Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
The sum of divisors of a number, excluding the number itself, is called its aliquot 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 including itself; in symbols, σ_{1}(n) = 2n where σ_{1} is the sumofdivisors function. For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 × 28.
This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called τέλειος ἀριθμός (perfect, ideal, or complete number). Euclid also proved a formation rule (IX.36) whereby is an even perfect number whenever is a prime of the form for prime —what is now called a Mersenne prime. Two millennia later, Euler proved that all even perfect numbers are of this form.^{[1]} This is known as the Euclid–Euler theorem.
It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first few perfect numbers are 6, 28, 496 and 8128 (sequence A000396 in the OEIS).
HistoryEdit
In about 300 BC Euclid showed that if 2^{p} − 1 is prime then 2^{p−1}(2^{p} − 1) is perfect. The first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus noted 8128 as early as around AD 100.^{[2]} In modern language, Nicomachus states without proof that every perfect number is of the form where is prime.^{[3]}^{[4]} He seems to be unaware that n 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 firstcentury 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 by Origen,^{[5]} and by Didymus the Blind, who adds the observation that there are only four perfect numbers that are less than 10,000. (Commentary on Genesis 1. 1419).^{[6]} St Augustine defines perfect numbers in 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 mathematician Ismail 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.^{[7]} The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician.^{[8]} In 1588, the Italian mathematician Pietro 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.^{[9]}^{[10]}^{[11]}
Even perfect numbersEdit
Unsolved problem in mathematics: Are there infinitely many perfect numbers? (more unsolved problems in mathematics)

Euclid proved that 2^{p−1}(2^{p} − 1) is an even perfect number whenever 2^{p} − 1 is prime (Elements, Prop. IX.36).
For example, the first four perfect numbers are generated by the formula 2^{p−1}(2^{p} − 1), with p a prime number, as follows:
 for p = 2: 2^{1}(2^{2} − 1) = 2 × 3 = 6
 for p = 3: 2^{2}(2^{3} − 1) = 4 × 7 = 28
 for p = 5: 2^{4}(2^{5} − 1) = 16 × 31 = 496
 for p = 7: 2^{6}(2^{7} − 1) = 64 × 127 = 8128.
Prime numbers of the form 2^{p} − 1 are known as Mersenne primes, after the seventeenthcentury monk Marin Mersenne, who studied number theory and perfect numbers. For 2^{p} − 1 to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2^{p} − 1 with a prime p are prime; for example, 2^{11} − 1 = 2047 = 23 × 89 is not a prime number.^{[12]} In fact, Mersenne primes are very rare—of the 2,610,944 prime numbers p up to 43,112,609,^{[13]} 2^{p} − 1 is prime for only 47 of them.
Although Nicomachus had stated (without proof) that all perfect numbers were of the form where is prime (though he stated this somewhat differently), Ibn alHaytham (Alhazen) circa AD 1000 conjectured only that every even perfect number is of that form.^{[14]} It was not until the 18th century that Leonhard Euler proved that the formula 2^{p−1}(2^{p} − 1) will yield all the even perfect numbers. Thus, there is a onetoone 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 the Euclid–Euler theorem.
An exhaustive search by the GIMPS distributed computing project has shown that the first 47 even perfect numbers are 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 and 43112609 (sequence A000043 in the OEIS).^{[15]}
Four higher perfect numbers have also been discovered, namely those for which p = 57885161, 74207281, 77232917, and 82589933, though there may be others within this range. As of December 2018^{[update]}, 51 Mersenne primes are known,^{[16]} and therefore 51 even perfect numbers (the largest of which is 2^{82589932} × (2^{82589933} − 1) with 49,724,095 digits). It is not known whether there are infinitely many perfect numbers, nor whether there are infinitely many Mersenne primes.
As well as having the form 2^{p−1}(2^{p} − 1), each even perfect number is the (2^{p} − 1)th triangular number (and hence equal to the sum of the integers from 1 to 2^{p} − 1) and the 2^{p−1}th hexagonal number. Furthermore, each even perfect number except for 6 is the ((2^{p} + 1)/3)th centered nonagonal number and is equal to the sum of the first 2^{(p−1)/2} odd cubes:
Even perfect numbers (except 6) are of the form
with each resulting triangular number T_{7} = 28, T_{31} = 496, T_{127} = 8128 (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with T_{2} = 3, T_{10} = 55, T_{42} = 903, T_{2730} = 3727815, ...^{[17]} This can be reformulated as follows: 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 the digital root) is obtained, always produces the number 1. For example, the digital root of 8128 is 1, because 8 + 1 + 2 + 8 = 19, 1 + 9 = 10, and 1 + 0 = 1. This works with all perfect numbers 2^{p−1}(2^{p} − 1) with odd prime p and, in fact, with all numbers of the form 2^{m−1}(2^{m} − 1) for odd integer (not necessarily prime) m.
Owing to their form, 2^{p−1}(2^{p} − 1), every even perfect number is represented in binary form as p ones followed by p − 1 zeros; for example,
 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}
and
 8128_{10} = 2^{12} + 2^{11} + 2^{10} + 2^{9} + 2^{8} + 2^{7} + 2^{6} = 1111111000000_{2}.
Thus every even perfect number is a pernicious number.
Every even perfect number is also a practical number (c.f. Related concepts).
Odd perfect numbersEdit
Unsolved problem in mathematics: Are there any odd perfect numbers? (more unsolved problems in mathematics)

It is unknown whether there is any odd perfect number, 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. Euler stated: "Whether (...) there are any odd perfect numbers is a most difficult question".^{[19]} More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist.^{[20]} All perfect numbers are also Ore's harmonic numbers, and it has been conjectured as well that there are no odd Ore's harmonic numbers other than 1.
Any odd perfect number N must satisfy the following conditions:
 N > 10^{1500}.^{[21]}
 N is not divisible by 105.^{[22]}
 N is of the form N ≡ 1 (mod 12) or N ≡ 117 (mod 468) or N ≡ 81 (mod 324).^{[23]}
 N is of the form
 where:
 q, p_{1}, ..., p_{k} are distinct primes (Euler).
 q ≡ α ≡ 1 (mod 4) (Euler).
 The smallest prime factor of N is less than (2k + 8) / 3.^{[24]}
 Either q^{α} > 10^{62}, or p_{j}^{2ej} > 10^{62} for some j.^{[21]}
 ^{[25]}^{[26]}
 .^{[27]}^{[28]}
 .^{[29]}
 The largest prime factor of N is greater than 10^{8}^{[30]} and less than ^{[31]}
 The second largest prime factor is greater than 10^{4}, and is less than .^{[32]}^{[33]}
 The third largest prime factor is greater than 100.^{[34]}
 N has at least 101 prime factors and at least 10 distinct prime factors.^{[21]}^{[35]} If 3 is not one of the factors of N, then N has at least 12 distinct prime factors.^{[36]}
Furthermore, several minor results are known concerning to the exponents e_{1}, ..., e_{k} in
 Not all e_{i} ≡ 1 (mod 3).^{[37]}
 Not all e_{i} ≡ 2 (mod 5).^{[38]}
 If all e_{i} ≡ 1 (mod 3) or 2 (mod 5), then the smallest prime factor of N must lie between 10^{8} and 10^{1000}.^{[38]}
 More generally, if all 2e_{i}+1 have a prime factor in a given finite set S, then the smallest prime factor of N must be smaller than an effectively computable constant depending only on S.^{[38]}
 If (e_{1}, ..., e_{k})= (1, ..., 1, 2, ..., 2) with t ones and u twos, then .^{[39]}
 (e_{1}, ..., e_{k}) ≠ (1, ..., 1, 3),^{[40]} (1, ..., 1, 5), (1, ..., 1, 6).^{[41]}
 If e_{1}= ...= e_{k}= e, then
 e cannot be 3,^{[42]} 5, 24,^{[43]} 6, 8, 11, 14 or 18.^{[41]}
 and N < 2^{42e2 + 8e + 3}.^{[44]}
In 1888, Sylvester stated:^{[45]}
...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.
Many of the properties proved about odd perfect numbers also apply to spoof odd perfect numbers, and Pace Nielsen has suggested that sufficient study of those numbers may lead to a proof that no odd perfect numbers exist. ^{[46]}
Minor resultsEdit
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 under Richard Guy's strong law of small numbers:
 The only even perfect number of the form x^{3} + 1 is 28 (Makowski 1962).^{[47]}
 28 is also the only even perfect number that is a sum of two positive cubes of integers (Gallardo 2010).^{[48]}
 The reciprocals of the divisors of a perfect number N must add up to 2 (to get this, take the definition of a perfect number, , and divide both sides by n):
 For 6, we have ;
 For 28, we have , etc.
 The number of divisors of a perfect number (whether even or odd) must be even, because N cannot be a perfect square.^{[49]}
 From these two results it follows that every perfect number is an Ore's harmonic number.
 The even perfect numbers are not trapezoidal numbers; that is, they cannot be represented as the difference of two positive nonconsecutive triangular numbers. There are only three types of nontrapezoidal numbers: even perfect numbers, powers of two, and the numbers of the form formed as the product of a Fermat prime with a power of two in a similar way to the construction of even perfect numbers from Mersenne primes.^{[50]}
 The number of perfect numbers less than n is less than , where c > 0 is a constant.^{[51]} In fact it is , using littleo notation.^{[52]}
 Every even perfect number ends in 6 or 28, base ten; and, with the only exception of 6, ends in 1, base 9.^{[53]}^{[54]} Therefore in particular the digital root of every even perfect number other than 6 is 1.
 The only squarefree perfect number is 6.^{[55]}
Related conceptsEdit
The sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are called deficient, and where it is greater than the number, abundant. These terms, together with perfect itself, come from Greek numerology. A pair of numbers which are the sum of each other's proper divisors are called amicable, and larger cycles of numbers are called sociable. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a practical number.
By definition, a perfect number is a fixed point of the restricted divisor function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence. All perfect numbers are also perfect numbers, or Granville numbers.
A semiperfect 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 called weird numbers.
See alsoEdit
NotesEdit
 ^ Caldwell, Chris, "A proof that all even perfect numbers are a power of two times a Mersenne prime".
 ^ Dickson, L. E. (1919). History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington. p. 4.
 ^ "Perfect numbers". wwwgroups.dcs.stand.ac.uk. Retrieved 9 May 2018.
 ^ 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.
 ^ Commentary on the Gospel of John 28.1.14, with further references in the Sources Chrétiennes edition: vol. 385, 58–61.
 ^ http://torreys.org/sblpapers2015/S2205_philonic_arithmological_exegesis.pdf
 ^ Roshdi Rashed, The Development of Arabic Mathematics: Between Arithmetic and Algebra (Dordrecht: Kluwer Academic Publishers, 1994), pp. 328–329.
 ^ Bayerische Staatsbibliothek, Clm 14908. See David Eugene Smith (1925). History of Mathematics: Volume II. New York: Dover. p. 21. ISBN 0486204308.
 ^ Dickson, L. E. (1919). History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington. p. 10.
 ^ Pickover, C (2001). Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning. Oxford: Oxford University Press. p. 360. ISBN 0195157990.
 ^ Peterson, I (2002). Mathematical Treks: From Surreal Numbers to Magic Circles. Washington: Mathematical Association of America. p. 132. ISBN 8883585372.
 ^ All factors of 2^{p} − 1 are congruent to 1 mod 2p. For example, 2^{11} − 1 = 2047 = 23 × 89, and both 23 and 89 yield a remainder of 1 when divided by 11. Furthermore, whenever p is a Sophie Germain prime—that is, 2p + 1 is also prime—and 2p + 1 is congruent to 1 or 7 mod 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, 251, ... OEIS: A002515.
 ^ "Number of primes <= 43112609". Wolfram Alpha. Retrieved 20181028.
 ^ O'Connor, John J.; Robertson, Edmund F., "Abu Ali alHasan ibn alHaytham", MacTutor History of Mathematics archive, University of St Andrews.
 ^ GIMPS Milestones Report. Retrieved 20180227
 ^ "GIMPS Home". Mersenne.org. Retrieved 20181221.
 ^ Weisstein, Eric W. "Perfect Number". MathWorld.
 ^ Dickson, L. E. (1919). History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington. p. 6.
 ^ http://www.math.harvard.edu/~knill/seminars/perfect/handout.pdf
 ^ Oddperfect.org. Archived 20061229 at the Wayback Machine
 ^ ^{a} ^{b} ^{c} Ochem, Pascal; Rao, Michaël (2012). "Odd perfect numbers are greater than 10^{1500}" (PDF). Mathematics of Computation. 81 (279): 1869–1877. doi:10.1090/S002557182012025634. ISSN 00255718. Zbl 1263.11005.
 ^ Kühnel, U (1949). "Verschärfung der notwendigen Bedingungen für die Existenz von ungeraden vollkommenen Zahlen". Mathematische Zeitschrift. 52: 201–211. doi:10.1515/crll.1941.183.98.^{[permanent dead link]}
 ^ Roberts, T (2008). "On the Form of an Odd Perfect Number" (PDF). Australian Mathematical Gazette. 35 (4): 244.
 ^ Grün, O (1952). "Über ungerade vollkommene Zahlen". Mathematische Zeitschrift. 55 (3): 353–354. doi:10.1007/BF01181133.
 ^ Chen, YongGao; Tang, CuiE (2014). "Improved upper bounds for odd multiperfect numbers". Bulletin of the Australian Mathematical Society. 89 (3): 353359.
 ^ Nielsen, PP (2003). "An upper bound for odd perfect numbers". Integers. 3: A14–A22. Archived from the original on 21 February 2003. Retrieved 30 March 2011.
 ^ Zelinsky, Joshua (25 May 2018). "An improvement of an inequality of Ochem and Rao concerning odd perfect numbers". Integers. 18. arXiv:1706.07009. Bibcode:2017arXiv170607009Z. Retrieved 23 May 2018.
 ^ 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/S002557182013027767.
 ^ Pomerance, Carl; Luca, Florian (2010). "On the radical of a perfect number". New York Journal of Mathematics. 16: 23–30. Retrieved 7 December 2018.
 ^ Goto, T; Ohno, Y (2008). "Odd perfect numbers have a prime factor exceeding 10^{8}" (PDF). Mathematics of Computation. 77 (263): 1859–1868. Bibcode:2008MaCom..77.1859G. doi:10.1090/S0025571808020509. 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.
 ^ 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..
 ^ 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/S0025571899011266. Retrieved 30 March 2011.
 ^ 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/S0025571899011278. Retrieved 30 March 2011.
 ^ Nielsen, PP (2015). "Odd perfect numbers, Diophantine equations, and upper bounds" (PDF). Mathematics of Computation. 84 (295): 2549–2567. doi:10.1090/S00255718201502941X. Retrieved 13 August 2015.
 ^ Nielsen, PP (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/S0025571807019904. Retrieved 30 March 2011.
 ^ McDaniel, Wayne L. (1970). "The nonexistence of odd perfect numbers of a certain form". Archiv der Mathematik. 21 (1): 52–53. doi:10.1007/BF01220877. ISSN 14208938. MR 0258723.
 ^ ^{a} ^{b} ^{c} 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/S002557182011025767. ISSN 00255718. 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 14468107. MR 0869751.
 ^ Kanold, HansJoachim (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 14355345. MR 0044579.
 ^ ^{a} ^{b} Cohen, G. L.; Williams, R. J. (1985). "Extensions of some results concerning odd perfect numbers" (PDF). Fibonacci Quarterly. 23 (1): 70–76. ISSN 00150517. 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/S00029939197202927405. ISSN 10886826. MR 0292740.
 ^ McDaniel, Wayne L.; Hagis, Peter, Jr. (1975). "Some results concerning the nonexistence of odd perfect numbers of the form " (PDF). Fibonacci Quarterly. 13 (1): 25–28. ISSN 00150517. 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/cm733932018. ISSN 17306302.
 ^ 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.
 ^ Nadis, Steve (10 September 2020). "Mathematicians Open a New Front on an Ancient Number Problem". Quanta Magazine. Retrieved 10 September 2020.
 ^ Makowski, A. (1962). "Remark on perfect numbers". Elem. Math. 17 (5): 109.CS1 maint: ref=harv (link)
 ^ Gallardo, Luis H. (2010). "On a remark of Makowski about perfect numbers". Elem. Math. 65: 121–126. doi:10.4171/EM/149.CS1 maint: ref=harv (link).
 ^ Yan, Song Y. (2012), Computational Number Theory and Modern Cryptography, John Wiley & Sons, Section 2.3, Exercise 2(6), ISBN 9781118188613.
 ^ Jones, Chris; Lord, Nick (1999). "Characterising nontrapezoidal numbers". The Mathematical Gazette. The Mathematical Association. 83 (497): 262–263. doi:10.2307/3619053. JSTOR 3619053.
 ^ Hornfeck, B (1955). "Zur Dichte der Menge der vollkommenen zahlen". Arch. Math. 6 (6): 442–443. doi:10.1007/BF01901120.
 ^ Kanold, HJ (1956). "Eine Bemerkung ¨uber die Menge der vollkommenen zahlen". Math. Ann. 131 (4): 390–392. doi:10.1007/BF01350108.
 ^ H. Novarese. Note sur les nombres parfaits Texeira J. VIII (1886), 11–16.
 ^ Dickson, L. E. (1919). History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington. p. 25.
 ^ Redmond, Don (1996). Number Theory: An Introduction to Pure and Applied Mathematics. Chapman & Hall/CRC Pure and Applied Mathematics. 201. CRC Press. Problem 7.4.11, p. 428. ISBN 9780824796969..
ReferencesEdit
 Euclid, Elements, Book IX, Proposition 36. See D.E. Joyce's website for a translation and discussion of this proposition and its proof.
 Kanold, H.J. (1941). "Untersuchungen über ungerade vollkommene Zahlen". Journal für die Reine und Angewandte Mathematik. 183: 98–109.
 Steuerwald, R. "Verschärfung einer notwendigen Bedingung für die Existenz einer ungeraden vollkommenen Zahl". S.B. Bayer. Akad. Wiss. 1937: 69–72.
Further readingEdit
 Nankar, M.L.: "History of perfect numbers," Ganita Bharati 1, no. 1–2 (1979), 7–8.
 Hagis, P. (1973). "A Lower Bound for the set of odd Perfect Prime Numbers". Mathematics of Computation. 27 (124): 951–953. doi:10.2307/2005530. JSTOR 2005530.
 Riele, H.J.J. "Perfect Numbers and Aliquot Sequences" in H.W. Lenstra and R. Tijdeman (eds.): Computational Methods in Number Theory, Vol. 154, Amsterdam, 1982, pp. 141–157.
 Riesel, H. Prime Numbers and Computer Methods for Factorisation, Birkhauser, 1985.
 Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 15–98. ISBN 1402025467. Zbl 1079.11001.
External linksEdit
 "Perfect number", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 David Moews: Perfect, amicable and sociable numbers
 Perfect numbers – History and Theory
 Weisstein, Eric W. "Perfect Number". MathWorld.
 OEIS sequence A000396 (Perfect numbers)
 OddPerfect.org A projected distributed computing project to search for odd perfect numbers.
 Great Internet Mersenne Prime Search (GIMPS)
 Perfect Numbers, math forum at Drexel.
 Grimes, James. "8128: Perfect Numbers". Numberphile. Brady Haran. Archived from the original on 20130531. Retrieved 20130402.