007). "Odd perfect numbers have at least nine distinct prime factors" (PDF). Mathematics of Computation. 76 (260): 2109–2126. arXiv:math/0602485. Bibcode:2007MaCom..76.2109N. doi:10.1090/S0025-5718-07-01990-4. S2CID 2767519. Archived from the original (PDF) on 2021-11-03. Retrieved 30 March 2011.

colgate.edu (Global: low place; English: low place)

math.colgate.edu

Bibby, Sean; Vyncke, Pieter; Zelinsky, Joshua (23 November 2021). "On the Third Largest Prime Divisor of an Odd Perfect Number" (PDF). Integers. 21. Retrieved 6 December 2021.
Zelinsky, Joshua (3 August 2021). "On the Total Number of Prime Factors of an Odd Perfect Number" (PDF). Integers. 21. Retrieved 7 August 2021.
Nielsen, Pace P. (2003). "An upper bound for odd perfect numbers". Integers. 3. doi:10.5281/zenodo.7607545. Retrieved 14 August 2025.
Graeme Clayton, Cody Hansen (2023). "On inequalities involving counts of the prime factors of an odd perfect number" (PDF). Integers. 23. arXiv:2303.11974. Retrieved 29 November 2023.

doi.org (Global: 2^nd place; English: 2^nd place)

Ochem, Pascal; Rao, Michaël (2012). "Odd perfect numbers are greater than 10¹⁵⁰⁰" (PDF). Mathematics of Computation. 81 (279): 1869–1877. doi:10.1090/S0025-5718-2012-02563-4. ISSN 0025-5718. Zbl 1263.11005.
Kühnel, Ullrich (1950). "Verschärfung der notwendigen Bedingungen für die Existenz von ungeraden vollkommenen Zahlen". Mathematische Zeitschrift (in German). 52: 202–211. doi:10.1007/BF02230691. S2CID 120754476.
Goto, T; Ohno, Y (2008). "Odd perfect numbers have a prime factor exceeding 10⁸" (PDF). Mathematics of Computation. 77 (263): 1859–1868. Bibcode:2008MaCom..77.1859G. doi:10.1090/S0025-5718-08-02050-9. Archived from the original (PDF) on 2011-08-07. Retrieved 30 March 2011.
Konyagin, Sergei; Acquaah, Peter (2012). "On Prime Factors of Odd Perfect Numbers". International Journal of Number Theory. 8 (6): 1537–1540. doi:10.1142/S1793042112500935.
Iannucci, DE (1999). "The second largest prime divisor of an odd perfect number exceeds ten thousand" (PDF). Mathematics of Computation. 68 (228): 1749–1760. Bibcode:1999MaCom..68.1749I. doi:10.1090/S0025-5718-99-01126-6. Retrieved 30 March 2011.
Zelinsky, Joshua (July 2019). "Upper bounds on the second largest prime factor of an odd perfect number". International Journal of Number Theory. 15 (6): 1183–1189. arXiv:1810.11734. doi:10.1142/S1793042119500659. S2CID 62885986..
Iannucci, DE (2000). "The third largest prime divisor of an odd perfect number exceeds one hundred" (PDF). Mathematics of Computation. 69 (230): 867–879. Bibcode:2000MaCom..69..867I. doi:10.1090/S0025-5718-99-01127-8. Retrieved 30 March 2011.
Nielsen, Pace P. (2015). "Odd perfect numbers, Diophantine equations, and upper bounds" (PDF). Mathematics of Computation. 84 (295): 2549–2567. doi:10.1090/S0025-5718-2015-02941-X. Archived from the original (PDF) on 8 July 2015. Retrieved 13 August 2015.
Nielsen, Pace P. (2007). "Odd perfect numbers have at least nine distinct prime factors" (PDF). Mathematics of Computation. 76 (260): 2109–2126. arXiv:math/0602485. Bibcode:2007MaCom..76.2109N. doi:10.1090/S0025-5718-07-01990-4. S2CID 2767519. Archived from the original (PDF) on 2021-11-03. Retrieved 30 March 2011.
Chen, Yong-Gao; Tang, Cui-E (2014). "Improved upper bounds for odd multiperfect numbers". Bulletin of the Australian Mathematical Society. 89 (3): 353–359. doi:10.1017/S0004972713000488.
Nielsen, Pace P. (2003). "An upper bound for odd perfect numbers". Integers. 3. doi:10.5281/zenodo.7607545. Retrieved 14 August 2025.
Ochem, Pascal; Rao, Michaël (2014). "On the number of prime factors of an odd perfect number". Mathematics of Computation. 83 (289): 2435–2439. doi:10.1090/S0025-5718-2013-02776-7.
Cohen, Graeme (1978). "On odd perfect numbers". Fibonacci Quarterly. 16 (6): 523-527. doi:10.1080/00150517.1978.12430277.
Suryanarayana, D. (1963). "On Odd Perfect Numbers II". Proceedings of the American Mathematical Society. 14 (6): 896–904. doi:10.1090/S0002-9939-1963-0155786-8.
McDaniel, Wayne L. (1970). "The non-existence of odd perfect numbers of a certain form". Archiv der Mathematik. 21 (1): 52–53. doi:10.1007/BF01220877. ISSN 1420-8938. MR 0258723. S2CID 121251041.
Fletcher, S. Adam; Nielsen, Pace P.; Ochem, Pascal (2012). "Sieve methods for odd perfect numbers" (PDF). Mathematics of Computation. 81 (279): 1753?1776. doi:10.1090/S0025-5718-2011-02576-7. ISSN 0025-5718. MR 2904601.
Cohen, G. L. (1987). "On the largest component of an odd perfect number". Journal of the Australian Mathematical Society, Series A. 42 (2): 280–286. doi:10.1017/S1446788700028251. ISSN 1446-8107. MR 0869751.
Kanold, Hans-Joachim [in German] (1950). "Satze uber Kreisteilungspolynome und ihre Anwendungen auf einige zahlentheoretisehe Probleme. II". Journal für die reine und angewandte Mathematik. 188 (1): 129–146. doi:10.1515/crll.1950.188.129. ISSN 1435-5345. MR 0044579. S2CID 122452828.
Cohen, G. L.; Williams, R. J. (1985). "Extensions of some results concerning odd perfect numbers" (PDF). Fibonacci Quarterly. 23 (1): 70–76. doi:10.1080/00150517.1985.12429857. ISSN 0015-0517. MR 0786364.
Hagis, Peter Jr.; McDaniel, Wayne L. (1972). "A new result concerning the structure of odd perfect numbers". Proceedings of the American Mathematical Society. 32 (1): 13–15. doi:10.1090/S0002-9939-1972-0292740-5. ISSN 1088-6826. MR 0292740.
McDaniel, Wayne L.; Hagis, Peter Jr. (1975). "Some results concerning the non-existence of odd perfect numbers of the form $p^{\alpha }M^{2\beta }$ " (PDF). Fibonacci Quarterly. 13 (1): 25–28. doi:10.1080/00150517.1975.12430680. ISSN 0015-0517. MR 0354538.
Yamada, Tomohiro (2019). "A new upper bound for odd perfect numbers of a special form". Colloquium Mathematicum. 156 (1): 15–21. arXiv:1706.09341. doi:10.4064/cm7339-3-2018. ISSN 1730-6302. S2CID 119175632.
Gallardo, Luis H. (2010). "On a remark of Makowski about perfect numbers". Elem. Math. 65 (3): 121–126. doi:10.4171/EM/149..
Jones, Chris; Lord, Nick (1999). "Characterising non-trapezoidal numbers". The Mathematical Gazette. 83 (497). The Mathematical Association: 262–263. doi:10.2307/3619053. JSTOR 3619053. S2CID 125545112.
Hornfeck, B (1955). "Zur Dichte der Menge der vollkommenen zahlen". Arch. Math. 6 (6): 442–443. doi:10.1007/BF01901120. S2CID 122525522.
Kanold, HJ (1956). "Eine Bemerkung ¨uber die Menge der vollkommenen zahlen". Math. Ann. 131 (4): 390–392. doi:10.1007/BF01350108. S2CID 122353640.

jstor.org (Global: 23^rd place; English: 15^th place)

Jones, Chris; Lord, Nick (1999). "Characterising non-trapezoidal numbers". The Mathematical Gazette. 83 (497). The Mathematical Association: 262–263. doi:10.2307/3619053. JSTOR 3619053. S2CID 125545112.

lirmm.fr (Global: low place; English: low place)

Ochem, Pascal; Rao, Michaël (2012). "Odd perfect numbers are greater than 10¹⁵⁰⁰" (PDF). Mathematics of Computation. 81 (279): 1869–1877. doi:10.1090/S0025-5718-2012-02563-4. ISSN 0025-5718. Zbl 1263.11005.
Fletcher, S. Adam; Nielsen, Pace P.; Ochem, Pascal (2012). "Sieve methods for odd perfect numbers" (PDF). Mathematics of Computation. 81 (279): 1753?1776. doi:10.1090/S0025-5718-2011-02576-7. ISSN 0025-5718. MR 2904601.

math.ca (Global: low place; English: low place)

fq.math.ca

Cohen, G. L.; Williams, R. J. (1985). "Extensions of some results concerning odd perfect numbers" (PDF). Fibonacci Quarterly. 23 (1): 70–76. doi:10.1080/00150517.1985.12429857. ISSN 0015-0517. MR 0786364.
McDaniel, Wayne L.; Hagis, Peter Jr. (1975). "Some results concerning the non-existence of odd perfect numbers of the form $p^{\alpha }M^{2\beta }$ " (PDF). Fibonacci Quarterly. 13 (1): 25–28. doi:10.1080/00150517.1975.12430680. ISSN 0015-0517. MR 0354538.

math.harvard.edu (Global: low place; English: low place)

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"The oldest open problem in mathematics" (PDF). Harvard.edu. Retrieved 16 June 2023.

mersenne.org (Global: low place; English: low place)

"GIMPS Milestones Report". Great Internet Mersenne Prime Search. Retrieved 28 July 2024.
"GIMPS Home". Mersenne.org. Retrieved 2024-10-21.

oddperfect.org (Global: low place; English: low place)

Oddperfect.org. Archived 2006-12-29 at the Wayback Machine

oeis.org (Global: 1,188^th place; English: 866^th place)

All factors of $2^{p}-1$ are congruent to $1{\pmod {2p}}$ . For example, $2^{1}1-1=2047=23\times 89$ , and both 23 and 89 yield a remainder of 1 when divided by 22. Furthermore, whenever $p$ is a Sophie Germain prime—that is, $2p+1$ is also prime—and $2p+1$ is congruent to $1{\pmod {8}}$ or $7{\pmod {8}}$ , then $2p+1$ will be a factor of $2^{p}-1,$ which is the case for $p=11,23,83,131,179,191,239,...$ (OEIS: A002515).
"A000396". On-Line Encyclopedia of Integer Sequences. Retrieved 2024-03-21.

quantamagazine.org (Global: 6,258^th place; English: 4,046^th place)

Nadis, Steve (10 September 2020). "Mathematicians Open a New Front on an Ancient Number Problem". Quanta Magazine. Retrieved 10 September 2020.

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api.semanticscholar.org

Kühnel, Ullrich (1950). "Verschärfung der notwendigen Bedingungen für die Existenz von ungeraden vollkommenen Zahlen". Mathematische Zeitschrift (in German). 52: 202–211. doi:10.1007/BF02230691. S2CID 120754476.
Zelinsky, Joshua (July 2019). "Upper bounds on the second largest prime factor of an odd perfect number". International Journal of Number Theory. 15 (6): 1183–1189. arXiv:1810.11734. doi:10.1142/S1793042119500659. S2CID 62885986..
Nielsen, Pace P. (2007). "Odd perfect numbers have at least nine distinct prime factors" (PDF). Mathematics of Computation. 76 (260): 2109–2126. arXiv:math/0602485. Bibcode:2007MaCom..76.2109N. doi:10.1090/S0025-5718-07-01990-4. S2CID 2767519. Archived from the original (PDF) on 2021-11-03. Retrieved 30 March 2011.
McDaniel, Wayne L. (1970). "The non-existence of odd perfect numbers of a certain form". Archiv der Mathematik. 21 (1): 52–53. doi:10.1007/BF01220877. ISSN 1420-8938. MR 0258723. S2CID 121251041.
Kanold, Hans-Joachim [in German] (1950). "Satze uber Kreisteilungspolynome und ihre Anwendungen auf einige zahlentheoretisehe Probleme. II". Journal für die reine und angewandte Mathematik. 188 (1): 129–146. doi:10.1515/crll.1950.188.129. ISSN 1435-5345. MR 0044579. S2CID 122452828.
Yamada, Tomohiro (2019). "A new upper bound for odd perfect numbers of a special form". Colloquium Mathematicum. 156 (1): 15–21. arXiv:1706.09341. doi:10.4064/cm7339-3-2018. ISSN 1730-6302. S2CID 119175632.
Jones, Chris; Lord, Nick (1999). "Characterising non-trapezoidal numbers". The Mathematical Gazette. 83 (497). The Mathematical Association: 262–263. doi:10.2307/3619053. JSTOR 3619053. S2CID 125545112.
Hornfeck, B (1955). "Zur Dichte der Menge der vollkommenen zahlen". Arch. Math. 6 (6): 442–443. doi:10.1007/BF01901120. S2CID 122525522.
Kanold, HJ (1956). "Eine Bemerkung ¨uber die Menge der vollkommenen zahlen". Math. Ann. 131 (4): 390–392. doi:10.1007/BF01350108. S2CID 122353640.

st-and.ac.uk (Global: 4,765^th place; English: 5,091^st place)

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"Perfect numbers". www-groups.dcs.st-and.ac.uk. Retrieved 9 May 2018.

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Rogers, Justin M. (2015). The Reception of Philonic Arithmological Exegesis in Didymus the Blind's Commentary on Genesis (PDF). Society of Biblical Literature National Meeting, Atlanta, Georgia.

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Goto, T; Ohno, Y (2008). "Odd perfect numbers have a prime factor exceeding 10⁸" (PDF). Mathematics of Computation. 77 (263): 1859–1868. Bibcode:2008MaCom..77.1859G. doi:10.1090/S0025-5718-08-02050-9. Archived from the original (PDF) on 2011-08-07. Retrieved 30 March 2011.

utm.edu (Global: 1,830^th place; English: 1,501^st place)

primes.utm.edu

Caldwell, Chris, "A proof that all even perfect numbers are a power of two times a Mersenne prime".

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Oddperfect.org. Archived 2006-12-29 at the Wayback Machine
Roberts, T (2008). "On the Form of an Odd Perfect Number" (PDF). Australian Mathematical Gazette. 35 (4): 244. Archived from the original (PDF) on 2013-05-14. Retrieved 2009-12-18.
Goto, T; Ohno, Y (2008). "Odd perfect numbers have a prime factor exceeding 10⁸" (PDF). Mathematics of Computation. 77 (263): 1859–1868. Bibcode:2008MaCom..77.1859G. doi:10.1090/S0025-5718-08-02050-9. Archived from the original (PDF) on 2011-08-07. Retrieved 30 March 2011.
Nielsen, Pace P. (2015). "Odd perfect numbers, Diophantine equations, and upper bounds" (PDF). Mathematics of Computation. 84 (295): 2549–2567. doi:10.1090/S0025-5718-2015-02941-X. Archived from the original (PDF) on 8 July 2015. Retrieved 13 August 2015.
Nielsen, Pace P. (2007). "Odd perfect numbers have at least nine distinct prime factors" (PDF). Mathematics of Computation. 76 (260): 2109–2126. arXiv:math/0602485. Bibcode:2007MaCom..76.2109N. doi:10.1090/S0025-5718-07-01990-4. S2CID 2767519. Archived from the original (PDF) on 2021-11-03. Retrieved 30 March 2011.

wolfram.com (Global: 1,091^st place; English: 1,169^th place)

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Weisstein, Eric W. "Perfect Number". mathworld.wolfram.com. Retrieved 2025-02-09. Perfect numbers are positive integers n such that n=s(n), where s(n) is the restricted divisor function (i.e., the sum of proper divisors of n), ...
Weisstein, Eric W. "Perfect Number". MathWorld.

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Ochem, Pascal; Rao, Michaël (2012). "Odd perfect numbers are greater than 10¹⁵⁰⁰" (PDF). Mathematics of Computation. 81 (279): 1869–1877. doi:10.1090/S0025-5718-2012-02563-4. ISSN 0025-5718. Zbl 1263.11005.
McDaniel, Wayne L. (1970). "The non-existence of odd perfect numbers of a certain form". Archiv der Mathematik. 21 (1): 52–53. doi:10.1007/BF01220877. ISSN 1420-8938. MR 0258723. S2CID 121251041.
Fletcher, S. Adam; Nielsen, Pace P.; Ochem, Pascal (2012). "Sieve methods for odd perfect numbers" (PDF). Mathematics of Computation. 81 (279): 1753?1776. doi:10.1090/S0025-5718-2011-02576-7. ISSN 0025-5718. MR 2904601.
Cohen, G. L. (1987). "On the largest component of an odd perfect number". Journal of the Australian Mathematical Society, Series A. 42 (2): 280–286. doi:10.1017/S1446788700028251. ISSN 1446-8107. MR 0869751.
Kanold, Hans-Joachim [in German] (1950). "Satze uber Kreisteilungspolynome und ihre Anwendungen auf einige zahlentheoretisehe Probleme. II". Journal für die reine und angewandte Mathematik. 188 (1): 129–146. doi:10.1515/crll.1950.188.129. ISSN 1435-5345. MR 0044579. S2CID 122452828.
Cohen, G. L.; Williams, R. J. (1985). "Extensions of some results concerning odd perfect numbers" (PDF). Fibonacci Quarterly. 23 (1): 70–76. doi:10.1080/00150517.1985.12429857. ISSN 0015-0517. MR 0786364.
Hagis, Peter Jr.; McDaniel, Wayne L. (1972). "A new result concerning the structure of odd perfect numbers". Proceedings of the American Mathematical Society. 32 (1): 13–15. doi:10.1090/S0002-9939-1972-0292740-5. ISSN 1088-6826. MR 0292740.
McDaniel, Wayne L.; Hagis, Peter Jr. (1975). "Some results concerning the non-existence of odd perfect numbers of the form $p^{\alpha }M^{2\beta }$ " (PDF). Fibonacci Quarterly. 13 (1): 25–28. doi:10.1080/00150517.1975.12430680. ISSN 0015-0517. MR 0354538.
Yamada, Tomohiro (2019). "A new upper bound for odd perfect numbers of a special form". Colloquium Mathematicum. 156 (1): 15–21. arXiv:1706.09341. doi:10.4064/cm7339-3-2018. ISSN 1730-6302. S2CID 119175632.

zbmath.org (Global: 2,089^th place; English: 1,141^st place)

Ochem, Pascal; Rao, Michaël (2012). "Odd perfect numbers are greater than 10¹⁵⁰⁰" (PDF). Mathematics of Computation. 81 (279): 1869–1877. doi:10.1090/S0025-5718-2012-02563-4. ISSN 0025-5718. Zbl 1263.11005.

Perfect number (English Wikipedia)

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