Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Perfect number

From Wikipedia, the free encyclopedia
Number equal to the sum of its proper divisors
For the 2012 film, seePerfect Number (film).
Illustration of the perfect number status of the number 6

Innumber theory, aperfect number is apositive integer that is equal to the sum of its positive properdivisors, that is, divisors excluding the number itself.[1] For instance, 6 has proper divisors 1, 2, and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, because 28 has proper divisors 1, 2, 4 , 7, 14, and 1 + 2 + 4 + 7 + 14 = 28.

The first seven perfect numbers are6,28,496,8128, 33550336, 8589869056, and 137438691328.[2]

The sum of proper divisors of a number is called itsaliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors; in symbols,σ1(n)=2n{\displaystyle \sigma _{1}(n)=2n} whereσ1{\displaystyle \sigma _{1}} is thesum-of-divisors function.

This definition is ancient, appearing as early asEuclid'sElements (Book VII, Definition 22) where it is calledτέλειος ἀριθμός (perfect,ideal, orcomplete number).Euclid also proved a formation rule (Book IX, Proposition 36) wherebyq(q+1)2{\textstyle {\frac {q(q+1)}{2}}} is an even perfect number wheneverq{\displaystyle q} is a primeof the form2p−1{\displaystyle 2^{p}-1} for positive integerp{\displaystyle p}—what is now called aMersenne prime. Two millennia later,Leonhard Euler proved that all even perfect numbers are of this form.[3] This is known as theEuclid–Euler theorem.

It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist.

History

[edit]

In about 300 BC Euclid showed that if 2p − 1 is prime then 2p−1(2p − 1) is perfect.The first four perfect numbers were the only ones known to earlyGreek mathematics, and the mathematicianNicomachus noted 8128 as early as around AD 100.[4] In modern language, Nicomachus states without proof thatevery perfect number is of the form2n−1(2n−1){\displaystyle 2^{n-1}(2^{n}-1)} where2n−1{\displaystyle 2^{n}-1} is prime.[5][6] He seems to be unaware thatn itself has to be prime. He also says (wrongly) that the perfect numbers end in 6 or 8 alternately. (The first 5 perfect numbers end with digits 6, 8, 6, 8, 6; but the sixth also ends in 6.)Philo of Alexandria in his first-century book "On the creation" mentions perfect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect. Philo is followed byOrigen,[7] and byDidymus the Blind, who adds the observation that there are only four perfect numbers that are less than 10,000. (Commentary on Genesis 1. 14–19).[8]Augustine of Hippo defines perfect numbers inThe City of God (Book XI, Chapter 30) in the early 5th century AD, repeating the claim that God created the world in 6 days because 6 is the smallest perfect number. The Egyptian mathematicianIsmail ibn Fallūs (1194–1252) mentioned the next three perfect numbers (33,550,336; 8,589,869,056; and 137,438,691,328) and listed a few more which are now known to be incorrect.[9] The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician.[10] In 1588, the Italian mathematicianPietro Cataldi identified the sixth (8,589,869,056) and the seventh (137,438,691,328) perfect numbers, and also proved that every perfect number obtained from Euclid's rule ends with a 6 or an 8.[11][12][13]

Even perfect numbers

[edit]
See also:Euclid–Euler theorem
Unsolved problem in mathematics
Are there infinitely many perfect numbers?
More unsolved problems in mathematics

Euclid proved that2p−1(2p−1){\displaystyle 2^{p-1}(2^{p}-1)} is an even perfect number whenever2p−1{\displaystyle 2^{p}-1} is prime inElements (Book IX, Proposition 36).

For example, the first four perfect numbers are generated by the formula2p−1(2p−1),{\displaystyle 2^{p-1}(2^{p}-1),} withp aprime number, as follows:p=2:21(22−1)=2×3=6p=3:22(23−1)=4×7=28p=5:24(25−1)=16×31=496p=7:26(27−1)=64×127=8128.{\displaystyle {\begin{aligned}p=2&:\quad 2^{1}(2^{2}-1)=2\times 3=6\\p=3&:\quad 2^{2}(2^{3}-1)=4\times 7=28\\p=5&:\quad 2^{4}(2^{5}-1)=16\times 31=496\\p=7&:\quad 2^{6}(2^{7}-1)=64\times 127=8128.\end{aligned}}}

Prime numbers of the form2p−1{\displaystyle 2^{p}-1} are known asMersenne primes, after the seventeenth-century monkMarin Mersenne, who studiednumber theory and perfect numbers. For2p−1{\displaystyle 2^{p}-1} to be prime, it is necessary thatp itself be prime. However, not all numbers of the form2p−1{\displaystyle 2^{p}-1} with a primep are prime; for example,211 − 1 = 2047 = 23 × 89 is not a prime number.[a] In fact, Mersenne primes are very rare: of the approximately 4 million primesp up to 68,874,199,2p−1{\displaystyle 2^{p}-1} is prime for only 48 of them.[14]

WhileNicomachus had stated (without proof) thatall perfect numbers were of the form2n−1(2n−1){\displaystyle 2^{n-1}(2^{n}-1)} where2n−1{\displaystyle 2^{n}-1} is prime (though he stated this somewhat differently),Ibn al-Haytham (Alhazen) circa AD 1000 was unwilling to go that far, declaring instead (also without proof) that the formula yielded only every even perfect number.[15] It was not until the 18th century thatLeonhard Euler proved that the formula2p−1(2p−1){\displaystyle 2^{p-1}(2^{p}-1)} indeed yields all the even perfect numbers. Thus, there is aone-to-one correspondence between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. This result is often referred to as theEuclid–Euler theorem.

An exhaustive search by theGIMPS distributed computing project has shown that the first 50 even perfect numbers are2p−1(2p−1){\displaystyle 2^{p-1}(2^{p}-1)} for

p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917OEIS: A000043.[14]

Two higher perfect numbers have also been discovered, namely those for whichp = 82589933 and 136279841. Although it is still possible there may be others within this range, initial but exhaustive tests by GIMPS have revealed no other perfect numbers forp below 138277717. As of October 2024[update], 52 Mersenne primes are known,[16] and therefore 52 even perfect numbers (the largest of which is2136279840 × (2136279841 − 1) with 82,048,640 digits). It isnot known whether there areinfinitely many perfect numbers, nor whether there are infinitely many Mersenne primes.

As well as having the form2p−1(2p−1){\displaystyle 2^{p-1}(2^{p}-1)}, each even perfect number is the(2p−1){\displaystyle (2^{p}-1)}-thtriangular number (and hence equal to the sum of the integers from 1 to2p−1{\displaystyle 2^{p}-1}) and the2p−1{\displaystyle 2^{p-1}}-thhexagonal number. Furthermore, each even perfect number except for 6 is the2p+13{\displaystyle {\tfrac {2^{p}+1}{3}}}-thcentered nonagonal number and is equal to the sum of the first2p−12{\displaystyle 2^{\frac {p-1}{2}}} odd cubes (odd cubes up to the cube of2p+12−1{\displaystyle 2^{\frac {p+1}{2}}-1}):

6=21(22−1)=1+2+3,28=22(23−1)=1+2+3+4+5+6+7=13+33496=24(25−1)=1+2+3+⋯+29+30+31=13+33+53+738128=26(27−1)=1+2+3+⋯+125+126+127=13+33+53+73+93+113+133+15333550336=212(213−1)=1+2+3+⋯+8189+8190+8191=13+33+53+⋯+1233+1253+1273{\displaystyle {\begin{alignedat}{3}6&=2^{1}(2^{2}-1)&&=1+2+3,\\[8pt]28&=2^{2}(2^{3}-1)&&=1+2+3+4+5+6+7\\&&&=1^{3}+3^{3}\\[8pt]496&=2^{4}(2^{5}-1)&&=1+2+3+\cdots +29+30+31\\&&&=1^{3}+3^{3}+5^{3}+7^{3}\\[8pt]8128&=2^{6}(2^{7}-1)&&=1+2+3+\cdots +125+126+127\\&&&=1^{3}+3^{3}+5^{3}+7^{3}+9^{3}+11^{3}+13^{3}+15^{3}\\[8pt]33550336&=2^{12}(2^{13}-1)&&=1+2+3+\cdots +8189+8190+8191\\&&&=1^{3}+3^{3}+5^{3}+\cdots +123^{3}+125^{3}+127^{3}\end{alignedat}}}

Even perfect numbers (except 6) are of the formT2p−1=1+(2p−2)×(2p+1)2=1+9×T(2p−2)/3{\displaystyle T_{2^{p}-1}=1+{\frac {(2^{p}-2)\times (2^{p}+1)}{2}}=1+9\times T_{(2^{p}-2)/3}}

with each resulting triangular numberT7 = 28,T31 = 496,T127 = 8128 (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting withT2 = 3,T10 = 55,T42 = 903,T2730 = 3727815, ...[17] It follows that by adding the digits of any even perfect number (except 6), then adding the digits of the resulting number, and repeating this process until a single digit (called thedigital root) is obtained, always produces the number 1. For example, the digital root of 8128 is 1, because8 + 1 + 2 + 8 = 19,1 + 9 = 10, and1 + 0 = 1. This works with all perfect numbers2p−1(2p−1){\displaystyle 2^{p-1}(2^{p}-1)} with odd primep and, in fact, withall numbers of the form2m−1(2m−1){\displaystyle 2^{m-1}(2^{m}-1)} for odd integer (not necessarily prime)m.

Owing to their form,2p−1(2p−1),{\displaystyle 2^{p-1}(2^{p}-1),} every even perfect number is represented in binary form asp ones followed byp − 1 zeros; for example:

610=22+21=11022810=24+23+22=11100249610=28+27+26+25+24=1111100002812810=212+211+210+29+28+27+26=11111110000002{\displaystyle {\begin{array}{rcl}6_{10}=&2^{2}+2^{1}&=110_{2}\\28_{10}=&2^{4}+2^{3}+2^{2}&=11100_{2}\\496_{10}=&2^{8}+2^{7}+2^{6}+2^{5}+2^{4}&=111110000_{2}\\8128_{10}=&\!\!2^{12}+2^{11}+2^{10}+2^{9}+2^{8}+2^{7}+2^{6}\!\!&=1111111000000_{2}\end{array}}}

Thus every even perfect number is apernicious number.

Every even perfect number is also apractical number.

Odd perfect numbers

[edit]
Unsolved problem in mathematics
Are there any odd perfect numbers?
More unsolved problems in mathematics

It is unknown whether any odd perfect numbers exist, though various results have been obtained. In 1496,Jacques Lefèvre stated that Euclid's rule gives all perfect numbers,[18] thus implying that no odd perfect number exists, but Euler himself stated: "Whether ... there are any odd perfect numbers is a most difficult question".[19] More recently,Carl Pomerance has presented aheuristic argument suggesting that indeed no odd perfect number should exist.[20] All perfect numbers are alsoharmonic divisor numbers, and it has been conjectured as well that there are no odd harmonic divisor numbers other than 1.

Any odd perfect numberN must satisfy the following conditions:

N=qαp12e1⋯pk2ek,{\displaystyle N=q^{\alpha }p_{1}^{2e_{1}}\cdots p_{k}^{2e_{k}},}
where:

Furthermore, several minor results are known about the exponentse1, ..., ek.

In 1888,Sylvester stated:[48]

... a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number]—its escape, so to say, from the complex web of conditions which hem it in on all sides—would be little short of a miracle.

On the other hand, several odd integers come close to being perfect. René Descartes observed that the numberD = 32 ⋅ 72 ⋅ 112 ⋅ 132 ⋅ 22021 = (3⋅1001)2 ⋅ (22⋅1001 − 1) = 198585576189 would be an odd perfect number if only22021 (= 192 ⋅ 61) were a prime number. The odd numbers with this property (they would be perfect if one of their composite factors were prime) are theDescartes numbers. Many of the properties proven about odd perfect numbers also apply to Descartes numbers, and Pace Nielsen has suggested that sufficient study of these numbers may lead to a proof that no odd perfect numbers exist.[49]

Minor results

[edit]

All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come underRichard Guy'sstrong law of small numbers:

Related concepts

[edit]
Euler diagram of numbers under 100:
  Weird
  Perfect

The sum ofproper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are calleddeficient, and where it is greater than the number,abundant. These terms, together withperfect itself, come from Greeknumerology. A pair of numbers which are the sum of each other's proper divisors are calledamicable, and larger cycles of numbers are calledsociable. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is apractical number.

By definition, a perfect number is afixed point of therestricted divisor functions(n) =σ(n) −n, and thealiquot sequence associated with a perfect number is a constant sequence. All perfect numbers are alsoS{\displaystyle {\mathcal {S}}}-perfect numbers, orGranville numbers.

Asemiperfect number is a natural number that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. Most abundant numbers are also semiperfect; abundant numbers which are not semiperfect are calledweird numbers.

See also

[edit]

Notes

[edit]
  1. ^All factors of2p−1{\displaystyle 2^{p}-1} are congruent to1mod 2p. For example,211 − 1 = 2047 = 23 × 89, and both 23 and 89 yield a remainder of 1 when divided by 22. Furthermore, wheneverp is aSophie Germain prime—that is,2p + 1 is also prime—and2p + 1 is congruent to 1 or 7 mod 8, then2p + 1 will be a factor of2p−1,{\displaystyle 2^{p}-1,} which is the case forp = 11, 23, 83, 131, 179, 191, 239, 251, ...OEIS: A002515.

References

[edit]
  1. ^Weisstein, Eric W."Perfect Number".mathworld.wolfram.com. Retrieved2025-02-09.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), ...
  2. ^"A000396 - OEIS".oeis.org. Retrieved2024-03-21.
  3. ^Caldwell, Chris,"A proof that all even perfect numbers are a power of two times a Mersenne prime".
  4. ^Dickson, L. E. (1919).History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington. p. 4.
  5. ^"Perfect numbers".www-groups.dcs.st-and.ac.uk. Retrieved9 May 2018.
  6. ^InIntroduction 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 atriangular number based on a Mersenne prime.
  7. ^Commentary on the Gospel of John 28.1.1–4, with further references in theSources Chrétiennes edition: vol. 385, 58–61.
  8. ^Rogers, Justin M. (2015).The Reception of Philonic Arithmological Exegesis in Didymus the Blind'sCommentary on Genesis(PDF).Society of Biblical Literature National Meeting, Atlanta, Georgia.
  9. ^Roshdi Rashed,The Development of Arabic Mathematics: Between Arithmetic and Algebra (Dordrecht: Kluwer Academic Publishers, 1994), pp. 328–329.
  10. ^Bayerische Staatsbibliothek, Clm 14908. SeeDavid Eugene Smith (1925).History of Mathematics: Volume II. New York: Dover. p. 21.ISBN 0-486-20430-8.{{cite book}}:ISBN / Date incompatibility (help)
  11. ^Dickson, L. E. (1919).History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington. p. 10.
  12. ^Pickover, C (2001).Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning. Oxford: Oxford University Press. p. 360.ISBN 0-19-515799-0.
  13. ^Peterson, I (2002).Mathematical Treks: From Surreal Numbers to Magic Circles. Washington: Mathematical Association of America. p. 132.ISBN 88-8358-537-2.
  14. ^ab"GIMPS Milestones Report".Great Internet Mersenne Prime Search. Retrieved28 July 2024.
  15. ^O'Connor, John J.;Robertson, Edmund F.,"Abu Ali al-Hasan ibn al-Haytham",MacTutor History of Mathematics Archive,University of St Andrews
  16. ^"GIMPS Home". Mersenne.org. Retrieved2024-10-21.
  17. ^Weisstein, Eric W."Perfect Number".MathWorld.
  18. ^Dickson, L. E. (1919).History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington. p. 6.
  19. ^"The oldest open problem in mathematics"(PDF).Harvard.edu. Retrieved16 June 2023.
  20. ^Oddperfect.org.Archived 2006-12-29 at theWayback Machine
  21. ^abcOchem, 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.
  22. ^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.
  23. ^Roberts, T (2008)."On the Form of an Odd Perfect Number"(PDF).Australian Mathematical Gazette.35 (4): 244. Archived fromthe original(PDF) on 2013-05-14. Retrieved2009-12-18.
  24. ^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 fromthe original(PDF) on 2011-08-07. Retrieved30 March 2011.
  25. ^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.
  26. ^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. Retrieved30 March 2011.
  27. ^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..
  28. ^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. Retrieved30 March 2011.
  29. ^Bibby, Sean; Vyncke, Pieter; Zelinsky, Joshua (23 November 2021)."On the Third Largest Prime Divisor of an Odd Perfect Number"(PDF).Integers.21. Retrieved6 December 2021.
  30. ^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 fromthe original(PDF) on 8 July 2015. Retrieved13 August 2015.
  31. ^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 fromthe original(PDF) on 2021-11-03. Retrieved30 March 2011.
  32. ^abZelinsky, Joshua (3 August 2021)."On the Total Number of Prime Factors of an Odd Perfect Number"(PDF).Integers.21. Retrieved7 August 2021.
  33. ^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.
  34. ^Nielsen, Pace P. (2003)."An upper bound for odd perfect numbers".Integers.3.doi:10.5281/zenodo.7607545. Retrieved14 August 2025.
  35. ^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.
  36. ^Graeme Clayton, Cody Hansen (2023)."On inequalities involving counts of the prime factors of an odd perfect number"(PDF).Integers.23.arXiv:2303.11974. Retrieved29 November 2023.
  37. ^Pomerance, Carl; Luca, Florian (2010)."On the radical of a perfect number".New York Journal of Mathematics.16:23–30. Retrieved7 December 2018.
  38. ^Cohen, Graeme (1978). "On odd perfect numbers".Fibonacci Quarterly.16 (6): 523-527.doi:10.1080/00150517.1978.12430277.
  39. ^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.
  40. ^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.
  41. ^abcFletcher, 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.
  42. ^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.
  43. ^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.
  44. ^abCohen, 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.
  45. ^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.
  46. ^McDaniel, Wayne L.; Hagis, Peter Jr. (1975)."Some results concerning the non-existence of odd perfect numbers of the formpαM2β{\displaystyle p^{\alpha }M^{2\beta }}"(PDF).Fibonacci Quarterly.13 (1):25–28.doi:10.1080/00150517.1975.12430680.ISSN 0015-0517.MR 0354538.
  47. ^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.
  48. ^The Collected Mathematical Papers of James Joseph Sylvester p. 590, tr. from "Sur les nombres dits de Hamilton",Compte Rendu de l'Association Française (Toulouse, 1887), pp. 164–168.
  49. ^Nadis, Steve (10 September 2020)."Mathematicians Open a New Front on an Ancient Number Problem".Quanta Magazine. Retrieved10 September 2020.
  50. ^Makowski, A. (1962). "Remark on perfect numbers".Elem. Math.17 (5): 109.
  51. ^Gallardo, Luis H. (2010)."On a remark of Makowski about perfect numbers".Elem. Math.65 (3):121–126.doi:10.4171/EM/149..
  52. ^Yan, Song Y. (2012),Computational Number Theory and Modern Cryptography, John Wiley & Sons, Section 2.3, Exercise 2(6),ISBN 9781118188613.
  53. ^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.
  54. ^Hornfeck, B (1955). "Zur Dichte der Menge der vollkommenen zahlen".Arch. Math.6 (6):442–443.doi:10.1007/BF01901120.S2CID 122525522.
  55. ^Kanold, HJ (1956). "Eine Bemerkung ¨uber die Menge der vollkommenen zahlen".Math. Ann.131 (4):390–392.doi:10.1007/BF01350108.S2CID 122353640.
  56. ^H. Novarese.Note sur les nombres parfaits Texeira J. VIII (1886), 11–16.
  57. ^Dickson, L. E. (1919).History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington. p. 25.
  58. ^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..

Sources

[edit]

Further reading

[edit]

External links

[edit]
Divisibility-based sets of integers
Overview
Divisibility of 60
Factorization forms
Constrained divisor sums
With many divisors
Aliquot sequence-related
Base-dependent
Other sets
Classes ofnatural numbers
Powers and related numbers
Of the forma × 2b ± 1
Other polynomial numbers
Recursively defined numbers
Possessing a specific set of other numbers
Expressible via specific sums
2-dimensional
centered
non-centered
3-dimensional
centered
non-centered
pyramidal
4-dimensional
non-centered
Combinatorial numbers
Divisor functions
Prime omega functions
Euler's totient function
Aliquot sequences
Primorial
Otherprime factor ordivisor related numbers
Numeral system-dependent numbers
Arithmetic functions
anddynamics
Digit sum
Digit product
Coding-related
Other
P-adic numbers-related
Digit-composition related
Digit-permutation related
Divisor-related
Other
Generated via asieve
Sorting related
Graphemics related
Authority control databasesEdit this at Wikidata
Retrieved from "https://en.wikipedia.org/w/index.php?title=Perfect_number&oldid=1336950628"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2026 Movatter.jp