A natural number is a perfect number (also perfect number ) called when the sum of all its (positive) factor is other than itself. An equivalent definition is: A perfect number is a number that is half the sum of all its positive factors (including yourself); H. . The smallest three perfect numbers are 6, 28 and 496. Example: The positive divisors of 28 are 1, 2, 4, 7, 14, 28 and all known perfect numbers are even and derived from Mersenne prime numbers . It is unknown whether there are also odd perfect numbers. Perfect numbers were already known in ancient Greece , and their most important properties were dealt with in the elements of Euclid . All even perfect numbers end in 6 or 8. Perfect numbers have often been the subject of numerical and numerological interpretations.
Just perfect numbers
The divisional sum of a number is necessarily less than, greater than or equal to . In the first case it is deficient , in the second case it is abundant and in the third case it is complete. In contrast to deficient and abundant numbers, perfect numbers are very rare. Euclid already stated that the first four perfect numbers from the term
by adding suitable numbers to:
- For :
- For :
- For :
- For :
Euclid proved that whenever is a prime number is a perfect number . These are the so-called Mersenne prime numbers . Almost 2000 years later Leonhard Euler was able to prove that all even perfect numbers n can be generated in this way: Even perfect numbers and Mersenne prime numbers are reversibly assigned to one another .
By January 2019, 51 Mersenne prime numbers were known; for the following exponents : 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1,279, 2,203, 2,281, 3,217, 4,253, 4,423, 9,689, 9,941, 11,213, 19,937, 21,701, 23,209, 44,497, 86,243, 110,503, 132,049, 216,091, 756,839, 859,433, 1,257,787, 1,398,269, 2,976,221, 3,021,377, 6,972,593, 13,466,917, 20,996,011, 24.036.583, 25.964.951, 30.402.457, 32.582.657, 37.156.667, 42.643.801, 43.112.609, 57.885.161, 74.207.281, 77.232.917, 82.589.933. (Follow A000043 in OEIS )
- It is not clear whether there are infinitely many perfect numbers.
- It is unclear whether there are infinitely many even perfect numbers. This question coincides with the question of whether there are infinitely many Mersenne prime numbers.
- It remains to be seen whether there is even an odd perfect number. If such a number exists, it has the following properties:
- It is greater than 10 1500 .
- It has the form or with a natural number . (Theorem by Jacques Touchard ).
- It has at least 8 different prime divisors .
- It has at least 11 different prime divisors if it is not divisible by 3.
- If the number of their different prime divisors is the smallest of them, then we have ( Otto Grün's theorem ).
- It is smaller than (theorem of DR Heath-Brown ).
- If it is less than 10 9118 , then it is divisible by with a prime number that is greater than 10 500 .
- It is not a square number .
Further properties of perfect numbers
Sum of the reciprocal factors
The sum of the reciprocal values of all divisors of a perfect number (including the number itself) is 2:
- The following applies to:
Illustration by Eaton (1995, 1996)
Every even perfect number n> 6 has the representation
- with and a nonnegative integer .
Conversely, one does not get a perfect number for every natural number.
- results and (perfect).
- results and (perfect).
- results and (not perfect).
Sum of the cubes of the first odd natural numbers
With the exception of 6, every even perfect number can be represented as
where is the exponent of the Mersenne prime number from the representation .
The following applies to each and :
- (Molecular formula of odd cube numbers).
In particular, this also applies to all powers of two and with :
You can substitute with odd :
The representation as the sum of cubic numbers is a property that only very indirectly is something with perfect numbers
- with p = 2, 3, 4, 5, 6, ...
has to do (only after removing the first perfect number n (p = 2) = 6 and assuming that there are no odd perfect numbers), but a property of the series of numbers
is. We also see why it cannot hold for the first perfect number ( is not odd and therefore not an integer).
Incidentally, this equation is fulfilled for numbers in addition to eight perfect numbers out of a total of 2,659,147,948,473 numbers.
Sum of the first natural numbers
Any even perfect number can be represented with a suitable natural number as
Or to put it another way: Every even perfect number is also a triangular number . As mentioned above, is always a Mersenne prime.
Any even perfect number can be represented with a suitable natural number as
An even perfect number appears in the dual system as a characteristic sequence of ones and zeros.
Due to its shape , it is represented in the base 2 number system as a sequence of ones and zeros:
An even perfect number appears in the quaternary system as a characteristic sequence of threes and zeros.
Due to its shape , it is represented in the base 4 number system as a sequence of one, three and zeros:
Generalization of the perfect numbers
A -perfect number is a number whose real divisors add up to -f times the number itself. The perfect numbers are then exactly the perfect numbers. All -perfect numbers with are trivially abundant numbers .
- As real divisors, 120 has the numbers 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40 and 60. The sum of these numbers gives what 120 is a perfect number .
Relationship with other number classes
Abundant and deficient numbers
Abundant numbers are natural numbers for which the sum of the real divisors is greater than the number itself. Deficient numbers are natural numbers for which this sum is smaller than the number itself.
The smallest abundant number is 12. The divisor sum results . The abundant numbers up to 100 are the following:
- 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, ... (sequence A005101 in OEIS )
The deficient numbers are almost all others:
- 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, ... (sequence A005100 in OEIS )
If a number is neither abundant nor deficient, it is a perfect number.
Friendly and sociable numbers
Two different natural numbers, where the sum of the real divisors of the first number is the second and that of the second number is the first, are called a friendly pair of numbers . The smaller of them is abundant and the larger is deficient.
- and form the smallest pair of friendly numbers.
Are needed more than two natural numbers to come back this way again to the original number, it is called sociable numbers (Engl. Sociable numbers ).
Example of 5 social numbers:
- 12,496, 14,288, 15,472, 14,536, 14,264
Pseudo perfect numbers
A natural number is called pseudo-perfect if it can be represented as the sum of several different real divisors.
- is pseudo perfect, but not perfect, because the divisor 2 is missing in the sum representation.
All pseudo perfect numbers are either perfect or abundant.
A real subset of the pseudo-perfect numbers is formed by the primarily pseudo-perfect numbers : Let be a composite number and the set of prime divisors of . The number is primarily called pseudo-perfect if:
Is equivalent to the following characterization: a composite number with the set of prime factors is pseudo fully come exactly then primarily when: . This shows the close relationship between the primarily pseudo-perfect numbers and the Giuga numbers , which are characterized by.
- 6 = 2 × 3
- 42 = 2 × 3 × 7
- 1806 = 2 × 3 × 7 × 43
- 47.058 = 2 × 3 × 11 × 23 × 31
- 2,214,502,422 = 2 × 3 × 11 × 23 × 31 × 47,059
- 52,495,396,602 = 2 × 3 × 11 × 17 × 101 × 149 × 3109
- 8,490,421,583,559,688,410,706,771,261,086 = 2 × 3 × 11 × 23 × 31 × 47,059 × 2,217,342,227 × 1,729,101,023,519
Sometimes the requirement that it must be composed is dispensed with , which then results in the addition of the numbers 1 and 2 to the list.
Properties of the primarily pseudo-perfect numbers:
- All primarily pseudo perfect numbers are free of squares .
- The number 6 is the only primarily pseudo-perfect number that is also perfect. All other primarily pseudo-perfect numbers are abundant.
- There are only finitely many primarily pseudo-perfect numbers with a given number of prime factors.
- It is not known whether there are an infinite number of primarily pseudo-perfect numbers.
Weird Numbers or weird numbers
A natural number is called weird (in German "strange") if it is abundant but not pseudo perfect. So it cannot be represented as the sum of some of its real divisors, even though the total of its real divisors exceeds the number .
Example: The number 70 is the smallest odd number. It cannot be written as the sum of numbers from the subset . The next strange numbers are 836, 4030, 5830, 7192, 7912, 9272, 10430. (Sequence A006037 in OEIS )
- There are an infinite number of strange numbers.
- All known strange numbers are even. It is unknown whether an odd odd number exists.
If both the number of divisors and the sum of the divisors of a natural number are perfect numbers, then they are called sublime. At the moment (2010) only two raised numbers are known: the 12 and a number with 76 digits (sequence A081357 in OEIS ).
Quasi-perfect numbers ( English quasiperfect numbers ) result as an obvious modification of the perfect numbers. This involves taking place all over the divider amount of a natural number only the non-trivial divisor, that is, all except divider and himself, and demands that the sum of which equals the number was. Accordingly, a is quasi-perfect if and only if the equation
So far (as of 2006) no quasi-perfect number is known. All that has been found is a series of necessary conditions that every quasi-perfect number has to satisfy, such as:
- has at least 7 different prime factors .
- For any divisor of the partial sum , the congruence relation or always applies .
Super perfect numbers
- The number 2 has the partial sum , 3 the partial sum . Because of 2 is super perfect.
- The perfect number 6 has the partial sum , 12 the partial sum . Therefore 6 is not super perfect.
Perfect numbers in late antiquity and the Middle Ages
The arithmetic properties of perfect numbers and sometimes their arithmological interpretation are part of the arithmetic subject matter in late antiquity and are passed on to the Latin Middle Ages by Boëthius in his Institutio arithmetica, which in turn is largely based on Nicomachus of Gerasa . Following his Greek model, Boëthius treats perfect numbers (numeri perfecti secundum partium aggregationem) as a subspecies of even numbers (numeri pares) and explains their calculation principle, which goes back to Euclid, in such a way that the terms in the series of even-even numbers ( numeri pariter pares: 2 n ) have to be added together until their sum results in a prime number: multiplying this prime number by the last added series element results in a perfect number. Boëthius demonstrates this method of calculation in the individual steps for the first three perfect numbers 6, 28 and 496 and also mentions the fourth perfect number 8128. Boëthius' supplementary observation of the regularity of the perfect numbers is based on this finding every decade (power of ten) occurred exactly once and ended in the "ones" on 6 or 8. In the centuries that followed, Boëthius' explanations formed the sum of the arithmetic knowledge of the perfect numbers, which was passed on in the tracts De arithmetica, in encyclopedias such as the Etymologiae Isidors and other didactic works, more or less completely, but without essential additions, up to and including With the discovery of the fifth perfect number (33550336) in the 15th century, it was recognized that the assumption about the regular distribution over the 'decades' is incorrect.
While Boëthius remains largely confined to the arithmetic subject matter when dealing with other types of numbers, the perfect numbers also give him reason for further, ethical considerations in which they use the abundant ( plus quam perfecti, also called superflui or abundantes ) and the deficit numbers ( inperfecti , also called deminuti or indigentes ): While these latter two types of numbers resemble human vices because they are very common just like these and are not subject to any particular order, the perfect numbers behave like virtue in that they have the right measure, the middle between excess and deficiency, to preserve, to be found extremely rarely and to submit to a fixed order. Boëthius also suggests an aesthetic preference for perfect numbers when he compares the abundant with monsters from mythology such as the three-headed Geryon , while he compares the deficient with deformities, which, like the one-eyed cyclops , are characterized by too little natural body parts are. In these comparisons, the Boëthius assumes already from its Greek original, stands in the background the idea that a number a of links (partes) composite body has, so are only available if the perfect numbers the members of the number in a balanced proportion to their body .
Their actual meaning for the medieval tradition unfolded the perfect numbers in the biblical exegesis , where the starting point was the interpretation of the six days of creation, on which God completed the works of his creation (“consummavit” in the Vetus Latina , “perfecit” in the Vulgate of Jerome ) formed in order to establish a connection between the arithmetic perfection of the six number and the perfection of the divine work of creation. In this tradition , the number six became a prime example for illustrating the view that divine creation is ordered according to measure, number and weight . Augustine , who in turn developed approaches from predecessors from Alexandrian exegesis, was decisive for the Latin world . Augustine expressed himself very often in his exegetical and homiletic works on the perfection of the number six, most extensively in his commentary De genesi ad litteram, where he not only explains the arithmetic facts and the theological question of whether or not God chose the number six because of its perfection He only gave it this perfection through his choice, but also demonstrates in the works of creation that the fulfillment of the number six through its parts (partes) 1, 2 and 3 is also reflected in the nature of the works of creation and corresponds to a latent ordo of creation:
- The first day of creation with the creation of light, which for Augustine also implies the creation of the heavenly intelligences, stands as a day of its own.
- It is followed by the two days on which the world structure, the fabrica mundi, was created: on the second day of creation first its upper area, the firmament of heaven, and on the third day of creation the lower area, the dry land and the sea.
- The last three days again form a group of their own, as those creatures were created on them that should move in this fabrica mundi and populate and adorn it: on the fourth day, initially again in the upper area, the celestial bodies, sun, moon and stars, on the fifth day then in the lower area the animals of the water and the air, and on the sixth day finally the animals of the land and finally, as the most perfect work, man.
The perfection of the number six, which Augustine also addresses as a triangular number , results from this factual interpretation in two ways: on the one hand in the sequence of days, but on the other hand also from the fact that the work of the first day is not assigned to any particular upper or lower area (symbolized here by the letter A), the works of the following days, on the other hand, belong either to the upper (B) or the lower (C) area, so that there is again a perfect order of 1, 2 and 3 days with the distribution .
Usually not with this detailed interpretation of the latent ordo, but at least in the general interpretation as arithmetic numerus perfectus , this understanding of the six-day work became the common property of medieval exegesis and the starting point for the interpretation of almost all other occurrences of the six number in the Bible and salvation history - see above Among other things, in the interpretation of the six world ages derived from the days of creation (Adam, Noah, Abraham, David, Babylonian captivity, Christ), which in turn as two "before the law" (ante legem), as three "under the law" (sub lege) and were interpreted as an age of grace (sub gratia) , in the interpretation of the six ages of man and in the interpretation of Holy Week - in which the passion of Christ is fulfilled on the sixth day from the sixth hour - and many other biblical and extra-biblical Senare more.
Medieval poets also occasionally tied in with this by using arithmetic understanding in its biblical and exegetical content for the structure of their works. Thus Alcuin a metric poem in six stanzas of six verses of Gundrada, a relative of Charlemagne , written and explained in an accompanying prose explanation that he had chosen the number six, so the moral perfectio to transport the recipient:
“Hoc carmen tibi cecini senario numero nobili, qui numerus perfectus est in partibus suis, te optans esse perfectum in sensibus tuis. Cuius numeri rationem, sicut et aliorum, sapientissimus imperator tuae perfacile ostendere potest sagacitati. "
“I sang this poem to you in the noble number six, which is perfect in its parts, because I want you to be perfect in your senses. What this and other numbers are all about, the very wisest emperor will be able to explain with ease to your eager understanding. "
Alkuin's pupil Hrabanus Maurus not only established similar relationships to the perfectio of the six number in several shorter poems , but also aligned the overall structure to the perfectio of 28 in his main poetic work, the Liber de laudibus sanctae crucis . This work consists of 28 figure poems (carmina figurata), each of which is accompanied by a prose explanation and a paraphrase in prose in the second book. The figure poems themselves are written in hexameters with the same number of letters within the poem and are written in the manuscripts without word spacing, so that the metric text appears as a rectangular block. Within this block, individual letters are then highlighted in color and by circles, which in turn can be reassembled into new texts, so-called versus intexti . In the prose explanation for the 28th and last of these figures, Hrabanus then also points out the reasons for his choice of the number 28:
"Continet autem totus liber iste viginti octo figuras metricas cum sequente sua prosa (...): qui numerus intra centenarium suis partibus perfectus est, ideo juxta hujus summam opus consummare volui, qui illam formam in eo cantavi quae consummatrix et perfectio rerum est."
“However, the entire book contains 28 metrical figures with their subsequent prose (...): This number is in the range of a hundred that is fulfilled by its parts, and that's why I wanted to complete this work in this sum, which I myself in which I sang about that form (ie the cross of Christ) which is the perfection and fulfillment of all things. "
As Burkhard Taeger (1972) discovered in modern times, the arithmetic understanding of the number also intervenes more deeply in the formal structure of the work. If you subdivide the 28 figure poems according to the number of their letters per verse, the result is a grouping of 1, 2, 4, 7 and 14 poems, so that the complete fulfillment of the 28 through their partes is also reflected in the internal structure of the work .
Evidence for poetic adaptations of the underlying numerical understanding can also be found in the later Middle Ages, and also in the fine arts, where one usually has to do without explanatory additions to the structure of the works, one can assume that Giotto's 28 frescoes about life of St. Francis in the Upper Basilica of Assisi with their number want to seal the perfection of the saint and the Christ-likeness of his life.
- Stanley J. Bezuszka: Even Perfect Numbers - An Update. In: Mathematics Teacher. Volume 74, 1981, pp. 460-463.
- Stanley J. Bezuszka, Margaret J. Kenney: Even Perfect Numbers: (Update )². In: Mathematics Teacher. Volume 90, 1997, pp. 628-633.
- Paul Erdős , János Surányi : Topics in the Theory of Numbers (= Undergraduate Texts in Mathematics ). 2nd Edition. Springer, New York 2003, ISBN 0-387-95320-5 (English, Hungarian: Válogatott fejezetek a számelméletből . Translated by Barry Guiduli).
- Otto Grün: About odd perfect numbers . In: Mathematical Journal . tape 55 , no. 3 , 1952, pp. 353-354 , doi : 10.1007 / BF01181133 .
- DR Heath-Brown: Odd perfect numbers . In: Mathematical Proceedings of the Cambridge Philosophical Society . tape 115 , no. 2 , 1994, p. 191–196 , doi : 10.1017 / S0305004100072030 ( full text [PDF; 117 kB ; accessed on June 16, 2017]).
- Ullrich Kühnel: tightening of the necessary conditions for the existence of odd perfect numbers. In: Mathematical Journal . Volume 52, No. 1, 1950, pp. 202-211, doi: 10.1007 / BF02230691 .
- József Sándor , Dragoslav S. Mitrinović, Borislav Crstici: Handbook of Number Theory. I . Springer Verlag, Dordrecht 2006, ISBN 978-1-4020-4215-7 .
- József Sándor , Borislav Crstici: Handbook of Number Theory. II . Kluwer Academic Publishers, Dordrecht / Boston / London 2004, ISBN 1-4020-2546-7 .
- Wacław Sierpiński : Elementary Theory of Numbers (= North-Holland Mathematical Library . Volume 31 ). 2nd revised and expanded edition. North-Holland (inter alia), Amsterdam (inter alia) 1988, ISBN 0-444-86662-0 .
- Eric W. Weisstein : Perfect Number . In: MathWorld (English).
- Perfect numbers and Mersenne numbers .
- List of known Mersenne prime numbers - PrimeNet. In: mersenne.org. Retrieved January 2, 2019 .
- Sierpiński: Elementary Theory of Numbers . S. 182 ff .
- Sándor Crstici: Handbook of Number Theory. II . S. 23 ff .
- Sándor Mitrinović-Crstici: Handbook of Number Theory. I . S. 100 ff .
- Pascal Ochem, Michaël Rao: Odd perfect numbers are greater than 10 1500 . In: Mathematics of Computation . tape 81 , no. 279 , January 1, 2012, ISSN 0025-5718 , p. 1869–1877 , doi : 10.1090 / S0025-5718-2012-02563-4 ( ams.org [accessed March 5, 2017]).
- Judy A. Holdener: A theorem of Touchard on the form of odd perfect numbers. In: Am. Math. Mon. 109, no. 7: 661-663 (2002).
- So you get the remainder 1 with whole number division by 12 or the remainder 9 with whole number division by 36.
- Sándor Crstici: Handbook of Number Theory. II . S. 36-37 .
- Sándor Mitrinović-Crstici: Handbook of Number Theory. I . S. 109-110 .
- Boëthius : De institutione arithmetica libri duo. Edited by Gottfried Friedleich (together with De institutione musica ), Leipzig 1867, Nachdr. Minerva GmbH, Frankfurt / Main 1966; on medieval reception, see Pearl Kibre: The Boethian De Institutione Arithmetica and the Quadrivium in the Thirteenth Century University Milieu at Paris. In: Michael Masi (Ed.): Boethius and the Liberal Arts: A Collection of Essays. Verlag Peter Lang, Bern / Frankfurt / Main / Las Vegas 1981 (= Utah Studies in Literature and Linguistics, 18), pp. 67–80; Michael Masi: The Influence of Boethius' De Arithmetica on Late Medieval Mathematics. Ibid, pp. 81-95.
- Nicomachus of Gerasa : Arithmetica introductio. Edited by Richard Hoche , Teubner Verlag, Leipzig 1866; for the tradition of perfect numbers in Byzantine arithmetic, see Nicole Zeegers-Vander Vorst: L'arithmétique d'un Quadrivium anonyme du XIe siècle. In: L'Antiquité classique . 32 (1963), pp. 129-161, especially pp. 144 f.
- Boëthius: De institutione arithmetica. Lib. I, cap. 19-20, ed. Friedlein 1867, pp. 39-45. For representations in the Latin tradition independent of Boëthius, see also Martianus Capella : De nvptiis Philologiae et Mercvrii. VII, 753, ed. by James Willis, Teubner Verlag, Leipzig 1983; Macrobius : Commentarii in Somnium Scipionis. I, vi, 12, ed. by Jakob Willis, Teubner Verlag, Leipzig 1963; and Cassiodorus : De artibus ac disciplinis liberalium litterarum. VII, PL 70, 1206. These present the same arithmetic understanding but go into less detail. Vitruvius offers a different reason for the perfectio of the number six : De architectura. III, i, 6 (edited and translated by Curt Fensterbusch, Wissenschaftliche Buchgesellschaft, Darmstadt 1964), which does not seem to have found any medieval successors.
- Isidore of Seville : Etymologiae. VII, v. 9-11, ed. by Wallace M. Lindsay, Clarendon Press, Leipzig 1911.
- Stanley J. Bezuszka: Even Perfect Numbers - An Update. In: Mathematics Teacher. 74: 460-463 (1981). On the history of the discovery of further perfect numbers up to 1997: Stanley J. Bezuszka, Margaret J. Kenney: Even Perfect Numbers: (Update) ². In: Mathematics Teacher. 90 (1997), pp. 628-633.
- Heinz Meyer: The number allegoresis in the Middle Ages: method and use. Wilhelm Finck Verlag, Munich 1975 (= Münstersche Mittelalter-Schriften 25. ) pp. 30–35; Heinz Meyer / Horst Suntrup (eds.): Lexicon of the meanings of medieval numbers. Wilhelm Finck Verlag, Munich 1987 (= MMS 56. ) Art. Sechs, Sp. 442–479.
- Augustine : De genesi ad litteram. IV, 1-7, CSEL 28.1 (1894), pp. 93-103; see also De trinitate. IV, iv-vi, CCSL 50 (1968), pp. 169-175.
- On the methodological prerequisites for the interpretation of numbers and numerical relationships in the structure of medieval literature, see Ernst Hellgardt : On the problem of symbol-determined and formally aesthetic number composition in medieval literature. CH Beck, Munich 1973 (= Munich Texts and Studies. Volume 45), ISBN 978-3-4060-2845-8 ; Otfried Lieberknecht: Allegorese and Philology: Reflections on the problem of the multiple sense of writing in Dante's Commedia. Franz Steiner Verlag, Stuttgart 1999 (= Text and Context, 14. ) P. 133 ff. ( Online version here. )
- Alcuin: Epistola 309 (Ad Gundradam). In: Epistolae (in quart) 4: Epistolae Karolini aevi (II). Published by Ernst Dümmler u. a. Berlin 1895, pp. 473–478 ( Monumenta Germaniae Historica , version) Translation quoted from Paul Klopsch (Hrsg.): Latin poetry of the Middle Ages. Reclam-Verlag, Stuttgart 1985 (= Reclams Universal-Bibliothek, 8088. ).
- Carmina. In: Poetae Latini medii aevi 2: Poetae Latini aevi Carolini (II). Published by Ernst Dümmler . Berlin 1884, pp. 154-258 ( Monumenta Germaniae Historica , ). E.g. Carm. XVIII, vv. 55-60 to Archbishop Otgar of Mainz , where the number of verses 66 is established with the desire that the receiver "perfect in manners and the perfect Mr belonging following servant" (perfectus moribus atque / Perfectum dominum rite sequens famulus) be may .
- Here, quoted from Mignes reprint of Wimpfeling's edition : Hrabanus Maurus: De laudibus sanctae crucis. PL 107, 133-294, cf. the new critical edition by Michel Perrin: In honorem sanctae crucis. CCCM 100 (1997) and the facsimile edition by Kurt Holter (ed.): Liber de laudibus Sanctae Crucis. Complete facsimile edition in the original format of Codex Vindobonensis 652 of the Austrian National Library. Akademische Druck- und Verlagsanstalt, Graz 1973 (= Codices selecti, 33. ).
- Hrabanus Maurus: De laudibus sanctae crucis. Lib. I, figura XXVIII, PL 107,264; as an example of the use of six as a perfect number, see also figura XXIII and the declaratio figurae, PL 107, 239–242.
- Burkhard Taeger: Number symbols in Hraban, in Hincmar - and in 'Heliand'? HC Beck, Munich 1972 (= Munich texts and investigations, 30. ).
- On Dante Alighieri see Otfried Lieberknecht: “Perfect numbers” in arithmetic, spiritual exegesis and literary number composition of the Middle Ages. Lecture, University of Kaiserslautern, special event History of Mathematics, February 18, 1998; ders .: Dante's Historical Arithmetics: The Numbers Six and Twenty-eight as “numeri perfecti secundum partium aggregationem” in Inferno XXVIII. Lecture, 32nd International Congress on Medieval Studies, Western Michigan University, Kalamazoo, 1997.
- See also Fritz Tschirch: Literary building works secrets. On the symbolic extent of medieval poetry. In: Ders .: Reflections. Investigations on the border between German studies and theology. Erich Schmidt Verlag, Berlin 1966, pp. 212–225, here p. 213 for number 28.