Academic Open Internet Journal

ISSN 1311-4360

www.acadjournal.com

Volume 17, 2006

 

 

 

Unique features of the first perfect number

 

Dr. David Kisets

The National Physical Laboratory of Israel (INPL), Danziger “A” bldg., Hebrew

University Givat-Ram, Jerusalem, 91904, Israel. E-mail: kisets@netvision.net.il

 

 

Abstract: The paper presents the results of investigation demonstrating remarkable properties of the six – the first perfect number, including the complete conformity of all its fractions to such conceptions of perfection as informational optimality, mathematical harmony and balance.

 

Keywords:  Informational optimality; Mathematical harmony; Perfect relations;

                     Proper fractions.

 

 

“Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect...”

               (Saint Augustine (354-430) - The City of God)

Introduction

The paper reveals in the series of so-called perfect numbers (6, 28, 496, 8128, . . .), which meet the Euclid’s requirement: if, for some k > 1,  2k is prime then 2k –1(2k – 1) is a perfect number, the first one possesses features enabling the six be to a far greater extent perfect number than all others.

All being specified indications of the numbers perfection [1], including the main additive property: a perfect number equals the sum of its divisors, excluding the number itself, are of purely numerical phenomenological nature dealing with integers only. However, even in such interpretation the 6, unlike other perfect numbers, possesses not only additive but also multiplicative property of perfection, namely the product of its divisors equals 6. One more remarkable phenomenon intrinsic just to the first perfect number is that the number of possible proper fractions (1/2, 1/3, 1/6, 2/3, 2/6, 3/6) also equals 6.

Nevertheless, the notion of perfection in the above interpretation is extremely poor by sense since it has nothing to do with at least one of such obvious aspects of perfection as optimality, harmony, and balance in a mathematical and, as possible, physical views. I intend to prove the direct connection of the first perfect number with these aspects. This demands to consider the divisors and the number as their paired relations, precisely as the above mentioned proper fractions, and compare them with the constants specific for the conceptions of informational optimality, harmony and balance.

Dimensional simulation of perfection

Before proceeding with the discussion of the first perfect number, the simple model describing a dimensional perfection can be considered. In terms of dimensional ratio (whatever object simulated) the notion of perfection may be associated with conceptions of harmony, balance and optimality. The optimality is considered here mainly in terms of information theory [2], using the method of eliminating the information redundancy based on the recently discovered principle of information cyclicity [3]. The mathematical harmony and balance represent here golden section and equality respectively. Thus, returning to the model, a dimension S can be divided into two (A and B) or proportional number of parts matching the conceptions through respective constants (one can call them informational constants) as follows:

       1) the informational optimality when the ratio of parts equals 1/2π, where π = 3.1416. Importantly, here the optimality is of different purposes: it may be the optimality of uncertainty-tolerance ratio: ρoa = B/S = 1/2π (optimum accuracy coefficient) or classification optimality: ρoc = B/A = 1/2π (optimum classification coefficient). In both cases ρo = 1/2π » 0.159 indicates on a redundancy absence or on a minimum possible information sufficiency (1 - ρo). The deriving of optimum accuracy and classification coefficients is briefly dealt with in the Appendix 1;

       2) the harmony (harmonious relation): fo = B/A = A/S = 0.618 – the fundamental mathematical constant known as the measure of harmony (named also golden section or golden mean or golden proportion or divine proportion);

 

3)     the balance (equality):  λ = A/S = B/S = 0.5  or λo = A/B = 1.

 

The peculiarity of ρo, as distinct from fo and λ, is in its universality regarding the determination of informatively permissible deviation from any of these constants in real usages of the conceptions concerned. Indeed, in terms of a tolerance optimality and by analogy with measurement error the deviation determined as ± 0.5ro multiplied by the respective constant could reasonably be attributed to any of the constant as the condition of proper usage of respective conception of perfection. Thus, the permissible ranges (PR) connecting to the constants and of their supplements (PRS) to 1 in the normalized presentation (S = 1) are determined as follows:

      ρo (1 ± 0.5 ρo) = 0.159 ± 0.013;       1 - ρo (1 ± 0.5 ρo) = 0.841 ± 0.013,           (1)                                                                 

      fo (1 ± 0.5 ρo) = 0.618 ± 0.049;        1 - fo (1 ± 0.5 ρo) =  0.382 ± 0.049,           (2)                                                                 

      λ (1 ± 0.5 ρo) = 0.5 ± 0.04;                1 - λ (1 ± 0.5 ρo) = 0.5 ± 0.04                  (3)

    

Irrespective of a system whose components represent or have been simulated as dimensional parts, if the relation of the parts match at least one of above expressions, this may be considered as a distinguishing feature of informational perfection.

 

Perfection of proper fractions for the first perfect number

 

The location of all possible proper fractions of the first perfect number within permissible ranges determined by means of information constants or their supplements is illustrated in Table 1.

 

Table 1

 Informational

    Constants

 

      PR or PRS

(permissible ranges)

 

   1/2

 

  1/3

 

  1/6

 

  2/6

 

   3/6

 

  2/3

         ρo

    0.146 ¸ 0.172

 

 

    ·

 

 

 

      1 - fo

    0.333 ¸ 0.431

 

    ·

 

    ·

 

   

  λ  or (1 – λ)

      0.46 ¸ 0.54

     ·

 

 

 

     ·

 

         fo

    0.569 ¸ 0.667

 

 

 

   

 

    ·

 

Clearly, there is no fraction created with the six that does not correspond to one of PR or PRS. At the same time, one can be convinced of the fact that there are no other so-called perfect numbers possessing such a property.

It should be noted that in the dimensional presentation the fraction 1/6 represents also exact optimum ratio in the classical example of simulating when making an open top box of greatest possible volume from a square piece of tin by cutting equal squares out of the corners and then folding up the tin to form the sides [4].

Conclusion  

The demonstration of remarkable properties of the first proper number both in phenomenological and informational views, as well as the referring to the physical example allows coming to a conclusion about the real comprehensive perfection of this unique number.    

Appendix 1: Determining of optimum information coefficients (information cyclicity)

 

The determination of optimum information coefficients as the fraction 1/2p, i.e. accuracy coefficient and classification coefficient, is carried out applying the conception of weights, which when presented in the normalized form conditionally are treated as formal analogs of probabilities.

 

Optimum accuracy coefficient

 

A dimension S may be divided onto n uniform constituents (S/n) with weights Kj = 2(n + 1 – j)/n(n + 1), representing a linear diagram (DK = Kj+1 – Kj = const); and the number j of informative constituents is determined using improved entropy function [3] as follows:

   j = exp (-) = exp {-           (4)

 

 When n ® ¥ the determination of informative parameters with this formula contains the fundamental estimation error of about 2% due to the inequality of weights in the redundant part (from j to n) of the linear diagram of weights. This part consists of m successively reducing subsystems of the complete groups of weights, and thus possesses own summary redundancy. This local redundancy causes both the optimization insufficiency and uncertainty in the main informative part (j=1¸j) of the system, and the redundant number of the subsystem parameters is equal to the product (n - j). The same number of weightiest subsystems parameters may be considered as the limit of possible addition to j, which is the interval of uncertainty in determining the informative parameters. The optimum number of parameters (jo) is within the limits from j to [j + (n - j)], and with estimation error of about 1% the optimum accuracy coefficient (ρoa) may be determined as follows:

    ρoa = 1 - jo/n = 1 – [j /n + 0.5(1 - j/n)* ] = 1/2p                       (5)

 

Optimum classification coefficient

 

Being converted into normalized form, non-dividable parts of dimension S represent the weights K1 = A/S and K2 = B/S that, in turn, represent relative contributions to any quality simulated by S. The optimum classification coefficient is evaluated for the necessary and sufficient number jo = exp(-K1 ln K1 – K2 ln K2) = 1.5 of components that is true for the most uncertain (50% confidence) situation about allowing or ignoring the lesser of two components [5]. Accordingly, the classification coefficient with high estimation accuracy is determined as follows:

    roc = arg {exp [- () ln () – () ln ()] = 1.5} =

         = arg {exp [- () ln () – () ln ()] = 1.5} = 1/2p                      (6)

 

References

[1] J.J. O'Connor and E.F Robertson.(2001) History topic: Perfect numbers. http://www- history.mcs.st-andrews.ac.uk/HistTopics/Perfect_numbers.html

[2] Shannon (1948) A mathematical theory of communication, Bell Syst. Tech, J.27 (1948).

[3] Kisets D.(1997) Optimum traceability type hierarchies. OIML Bulletin, 1997,XXXVIII  (2).

[4] W.A. Granville, Percey F. Smith, and W.R. Longley. Elements of Calculus. Ginn and Company, 1946, p.p. 58,59.

[5] Kisets D. (2003). Information sufficiency of an uncertainty budget. Proceedings of 2nd

 InternationalConference on Metrology, November 2003, Eilat, Israel.

 

 

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